Research ArticleAnalyses andEncryption Implementationof aNewChaotic SystemBased on Semitensor Product

Rui Wang 12 Peifeng Du23 Wenqi Zhong24 Han Han25 and Hui Sun 2

1Tianjin Key Laboratory for Civil Aircraft Airworthiness and Maintenance Civil Aviation University of ChinaTianjin 300300 China2College of Information Engineering and Automation Civil Aviation University of China Tianjin 300300 China3School of Electronics and Communication Engineering (SECE) Sun Yat-sen University Guangzhou 510006 China4School of Automation Northwestern Polytechnical University Xirsquoan 710072 China5North China Research Institute of Electro-Optics Beijing 100015 China

Correspondence should be addressed to Rui Wang wrhappyfuturehotmailcom and Hui Sun shhappy1hotmailcom

Received 4 January 2020 Revised 27 April 2020 Accepted 21 May 2020 Published 17 July 2020

Guest Editor Viet-+anh Pham

Copyright copy 2020 Rui Wang et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Semitensor product theory can deal with matrices multiplication with different numbers of columns and rows +erefore a newchaotic system for different high dimensions can be created by employing a semitensor product of chaotic systems with differentdimensions so that more channels can be selected for encryption +is paper proposes a new chaotic system generated bysemitensor product applied onQi and Lorenz systems+e corresponding dynamic characteristics of the new system are discussedin this paper to verify the existences of different attractors +e detailed algorithms are illustrated in this paper +e FPGAhardware encryption implementations are also elaborated and conducted Correspondingly the randomness tests are realized aswell and compared to that of the individual Qi system and Lorenz system the proposed system in this paper owns the betterrandomness characteristic +e statistical analyses and differential and correlation analyses are also discussed

1 Introduction

With the coming of 5G technology more and more in-formation is transformed by video and video informationsecurity becomes more and more important in practicalapplications especially for long-distance transmission [1 2]+e existing traditional encryption methods such as DESand AES however cannot meet the high requirements forreal time [3] +erefore it is necessary to focus on makingprogresses in the encryption technology in order to meet thereal-time requirements

People never stop studying the chaotic systems sinceLorenz proposed the first chaotic system Except for thetypical chaotic systems such as Chen Lv and Qi systemssome new different types of chaotic systems are generated aswell such as multistable chaotic hyperjerk system [4] a classof factional-order partial differential systems [5] multistablemodified fourth-order autonomous Chuarsquos system [6]

coexisting chaotic attractors chaotic systems [7ndash9] andchaotic system generation with memristors [10]+e authorsin reference [11] present a MDMBCAs design methodwithout reconstructing nonlinear function Correspond-ingly chaotic systems are employed in different areas such asmodeling neurodegenerative disease [12] and image andvideo encryptions [13ndash18] It is known that the character-istics of chaotic systems such as pseudorandomness andsensitivity to initial values meet the requirements of en-cryption discussed in the Shannonrsquos epoch-making paperldquoCommunication +eory of Secrecy Systemsrdquo Conse-quently it is hot for researchers to focus on the image andvideo data encryption where the encrypted sequences aregenerated by chaotic systems in order to satisfy the need forremote communications and other applications Multiplehardware platforms are implemented on encryptions such asFPGA ARM or circuits implementation [19ndash32] +usdifferent encryption methods based on various hardware

HindawiComplexityVolume 2020 Article ID 1230804 13 pageshttpsdoiorg10115520201230804

platforms are proposed in many articles [19] +e authors inreference [20] propose chaos encryption and decryptionoperated on FPGA and tested by TESTU01 +e study in[21 22] implements scrambling and antisqueezing of RGBthree primary color pixel position and video chaotic en-cryption and decryption of pixel value on arm and a digitalprogrammable audio encryption based on chaos system onFPGA +e effectiveness of chaotic secure communicationsystem method is proved by using adder and multiplier ofFPGA [23] A generalized improved chaotic transformationmapping is proposed in [24] Based on this mapping thespeech encryption of position transformation network isimplemented on FPGA Meanwhile [25] realizes FPGAcircuit output of three-dimensional chaotic system withoutbalancing points on FPGA+e study in [26] implements themultibutterfly chaotic attractor problem on FPGA +estudy in [27] proposes a Kolmogorov-type three-dimen-sional chaotic system and implements the chaotic system onFPGA+e study in [28] implements an application of high-dimensional digital chaos system (HDDCS) in image en-cryption in a limited precision range on FPGA +e study in[29] proposes a fractional order three-dimensional chaoticsystem with four wing chaotic attractors implemented onFPGA+e study in [30] proposes a sinusoidal chaotic model(SCM) and uses FPGA to implement chaotic mapping toverify its complexity and larger chaotic range SOPC tech-nology is used to realize the video processing of FPGA andthe data receiving and sending of ARM [15] A method ofgenerating pseudorandom number based on chaotic systemis proposed and implemented on FPGA [31] In addition toFPGA and arm [32 33] also use improved modular circuitdesign in hyperchaotic system +e study in [34] studies thesecurity of the latest three-dimensional chaotic self-syn-chronization flow secret key and a single secret key algo-rithm +e study in [35] proposes a method for a high-dimension chaotic system implemented on FPGA and alsoprovides comparison among different methods such as RealDomain Chaotic System (RDCS) Integer Domain ChaoticSystem (IDCS) Chaotic Bitwise Dynamical System (CBDS)and Higher-Dimensional Digital Chaotic Systems (HDDCS)implemented on FPGA Compared to these methods theproposedmethod can solve the dynamical degradation issue+e study in [36] discusses the Orthogonal Frequency Di-vision Multiplexing-Passive Optical Network (OFDM-PON) method which initiates a method for real-time videoencryption with chaotic systems Chen et al design anencryption algorithm using chaotic control methods andimplement this method on FPGA and ARM hardwareplatforms Furthermore the comparisons of the encryptionmethod based on ARM and FPGA are discussed by mixingthe advantages of each platform to achieve better real-timeperformance [15 23 24]

Semitensor product is a matrix operation first proposedby Cheng et al [37] +is method breaks the restriction ofmatrix product that is the column dimension of the frontmatrix must be the same with the row dimension of the backmatrix +en semitensor product realizes the multiplicationfor matrices with different dimensions +erefore thismethod makes the matrices product more easily and can be

applied in much wider areas Semitensor product method isalso extended in nonlinear issues and multiple areas suchBoolean network control game theory compressed sensingand data fusion [38ndash40] +e study in [41] provides theliterature review for the applications of semitensor productin engineering areas +is paper is inspired by the typicalcharacteristics of semitensor product method mentionedpreviously which provides a new idea to generate chaoticsystems +e new chaotic systems can be employed in real-time video encryption areas as well

+e main contribution of this paper is to employ theunique characteristic of the semitensor product to form anew chaotic system with different-order chaotic systems Qiand Lorenz systems in order to enhance the randomness ofthe sequence +en the dynamic characteristics of the newchaotic system are analyzed and the system is applied invideo encryption When compared to the individual chaoticsystem the new one constructed by semitensor product hasthe overwhelmed pseudorandomness

+e rest of the paper is arranged as follows Section 2presents a new chaotic system formed by semitensor producttheory Furthermore the corresponding dynamic charac-teristics of the system are analyzed Section 3 provides thedetailed encryption implementation based on FPGA withthe new chaotic system NIST test and the correspondingstochastic analysis are conducted as well in this section +econclusion of the paper is drawn in Section 4

2 A New Chaotic System Based on theSemitensor Product Theory

21 Preliminaries of Semitensor Product Normally thesemitensor product operation includes left semitensorproduct operation and right semitensor product operationLeft semitensor product operation meets the multiple-di-mension condition that is n ttimes p (t isinΝ+) Or if p stimes n(s isinΝ+) the operation is right semitensor product operation

Lemma 1 Assume matrix A is mtimes n matrix B is p times q andn ttimes p (t isinΝ+) [42] Let A be divided into a blocking matrix[A1 At] where Ai (i 1 t) is an mtimes p matrix gtenthe left semitensor product is defined as

Amtimesn ⋉B

ptimesq A

mtimesp1 B

ptimesq A

mtimesp2 B

ptimesq A

mtimesp3 B

ptimesq A

mtimespt B

ptimesq1113872 1113873

mtimestq

(1)

where ldquo⋉ rdquo is the left semitensor product

Lemma 2 If A isinMmtimestp and B isinMptimesq then

A⋉B A Botimes It( 1113857 (2)

where otimes is the Kronecker product of matrices [42]

22 A New Chaotic System Generated by Semitensor Productgteory Qi system is a hyperchaotic system with two pos-itive Lyapunov exponents 3 and 13 under certain condi-tions [28] And it could be used for secure communicationdue to its large positive Lyapunov exponents It is knownthat Qi system is described as follows

2 Complexity

_xQ1 a xQ2 minus xQ11113872 1113873 + xQ2xQ3

_xQ2 b xQ1 + xQ21113872 1113873 minus xQ1xQ3

_xQ3 minuscxQ3 minus exQ4 + xQ1xQ2

_xQ4 minusdxQ4 + fxQ3 + xQ1xQ2

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)

where xQi (i 1 2 3 4) is the state variable and a b c d and fare the related system parameters System (3) is a hyper-chaotic system when 49le ale55 20le ble 24 c 13 d 8e 33 and f 30

+e first chaotic system under study is a Lorenz chaoticsystem [28] +e dynamics of the system are shown in

_x σ(y minus x)

_y rx minus y minus xz

_z xy minus βz

⎧⎪⎪⎨

⎪⎪⎩(4)

where x y and z are state variables and σ r and β are therelated system parameters+e typical system parameters forLorenz chaotic system are selected as σ 10 r 28 andβ 83

It is relaxed for semitensor product operation only tosatisfy the multiple-dimension condition +erefore dif-ferent numbers state variables of systems can be selectedand conduct semitensor operation For example this paperselects a two-dimension state variable (xy)T of Lorenzsystem and a four-dimension state variable (xQ1xQ2xQ3xQ4)

T of Qi system to operate semitensor product +e result isshown as follows

x

y1113890 1113891⋉

xQ1

xQ2

xQ3

xQ4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

x

xQ1

xQ2

xQ3

xQ4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

y

xQ1

xQ2

xQ3

xQ4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

(5)

It is observed that the result of equation (5) is eight-dimension column vector which is equivalent to the newsystem state variable vector (x1x2x3x4x5x6x7x8)T that is

x1

x2

x3

x4

x5

x6

x7

x8

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xxQ1

xxQ2

xxQ3

xxQ4

yxQ1

yxQ2

yxQ3

yxQ4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

Differentiate each state variable in equation (6) andsubstitute equations (3) and (4) into the result +en thefollowing equation can be derived

_x1

_x2

_x3

_x4

_x5

_x6

_x7

_x8

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

_xxQ1 + x _xQ1

_xxQ2 + x _xQ2

_xxQ3 + x _xQ3

_xxQ4 + x _xQ4

_yxQ1 + y _xQ1

_yxQ2 + y _xQ2

_yxQ3 + y _xQ3

_yxQ4 + y _xQ4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

σ(y minus x)xQ1 + ax xQ2 minus xQ11113872 1113873 + xxQ2xQ3

σ(y minus x)xQ2 + bx xQ1 + xQ21113872 1113873 minus xxQ1xQ3

σ(y minus x)xQ3 minus cxxQ3 minus exxQ4 + xxQ1xQ2

σ(y minus x)xQ4 minus dx xQ4 + fxxQ3 + xxQ1xQ2

(rx minus y minus xz)xQ1 + ay xQ2 minus xQ11113872 1113873 + yxQ2xQ3

(rx minus y minus xz)xQ2 + by xQ1 + xQ21113872 1113873 minus yxQ1xQ3

(rx minus y minus xz)xQ3 minus cyxQ3 minus eyxQ4 + yxQ1xQ2

(rx minus y minus xz)xQ4 minus dy xQ4 + fyxQ3 + yxQ1xQ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

σ x5 minus x1( 1113857 + a x2 minus x1( 1113857 +x2x3

x

σ x6 minus x2( 1113857 + b x2 + x1( 1113857 minusx1x3

x

σ x7minusx3( 1113857 minus cx3 minus ex4 +x1x2

x

σ x8minusx4( 1113857 minus dx4 + fx3 +x1x2

x

rx1 minus x5 minus x1z + a x6 minus x5( 1113857 +x6x7

x

rx2 minus x6 minus x2z + b x5 + x6( 1113857 minusx5x7

x

rx3 minus x7 minus x3z minus cx7 minus ex8 +x5x6

x

rx4 minus x8 minus x4z minus dx8 + fx7 +x5x6

x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

It is obvious that the dynamics of (7) still include threestate variables of Lorenz system x y and z+erefore insert

equation (4) into equation (7) and then form a completeeleven-dimension system as illustrated in the following

Complexity 3

_x1 σ x5 minus x1( 1113857 + a x2 minus x1( 1113857 +x2x3

x

_x2 σ x6 minus x2( 1113857 + b x2 + x1( 1113857 minusx1x3

x

_x3 σ x7minusx3( 1113857 minus cx3 minus ex4 +x1x2

x

_x4 σ x8minusx4( 1113857 minus dx4 + fx3 +x1x2

x

_x5 rx1 minus x5 minus x1z + a x6 minus x5( 1113857 +x6x7

y

_x6 rx2 minus x6 minus x2z + b x5 + x6( 1113857 minusx5x7

y

_x7 rx3 minus x7 minus x3z minus cx7 minus ex8 +x5x6

y

_x8 rx4 minus x8 minus x4z minus dx8 + fx7 +x5x6

y

_x σ(y minus x)

_y rx minus y minus xz

_z xy minus βz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

As seen from equation (8) if one substitutes x⟶ minusxy⟶ minusy and z⟶ z _z (minusx)(minusy) minus βz xy minus βz itproves that it is symmetric with respect to z variable for xand y

23 Numerical Analysis of the New System +e paper ana-lyzes some dynamics characteristics of the new system in-cluding symmetry dissipativity equilibrium pointequilibria bifurcation diagram Lyapunov diagram andphase portraits

231 Symmetry As described in system (8) the system issymmetry with respect to z-axis since the system is invariantunder the coordinate transformations (x1 x2 x3 x4 x5 x6

x7 x8 x y z)⟶ (minusx1 minusx2 minusx3 minusx4 minusx5 minusx6 minusx7 minusx8

minus x minusy z)

232 Dissipativity +e divergence of system (12) is given by

nabla middot f zf1

zx1+

zf2

zx2+ middot middot middot +

zf8

zx8+

zf9

zx+

zf10

zy+

zf11

zz

minus5σ minus 2a + 2b minus 2c minus 2 d minus 5 minus β

(9)

and when minus5σ minus 2a + 2b minus 2c minus 2d minus 5 minus βlt 0 the systemundergoes dissipation

233 Equilibria As shown in system (8) x y and z couldnot be zero when calculating equilibria +en the equilibriaof system (8) are (0 0 0 0 0 0 0 0 plusmn

β(r minus 1)

1113968

plusmnβ(r minus 1)

1113968 r minus 1) One has

J12

minusσ minus a a 0 0 σ 0 0 0 0 0 0

b b minus σ 0 0 0 σ 0 0 0 0 0

0 0 minusσ minus c minuse 0 0 σ 0 0 0 0

0 0 f minusσ minus d 0 0 0 σ 0 0 0

1 0 0 0 minus1 minus a a 0 0 0 0 0

0 1 0 0 b b minus 1 0 0 0 0 0

0 0 1 0 0 0 minus1 minus c minuse 0 0 0

0 0 0 1 0 0 f minus1 minus d 0 0 0

0 0 0 0 0 0 0 0 minusσ 0 0

0 0 0 0 0 0 0 0 1 minus1 0

0 0 0 0 0 0 0 0 plusmnβ(r minus 1)

1113968plusmn

β(r minus 1)

1113968β

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

equilibria

(10)

+e corresponding polynomial is

f(λ) λ(λ + β)(λ minus 1)f1(λ) (11)where f1(λ) is an eighth-order polynomial It is obvious thatat least 0 1 and minusβ are eigenvalues of system (8) for the theseequilibrium points therefore not each real part of the

4 Complexity

eigenvalues is negative +en it can be concluded that theseare not stable equilibrium points

234 Bifurcation Diagram Lyapunov Diagram and PhasePortraits It is known that when 49le ale 55 20le ble 24c 13 d 8 e 33 and f 30 Qi system is a hyperchaoticsystem When σ 10 r 28 and β 83 Lorenz system is achaotic system +erefore the paper selects the parametersa 50 c 13 d 8 e 33 f 30 σ 10 r 28 and β 83and varies b to analyze the bifurcation of system (12) asshown in Figure 1(a) As the bifurcation diagram shows thesystem demonstrates the chaotic characteristics whenb isin [minus5 26] +e corresponding Lyapunov diagram is il-lustrated in Figure 1(b) Furthermore partial phase portraitsof system (7) for different initials when b 24 are shown inFigures 1(c) One has

_x1 10 x5 minus x1( 1113857 + 50 x2 minus x1( 1113857 +x2x3

x

_x2 10 x6 minus x2( 1113857 + b x2 + x1( 1113857 minusx1x3

x

_x3 10 x7minusx3( 1113857 minus 13x3 minus 33x4 +x1x2

x

_x4 10 x8minusx4( 1113857 minus 8x4 + 30x3 +x1x2

x

_x5 28x1 minus x5 minus x1z + 50 x6 minus x5( 1113857 +x6x7

y

_x6 28x2 minus x6 minus x2z + b x5 + x6( 1113857 minusx5x7

y

_x7 28x3 minus x7 minus x3z minus 13x7 minus 33x8 +x5x6

y

_x8 28x4 minus x8 minus x4z minus 8x8 + 30x7 +x5x6

y

_x 10(y minus x)

_y 28x minus y minus xz

_z xy minus83z

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

Figures 1(c)ndash1(f) illustrate different phase portraits in-cluding x1 versus x2 x2 versus x4 x3 versus x and x versuszwhen b 24 for two initial value sets the initial values for theblue line phase portraits are 001418 004217 009157007922 009594 006557 000357 008491 009339006787 and 007577 and those for the red line phase

portraits are 001417 004218 009156 007921 009593006558 000356 008492 009338 006788 and 007576+ese portraits demonstrate that system (12) has obviouschaotic attractors and approaches periodic characteristics asinitial values changes

3 Encryption Implementation with the NewChaotic System Based on FPGA

+is paper employs the random sequence of system (12) asthe random sequence to encrypt video data and realize thehardware implement on FPGA Figure 2 is the FPGAhardware diagram used for the encryption +e maincomponents are HDMI ZYNQ JTAG and source interface+e video is collected from JTAG then the encryption al-gorithm is performed in ZYNQ powered by 5V DC and theoutputs will be shown in the monitor through HDMI

+e encryption algorithm is described in the followingand the corresponding block diagram is demonstrated inFigure 3

Step 1 to generate the random sequences for each statevariable for both discretized Qi system and Lorenzsystem respectivelyStep 2 to generate the random sequence for the newsystem (12) constructed by semitensor product oper-ation on (xQ1 xQ2 xQ3 xQ4)T and (x y)TStep 3 to generate the sequence xi(xi1 xi2 xi3 xi32)by the new system (i 1 2 8 j 1 2 32) wherexij is a binary number i represents the number of statevariables and j is the bit number for each state variableChoose a sequence xi with fixed bits from t to q that is

ci(n) xi mod 2q( 1113857mod 2tminusq

1113872 1113873 (i 1 2 amp 8 1le tlt qle 32)

(13)

Make an XOR operation on ci (n) and divide video databased on pixels that is

m(n) xi mod 2tminus11113872 1113873mod 2tminusq

1113872 1113873oplus (m) (14)

where oplus is the XOR operation

31 Discretization for the New System and Its ImplementationBasedonFPGA In the hardware experiment it is impossibleto implement the continuous Lorenz and Qi chaotic systemsbecause of limitation of the bit width in FPGA +erefore itis necessary to discretize continuous system first Multiplemethods can be used to discretize a differential equationsuch as Euler method improved Euler method and Runge-Kutta method To meet the requirement of real-time per-formance and the limitation of hardware implementationEuler method is used to discretize the differential equationsdue to its low computation complexity First Euler methodis used to discretize Qi and Lorenz systems respectively +ecorresponding process of Qi system is proposed as follows

Complexity 5

8000

6000

4000

2000

0

ndash2000

ndash4000ndash5 0 5 10 15 20 25

b

(a)

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

Lyap

unov

expo

nent

t

ndash5 0 5 10 15 20 25b

(b)

ndash8000 ndash6000 ndash4000 ndash2000 0 2000 4000 6000ndash6000

ndash4000

ndash2000

0

2000

4000

6000

x 2

x1

(c)

x 4

ndash6000 ndash4000 ndash2000 0 2000 4000 6000x2

ndash4

ndash3

ndash2

ndash1

0

1

2

3 times104

(d)

Figure 1 Continued

6 Complexity

xQ1(n + 1) minus xQ1(n)1113872 1113873

τ a xQ2(n) minus xQ1(n)) + xQ2(n)xQ3(n)1113872 1113873

xQ2(n + 1) minus xQ2(n)1113872 1113873

τ b xQ1(n) + xQ2(n)) minus xQ1(n)xQ3(n)1113872 1113873

xQ3(n + 1) minus xQ3(n)1113872 1113873

τ minuscxQ3 (n) minus exQ4(n) + xQ1(n)xQ2(n)1113872 1113873

xQ4(n + 1) minus xQ4(n)1113872 1113873

τ minusdxQ4 (n) + fxQ3(n) + xQ1(n)xQ2(n)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

+en the iteration equations of Qi system are shown inxQ1(n + 1) aτxQ2(n) +(1 minus aτ)xQ1(n) + τxQ2(n)xQ3(n)

xQ2(n + 1) bτxQ1(n) +(1 + bτ)xQ2(n) minus τxQ1(n)xQ3(n)

xQ3(n + 1) (1 minus cτ)xQ3(n) minus eτxQ4(n) + τxQ1(n)xQ2(n)

xQ4(n + 1) (1 minus dτ)xQ4(n) + fτxQ3(n) + τxQ1(n)xQ2(n)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)

Similarly the discrete Lorenz system isx(n + 1) minus x(n)

τ σ(y(n) minus x(n))

y(n + 1) minus y(n)

τ rx(n) minus y(n) minus x(n)z(n)

z(n + 1) minus z(n)

τ x(n)y(n) minus βz(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Correspondingly the iteration equations of Lorenz systemare

x3

ndash15 ndash1 ndash05 0 05 1 15 2times104

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

x

(e)

ndash20 ndash10 0 10 20x

0

10

20

30

40

50

z

(f )

Figure 1 Bifurcation diagram along with b variation and partial phase portraits for different initial values (blue line for the first initial valueset and red line for the second initial value set)

Source interface

ZYNQ

HDMI JTAG

Figure 2 FPGA hardware diagram

Semitensor

Video data m (n)

c (n)

p (n)

lfloorze (n + 1)rfloor

lfloorye (n + 1)rfloor

lfloorxe (n + 1)rfloor

lfloorxeQ1 (n + 1)rfloor

lfloorxeQ2 (n + 1)rfloor

lfloorxeQ3 (n + 1)rfloor

lfloorxeQ4 (n + 1)rfloor

lfloorxi (n)rfloor i = 1 2 8

mod (mod (lfloorxi (n)rfloor 2tndash1) 2tndashq)

Figure 3 Block diagram of encryption algorithm

Complexity 7

x(n + 1) στy(n) +(1 minus στ)x(n)

y(n + 1) rτx(n) +(1 minus τ)y(n) minus τx(n)z(n)

z(n + 1) τx(n)y(n) +(1 minus βτ)z(n)

⎧⎪⎪⎨

⎪⎪⎩(18)

In general FPGA can store float data and fixed-pointdata Since fixed-point data require less computing resourcesthan that of float data this paper uses 64-bit fixed-point

number to represent the data+e detailed data format of 64-bit fixed-point numbers is shown in Figure 4

In Figure 4 I represents the integer part of 64-bit fixed-point numbers and f is the fractional part

As mentioned before because of the limitation of bitwidth in FPGA all data are truncated numbers in hardwareimplementation +erefore the Qi and Lorenz systembecomes

lfloorxQ1(n + 1)rfloor aτlfloorxQ2(n)rfloor +(1 minus aτ)lfloorxQ1(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor bτlfloorxQ1(n)rfloor +(1 + bτ)lfloorxQ2(n)rfloor minus τlfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor (1 minus cτ)lfloorxQ3(n)rfloor minus eτlfloorxQ4(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor (1 minus dτ)lfloorxQ4(n)rfloor + fτlfloorxQ3(n)rfloor + τlfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

lfloorx(n + 1)rfloor στlfloory(n)rfloor +(1 minus στ)lfloorx(n)rfloor

lfloory(n + 1)rfloor rτlfloorx(n)rfloor +(1 minus τ)lfloory(n)rfloor minus τlfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor τlfloorx(n)rfloorlfloory(n)rfloor +(1 minus βτ)lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(20)

Let the iteration step be τ 000001 and use the sameparameters in system (12) +en substitute them into (19)

and (20) respectively +erefore Qi system and Lorenzsystem are changed as follows

lfloorxQ1(n + 1)rfloor 09995lfloorxQ2(n)rfloor + 00005lfloorxQ1(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor 00002lfloorxQ1(n)rfloor + 10002lfloorxQ2(n)rfloor minus 000001lfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor 099987lfloorxQ3(n)rfloor minus 000033lfloorxQ4(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor 000008lfloorxQ4(n)rfloor + 09997lfloorxQ3(n)rfloor + 000001lfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)

lfloorx(n + 1)rfloor 00001lfloory(n)rfloor + 09999lfloorx(n)rfloor

lfloory(n + 1)rfloor 000028lfloorx(n)rfloor + 099999lfloory(n)rfloor minus 000001lfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor 000001lfloorx(n)rfloorlfloory(n)rfloor + 09999733lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(22)

To iterate Qi system and Lorenz system and makesemitensor product operation on these two systems aftereach iteration respectively the discretized first 8 statevariables of the new system are obtained

x1(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x2(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x3(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x4(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

x5(n + 1) lfloory(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x6(n + 1) lfloory(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x7(n + 1) lfloory(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x8(n + 1) lfloory(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

where xi(n+ 1) and xi (n) are system state variables lfloory(n +

1)rfloor is the approximate value of y (n+ 1) using fixed-pointnumber

32 Implementation and Analysis of Encryption Algorithm ofthe New Chaotic System In order to ensure the randomnessof the random sequence therefore select the low bits from tto q as shown in Figure 5 +e positions of these bits are notclose to those of sign and exponent bits +en the chosenrandom encryption sequence ci (n) is shown in equation(17) +is paper selects t 1 and q 6

ci (n) modlfloorxi(n)rfloor2tminus 11113888 1113889 2qminust+1

1113888 1113889 i 1 2 8

(24)Random sequences which are selected from eight states

based on the method mentioned previously are combined togenerate the random sequence c (n)

c(n) c1(n) c2(v) c8(n)( 1113857 (25)

In order to resist the differential attack and decrease thecorrelation between adjacent random sequences the paperselects the very first iteration sequence among every N it-erations and stacks these selected sequences to construct a

8 Complexity

random sequence c (n) as shown in equation (18) +is canimprove the randomness of the random sequence

Next the random sequence c (n) conducts XOR op-eration with the divided video data Since a frame video dataincludes tricolor integer sequences R (n) G (n) and B (n)these three sequences will be encrypted simultaneously afterchanging the random sequence c (n) into three columnsevenly c1 (n) c2 (n) and c3 (n)

p1(n) c1(n)oplusR(n)

p2(n) c2(n)oplusG(n)

p3(n) c3(n)oplusB(n)

(26)

where p1 (n) p2 (n) and p3 (n) are encrypted sequences andoplus is an XOR operation One has

1113954R(n) modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplusp1(n)

modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusR(n)

1113954G(n) modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp2(n)rfloor

modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusG(n)

1113954B(n) modlfloor1113954x3(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp3(n)rfloor

mod mod lfloor1113954x3(n)rfloor 2t1113872 1113873 2tminus q

1113872 1113873oplus c3(n)oplusB(n)

(27)

where lfloor1113954xpprime(n)rfloor pprime 1 2 3 are receiver terminal sequences

33 Analysis for NIST Test NIST test is provided by NationalInstitute of Standards and Technology and it is a standard to testthe randomness of a random sequence According to the en-cryption algorithm in this paper c (n) in equation (25) should betested by NIST standard +e comparisons of the random se-quence among the new system Qi system and Lorenz system c(n) cL (n) and cQ (n) are conducted which are obtained fromserial interfaces +e results of the tests are shown in Table 1

As shown in Table 1 all the test results for the randomsequences of the new system meet the NIST test index

standards Partial test results are larger than 08 which meansthese random indexes are quite close to those of the realrandom sequences +e randomness indexes and some othertest results are better than those generated from Lorenz systemand Qi system such as frequency block frequency cumulativesums nonoverlapping template approximate entropy randomexcursions random excursions variant and linear complexity

34 Statistical Analyses Vivado IDE is used to conduct thehardware simulation +e paper also performs the statisticsanalysis for the encrypted video data generated by hardwareFigure 6(a) is one picture of a video before encryptionFigure 6(b) is the encrypted picture of a video

Figure 7 demonstrates the comparisons of statisticshistogram between the original and encrypted pictures

Figure 7 demonstrates the comparisons of statisticshistograms between the original and encrypted pictures Asillustrated in Figure 7(a) the difference of the pixels dis-tribution is obvious However distribution of differentpixels for the encrypted picture shown in Figure 7(b) is theapproximately uniform distribution It can be concludedthat the proposed encryption algorithm for the new systemcan better resist statistic attack effectively

35 Differential Analysis Differential attack is used tomeasure the sensitivity of plaintext change for the encryp-tion algorithm and commonly uses NPCR (Number of PixelsChange Rate) and UACI (Unified Average Changing In-tensity) as indexes defined as follows

NPRC 1113936efD(e f)

W times Htimes 100

UACI 1

W times Htimes 1113944

ef

C(e f) minus Cprime(e f)1113868111386811138681113868

1113868111386811138681113868

255times 100

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

NPRC 1113944m

NPRC (m)

UACI 1113944m

UACI (m)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(29)

where C (e f) is the pixel value before encryption and Cprime (ef ) is the pixel value after encryption If C (e f )Cprime (e f ) D(e f ) 0 else 1 NPRC and UACI calculated by (29) and theproposed system and encryption algorithm are 9960 and1228 for the first-time encryption respectively +ereforethe ability to resist differential attack improves to someextent In video encryption application the requirement forencryption speed is more concerned

36 Correlation Analysis Correlation analysis is used tocheck whether the neighbor pixels are close or not +ispaper analyzes the correlation for Figure 6 +e paper selects5000 random pixels from the original and the encryptedimages and analyzes the correlation among these random-pixel pairs as shown in Figure 8 As Figure 8 illustrates the

ffffffff ffffffff ffffffff ffffffff ffffffff4024

IIIIIIII IIIIIIII IIIIIIII

Figure 4 Data format for 64-bit fixed-point numbers

q t 164 63

hellip hellip hellip

Figure 5 +e schematics of numbered data bits

Complexity 9

Table 1 NIST test results for the random sequences of the new system Lorenz system and Qi system (the number of sequences is 100 andtheir lengths are 1000000)

Statistical test P value of cL (n) P value of cQ (n) P value of c (n) Test results of c (n) ProportionFrequency 0000000 0236810 0474986 radic 100100Block frequency (m 128) 0289667 0699313 0946308 radic 99100Cumulative sums 0000000 0108791 0779188 radic 100100Runs 0000000 0554420 0075719 radic 100100Longest run 0419021 0383827 0289667 radic 100100Rank 0851383 0996335 0289667 radic 100100FFT 0911413 0911413 0213309 radic 99100Nonoverlapping template (m 9) 0000003 0181557 0983453 radic 100100Overlapping template (m 9) 0181557 0935716 0924076 radic 100100Universal 0616305 0289667 0014550 radic 97100Approximate entropy (m 10) 0000000 0798139 0816537 radic 99100Random excursions 0008879 0319084 0739918 radic 6464Random excursions variant 0213309 0289667 0949602 radic 6464Linear complexity (M 500) 0010988 0013569 0108791 radic 99100Serial (m 16) 0616305 0028817 0262249 radic 100100

(a) (b)

(c)

Figure 6+e original and encrypted pictures of a video and the corresponding encrypted video data through FPGA (a)+e original pictureof a video (b) +e encrypted picture of a video (c) Encrypted video data shown on a monitor

0

05

1

15

2

0 50 100 150 200 250

104

(a)

0

05

1

15

2

0 50 100 150 200 250

104

(b)

Figure 7 Histogram between the original and encrypted pictures (a) Histogram of the original picture (b) Histogram of the encrypted picture

10 Complexity

correlation dramatically decreases when comparing twofigures before and after encryption as shown in Figure 6

4 Conclusions

+is paper proposed a new chaotic system generated byusing semitensor product on two chaotic systems and therelated dynamic characteristics are analyzed +e new sys-tem is employed in video encryption as well and the pro-posed method can generate 8 or even 12 state variables whencompared to Qi system and Lorenz system which onlygenerate 7 state variables at most in one iteration period+eproposed method can improve the speed of random se-quence generation +e NIST test results demonstrate thatthe pseudorandomness of new system is better than that ofsingle Qi system and single Lorenz system

+e proposed encryption algorithm based on semitensorproduct can be used in other chaotic systems +e synchro-nization of the new system can be implemented by synchro-nizing two original chaotic systems separately In this paperFPGA is used to implement the generation of the new chaoticsystem and to encrypt video data +e corresponding statisticsand differential and correlation analyses were also conductedwhich demonstrates that the new system has obvious advan-tages such as better random features better resistance todifferential attacks and lower pixel correlation for encryptedimages +e future work will focus on the decryption of videoinformation by the proposed chaotic system generated by thesemitensor product method in hardware

Data Availability

+e figuresrsquo zip data used to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities-Civil Aviation Universityof China (3122019044)

References

[1] J Li J Li X Chen C Jia and W Lou ldquoIdentity-based en-cryption with outsourced revocation in cloud computingrdquoIEEE Transactions on Computers vol 64 no 2 pp 425ndash4372015

[2] J Li Y K Li X Chen P P C Lee and W Lou ldquoA hybridcloud approach for secure authorized deduplicationrdquo IEEETransactions on Parallel and Distributed Systems vol 26no 5 pp 1206ndash1216 2015

[3] M Feki B Robert G Gelle and M Colas ldquoSecure digitalcommunication using discrete-time chaos synchronizationrdquoChaos Solitons amp Fractals vol 18 no 4 pp 881ndash890 2003

[4] V T Pham S Vaidyanathan C Volos S Jafari andT Kapitaniak ldquoA new multi-stable chaotic hyperjerk systemits special features circuit realization control and synchro-nizationrdquo Archives of Control Sciences vol 30 no 1pp 23ndash45 2020

[5] A Ouannas X Wang V T Pham G Grassi andV V Huynh ldquoSynchronization results for a class of frac-tional-order spatiotemporal partial differential systems basedon fractional lyapunov approachrdquo Boundary Value Problemsvol 2019 no 1 2019

[6] F Yu H Shen L Liu et al ldquoCCII and FPGA realization amultistable modified fourth-order autonomous Chuarsquos

Value of pixel (x y)

0

50

100

150

200

250Va

lue o

f pix

el (x

+ 1

y)

0 50 100 150 200 250

(a)

0 50 100 150 200 250Value of pixel (x y)

0

50

100

150

200

250

Valu

e of p

ixel

(x +

1 y

)(b)

Figure 8 Correlation analyses for the original and encrypted figures in Figure 6 (a) Correlation analysis for the original figure as shown inFigure 6(a) (b) Correlation analysis for the encrypted figure as shown in Figure 6(b)

Complexity 11

chaotic system with coexisting multiple attractorsrdquo Com-plexity vol 2020 Article ID 5212601 17 pages 2020

[7] Q Lai P D K Kuate F Liu and H H Lu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs vol 67 no 6 2019

[8] Q Lai A Akgul C B Li G H Xu and U Cavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and s-box designrdquo Entropy vol 20 no 12018

[9] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[10] R Wang M Li Z Gao and H Sun ldquoA new memristor-based5D chaotic system and circuit implementationrdquo Complexityvol 2018 Article ID 6069401 12 pages 2018

[11] Q Hong Y Li X Wang and Z Zeng ldquoA versatile pulsecontrol method to generate arbitrary multidirection multi-butterfly chaotic attractorsrdquo IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol 38 no 8pp 1480ndash1492 2019

[12] P S Shabestari Z Rostami V T Pham F E Alsaadi andT Hayat ldquoModeling of neurodegenerative diseases usingdiscrete chaotic systemsrdquo Communications in gteoreticalPhysics vol 71 no 10 pp 1241ndash1245 2019

[13] Q Jiang J Jiang and J Li ldquoAn image encryption algorithmbased on high-dimensional chaotic systemsrdquo in Proceedings ofthe IEEE International Conference on Signal ProcessingCommunications and Computing pp 1ndash4 Hong Kong ChinaAugust 2016

[14] S Singh M Ahmad and D Malik ldquoBreaking an imageencryption scheme based on chaotic synchronization phe-nomenonrdquo in Proceedings of the 2016 Ninth InternationalConference on Contemporary Computing (IC3) Noida IndiaAugust 2016

[15] P Chen S Yu B Chen L Xiao and J Lu ldquoDesign and SOPC-based realization of a video chaotic secure communicationschemerdquo International Journal of Bifurcation and Chaosvol 28 no 13 Article ID 1850160 24 pages 2018

[16] P R Sankpal and P A Vijaya ldquoImage encryption Usingchaotic maps a surveyrdquo in Proceedings of the 2014 Fifth In-ternational Conference on Signal and Image Processingpp 102ndash107 IEEE Bangalore India January 2014

[17] A Rani and B Raman ldquoAn image copyright protectionsystem using chaotic mapsrdquo Multimedia Tools and Applica-tions vol 76 no 2 pp 3121ndash3138 2016

[18] N Shyamala and K Anusudha ldquoReversible chaotic encryp-tion techniques for imagesrdquo in Proceedings of the 2017 FourthInternational Conference on Signal Processing Communica-tion and Networking (ICSCN) Chennai India March 2017

[19] S Chen S Yu J Lu G Chen and J He ldquoDesign and FPGA-based realization of a chaotic secure video communicationsystemrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 28 no 9 pp 2359ndash2371 2018

[20] Z Lin S Yu J Lu S Cai and G Chen ldquoDesign and ARM-embedded implementation of a chaotic map-based real-timesecure video communication systemrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 25 no 7pp 1203ndash1216 2015

[21] D Chang Z Li M Wang and Y Zeng ldquoA novel digital pro-grammable multi-scroll chaotic system and its application inFPGA-based audio secure communicationrdquo AEUmdashInternational

Journal of Electronics and Communications vol 88 pp 20ndash292018

[22] F J Farsana and K Gopakumar ldquoPrivate key encryption ofspeech signal based on three dimensional chaotic maprdquo inProceedings of the International Conference on Communica-tion amp Signal Processing pp 2197ndash2201 Chennai India April2017

[23] H Jia Z Guo S Wang and Z Chen ldquoMechanics analysis andhardware implementation of a new 3D chaotic systemrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 13Article ID 1850161 14 pages 2018

[24] E Tlelo-Cuautle L G De La Fraga V-T Pham C VolosS Jafari and A D J Quintas-Valles ldquoDynamics FPGA re-alization and application of a chaotic system with an infinitenumber of equilibrium pointsrdquo Nonlinear Dynamics vol 89no 2 pp 1129ndash1139 2017

[25] A Akgul H Calgan I Koyuncu I Pehlivan andA Istanbullu ldquoChaos-based engineering applications with a3D chaotic system without equilibrium pointsrdquo NonlinearDynamics vol 84 no 2 pp 481ndash495 2016

[26] Q Lai X Zhao K Rajagopal G Xu A Akgul andE Guleryuz ldquoDynamic analyses FPGA implementation andengineering applications of multi-butterfly chaotic attractorsgenerated from generalised sprott c systemrdquo Pramana vol 90no 1 p 12 2018

[27] W S Sayed M F Tolba A G Radwan and S K Abd-El-Hafiz ldquoFPGA realization of a speech encryption system basedon a generalized modified chaotic transition map and bitpermutationrdquo Multimedia Tools and Applications vol 78no 12 pp 16097ndash16127 2019

[28] Q Wang S Yu C Li et al ldquo+eoretical design and FPGA-based implementation of higher-dimensional digital chaoticsystemsrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 63 no 3 pp 401ndash412 2016

[29] E Dong Z Wang Z Chen and Z Wang ldquoTopologicalhorseshoe analysis and field-programmable gate arrayimplementation of a fractional-order four-wing chaoticattractorrdquo Chinese Physics B vol 27 no 1 pp 300ndash306 2018

[30] Z Hua B Zhou and Y Zhou ldquoSine chaotification model forenhancing chaos and its hardware implementationrdquo IEEETransactions on Industrial Electronics vol 66 no 2 pp 1273ndash1284 2019

[31] I Ozturk and R Kılıccedil ldquoA novel method for producing pseudorandom numbers from differential equation-based chaoticsystemsrdquo Nonlinear Dynamics vol 80 no 3 pp 1147ndash11572015

[32] R Wang Q Xie Y Huang H Sun and Y Sun ldquoDesign of aswitched hyperchaotic system and its applicationrdquo Interna-tional Journal of Computer Applications in Technology vol 57no 3 pp 207ndash218 2018

[33] R Wang H Sun J Wang L Wang and Y Wang ldquoAppli-cations of modularized circuit designs in a new hyper-chaoticsystem circuit implementationrdquo Chinese Physics B vol 24no 4 Article ID 020501 2015

[34] Z Lin S Yu and J Li ldquoChosen ciphertext attack on a chaoticstream cipherrdquo in Proceedings of the 30th Chinese Control andDecision Conference pp 1238ndash1242 Shenyang China June2018

[35] G Qi G Chen M A Van Wyk and B J Van Wyk ldquoOn anew hyperchaotic systemrdquo Physics Letters A vol 372 no 2pp 124ndash136 2008

[36] Z Hu and C-K Chan ldquoA real-valued chaotic orthogonalmatrix transform-based encryption for OFDM-PONrdquo IEEE

12 Complexity

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13

_xQ1 a xQ2 minus xQ11113872 1113873 + xQ2xQ3

_xQ2 b xQ1 + xQ21113872 1113873 minus xQ1xQ3

_xQ3 minuscxQ3 minus exQ4 + xQ1xQ2

_xQ4 minusdxQ4 + fxQ3 + xQ1xQ2

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(3)

where xQi (i 1 2 3 4) is the state variable and a b c d and fare the related system parameters System (3) is a hyper-chaotic system when 49le ale55 20le ble 24 c 13 d 8e 33 and f 30

+e first chaotic system under study is a Lorenz chaoticsystem [28] +e dynamics of the system are shown in

_x σ(y minus x)

_y rx minus y minus xz

_z xy minus βz

⎧⎪⎪⎨

⎪⎪⎩(4)

where x y and z are state variables and σ r and β are therelated system parameters+e typical system parameters forLorenz chaotic system are selected as σ 10 r 28 andβ 83

It is relaxed for semitensor product operation only tosatisfy the multiple-dimension condition +erefore dif-ferent numbers state variables of systems can be selectedand conduct semitensor operation For example this paperselects a two-dimension state variable (xy)T of Lorenzsystem and a four-dimension state variable (xQ1xQ2xQ3xQ4)

T of Qi system to operate semitensor product +e result isshown as follows

x

y1113890 1113891⋉

xQ1

xQ2

xQ3

xQ4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

x

xQ1

xQ2

xQ3

xQ4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

y

xQ1

xQ2

xQ3

xQ4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

T

(5)

It is observed that the result of equation (5) is eight-dimension column vector which is equivalent to the newsystem state variable vector (x1x2x3x4x5x6x7x8)T that is

x1

x2

x3

x4

x5

x6

x7

x8

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xxQ1

xxQ2

xxQ3

xxQ4

yxQ1

yxQ2

yxQ3

yxQ4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

Differentiate each state variable in equation (6) andsubstitute equations (3) and (4) into the result +en thefollowing equation can be derived

_x1

_x2

_x3

_x4

_x5

_x6

_x7

_x8

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

_xxQ1 + x _xQ1

_xxQ2 + x _xQ2

_xxQ3 + x _xQ3

_xxQ4 + x _xQ4

_yxQ1 + y _xQ1

_yxQ2 + y _xQ2

_yxQ3 + y _xQ3

_yxQ4 + y _xQ4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

σ(y minus x)xQ1 + ax xQ2 minus xQ11113872 1113873 + xxQ2xQ3

σ(y minus x)xQ2 + bx xQ1 + xQ21113872 1113873 minus xxQ1xQ3

σ(y minus x)xQ3 minus cxxQ3 minus exxQ4 + xxQ1xQ2

σ(y minus x)xQ4 minus dx xQ4 + fxxQ3 + xxQ1xQ2

(rx minus y minus xz)xQ1 + ay xQ2 minus xQ11113872 1113873 + yxQ2xQ3

(rx minus y minus xz)xQ2 + by xQ1 + xQ21113872 1113873 minus yxQ1xQ3

(rx minus y minus xz)xQ3 minus cyxQ3 minus eyxQ4 + yxQ1xQ2

(rx minus y minus xz)xQ4 minus dy xQ4 + fyxQ3 + yxQ1xQ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

σ x5 minus x1( 1113857 + a x2 minus x1( 1113857 +x2x3

x

σ x6 minus x2( 1113857 + b x2 + x1( 1113857 minusx1x3

x

σ x7minusx3( 1113857 minus cx3 minus ex4 +x1x2

x

σ x8minusx4( 1113857 minus dx4 + fx3 +x1x2

x

rx1 minus x5 minus x1z + a x6 minus x5( 1113857 +x6x7

x

rx2 minus x6 minus x2z + b x5 + x6( 1113857 minusx5x7

x

rx3 minus x7 minus x3z minus cx7 minus ex8 +x5x6

x

rx4 minus x8 minus x4z minus dx8 + fx7 +x5x6

x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

It is obvious that the dynamics of (7) still include threestate variables of Lorenz system x y and z+erefore insert

equation (4) into equation (7) and then form a completeeleven-dimension system as illustrated in the following

Complexity 3

_x1 σ x5 minus x1( 1113857 + a x2 minus x1( 1113857 +x2x3

x

_x2 σ x6 minus x2( 1113857 + b x2 + x1( 1113857 minusx1x3

x

_x3 σ x7minusx3( 1113857 minus cx3 minus ex4 +x1x2

x

_x4 σ x8minusx4( 1113857 minus dx4 + fx3 +x1x2

x

_x5 rx1 minus x5 minus x1z + a x6 minus x5( 1113857 +x6x7

y

_x6 rx2 minus x6 minus x2z + b x5 + x6( 1113857 minusx5x7

y

_x7 rx3 minus x7 minus x3z minus cx7 minus ex8 +x5x6

y

_x8 rx4 minus x8 minus x4z minus dx8 + fx7 +x5x6

y

_x σ(y minus x)

_y rx minus y minus xz

_z xy minus βz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

As seen from equation (8) if one substitutes x⟶ minusxy⟶ minusy and z⟶ z _z (minusx)(minusy) minus βz xy minus βz itproves that it is symmetric with respect to z variable for xand y

23 Numerical Analysis of the New System +e paper ana-lyzes some dynamics characteristics of the new system in-cluding symmetry dissipativity equilibrium pointequilibria bifurcation diagram Lyapunov diagram andphase portraits

231 Symmetry As described in system (8) the system issymmetry with respect to z-axis since the system is invariantunder the coordinate transformations (x1 x2 x3 x4 x5 x6

x7 x8 x y z)⟶ (minusx1 minusx2 minusx3 minusx4 minusx5 minusx6 minusx7 minusx8

minus x minusy z)

232 Dissipativity +e divergence of system (12) is given by

nabla middot f zf1

zx1+

zf2

zx2+ middot middot middot +

zf8

zx8+

zf9

zx+

zf10

zy+

zf11

zz

minus5σ minus 2a + 2b minus 2c minus 2 d minus 5 minus β

(9)

and when minus5σ minus 2a + 2b minus 2c minus 2d minus 5 minus βlt 0 the systemundergoes dissipation

233 Equilibria As shown in system (8) x y and z couldnot be zero when calculating equilibria +en the equilibriaof system (8) are (0 0 0 0 0 0 0 0 plusmn

β(r minus 1)

1113968

plusmnβ(r minus 1)

1113968 r minus 1) One has

J12

minusσ minus a a 0 0 σ 0 0 0 0 0 0

b b minus σ 0 0 0 σ 0 0 0 0 0

0 0 minusσ minus c minuse 0 0 σ 0 0 0 0

0 0 f minusσ minus d 0 0 0 σ 0 0 0

1 0 0 0 minus1 minus a a 0 0 0 0 0

0 1 0 0 b b minus 1 0 0 0 0 0

0 0 1 0 0 0 minus1 minus c minuse 0 0 0

0 0 0 1 0 0 f minus1 minus d 0 0 0

0 0 0 0 0 0 0 0 minusσ 0 0

0 0 0 0 0 0 0 0 1 minus1 0

0 0 0 0 0 0 0 0 plusmnβ(r minus 1)

1113968plusmn

β(r minus 1)

1113968β

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

equilibria

(10)

+e corresponding polynomial is

f(λ) λ(λ + β)(λ minus 1)f1(λ) (11)where f1(λ) is an eighth-order polynomial It is obvious thatat least 0 1 and minusβ are eigenvalues of system (8) for the theseequilibrium points therefore not each real part of the

4 Complexity

eigenvalues is negative +en it can be concluded that theseare not stable equilibrium points

234 Bifurcation Diagram Lyapunov Diagram and PhasePortraits It is known that when 49le ale 55 20le ble 24c 13 d 8 e 33 and f 30 Qi system is a hyperchaoticsystem When σ 10 r 28 and β 83 Lorenz system is achaotic system +erefore the paper selects the parametersa 50 c 13 d 8 e 33 f 30 σ 10 r 28 and β 83and varies b to analyze the bifurcation of system (12) asshown in Figure 1(a) As the bifurcation diagram shows thesystem demonstrates the chaotic characteristics whenb isin [minus5 26] +e corresponding Lyapunov diagram is il-lustrated in Figure 1(b) Furthermore partial phase portraitsof system (7) for different initials when b 24 are shown inFigures 1(c) One has

_x1 10 x5 minus x1( 1113857 + 50 x2 minus x1( 1113857 +x2x3

x

_x2 10 x6 minus x2( 1113857 + b x2 + x1( 1113857 minusx1x3

x

_x3 10 x7minusx3( 1113857 minus 13x3 minus 33x4 +x1x2

x

_x4 10 x8minusx4( 1113857 minus 8x4 + 30x3 +x1x2

x

_x5 28x1 minus x5 minus x1z + 50 x6 minus x5( 1113857 +x6x7

y

_x6 28x2 minus x6 minus x2z + b x5 + x6( 1113857 minusx5x7

y

_x7 28x3 minus x7 minus x3z minus 13x7 minus 33x8 +x5x6

y

_x8 28x4 minus x8 minus x4z minus 8x8 + 30x7 +x5x6

y

_x 10(y minus x)

_y 28x minus y minus xz

_z xy minus83z

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

Figures 1(c)ndash1(f) illustrate different phase portraits in-cluding x1 versus x2 x2 versus x4 x3 versus x and x versuszwhen b 24 for two initial value sets the initial values for theblue line phase portraits are 001418 004217 009157007922 009594 006557 000357 008491 009339006787 and 007577 and those for the red line phase

portraits are 001417 004218 009156 007921 009593006558 000356 008492 009338 006788 and 007576+ese portraits demonstrate that system (12) has obviouschaotic attractors and approaches periodic characteristics asinitial values changes

3 Encryption Implementation with the NewChaotic System Based on FPGA

+is paper employs the random sequence of system (12) asthe random sequence to encrypt video data and realize thehardware implement on FPGA Figure 2 is the FPGAhardware diagram used for the encryption +e maincomponents are HDMI ZYNQ JTAG and source interface+e video is collected from JTAG then the encryption al-gorithm is performed in ZYNQ powered by 5V DC and theoutputs will be shown in the monitor through HDMI

+e encryption algorithm is described in the followingand the corresponding block diagram is demonstrated inFigure 3

Step 1 to generate the random sequences for each statevariable for both discretized Qi system and Lorenzsystem respectivelyStep 2 to generate the random sequence for the newsystem (12) constructed by semitensor product oper-ation on (xQ1 xQ2 xQ3 xQ4)T and (x y)TStep 3 to generate the sequence xi(xi1 xi2 xi3 xi32)by the new system (i 1 2 8 j 1 2 32) wherexij is a binary number i represents the number of statevariables and j is the bit number for each state variableChoose a sequence xi with fixed bits from t to q that is

ci(n) xi mod 2q( 1113857mod 2tminusq

1113872 1113873 (i 1 2 amp 8 1le tlt qle 32)

(13)

Make an XOR operation on ci (n) and divide video databased on pixels that is

m(n) xi mod 2tminus11113872 1113873mod 2tminusq

1113872 1113873oplus (m) (14)

where oplus is the XOR operation

31 Discretization for the New System and Its ImplementationBasedonFPGA In the hardware experiment it is impossibleto implement the continuous Lorenz and Qi chaotic systemsbecause of limitation of the bit width in FPGA +erefore itis necessary to discretize continuous system first Multiplemethods can be used to discretize a differential equationsuch as Euler method improved Euler method and Runge-Kutta method To meet the requirement of real-time per-formance and the limitation of hardware implementationEuler method is used to discretize the differential equationsdue to its low computation complexity First Euler methodis used to discretize Qi and Lorenz systems respectively +ecorresponding process of Qi system is proposed as follows

Complexity 5

8000

6000

4000

2000

0

ndash2000

ndash4000ndash5 0 5 10 15 20 25

b

(a)

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

Lyap

unov

expo

nent

t

ndash5 0 5 10 15 20 25b

(b)

ndash8000 ndash6000 ndash4000 ndash2000 0 2000 4000 6000ndash6000

ndash4000

ndash2000

0

2000

4000

6000

x 2

x1

(c)

x 4

ndash6000 ndash4000 ndash2000 0 2000 4000 6000x2

ndash4

ndash3

ndash2

ndash1

0

1

2

3 times104

(d)

Figure 1 Continued

6 Complexity

xQ1(n + 1) minus xQ1(n)1113872 1113873

τ a xQ2(n) minus xQ1(n)) + xQ2(n)xQ3(n)1113872 1113873

xQ2(n + 1) minus xQ2(n)1113872 1113873

τ b xQ1(n) + xQ2(n)) minus xQ1(n)xQ3(n)1113872 1113873

xQ3(n + 1) minus xQ3(n)1113872 1113873

τ minuscxQ3 (n) minus exQ4(n) + xQ1(n)xQ2(n)1113872 1113873

xQ4(n + 1) minus xQ4(n)1113872 1113873

τ minusdxQ4 (n) + fxQ3(n) + xQ1(n)xQ2(n)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

+en the iteration equations of Qi system are shown inxQ1(n + 1) aτxQ2(n) +(1 minus aτ)xQ1(n) + τxQ2(n)xQ3(n)

xQ2(n + 1) bτxQ1(n) +(1 + bτ)xQ2(n) minus τxQ1(n)xQ3(n)

xQ3(n + 1) (1 minus cτ)xQ3(n) minus eτxQ4(n) + τxQ1(n)xQ2(n)

xQ4(n + 1) (1 minus dτ)xQ4(n) + fτxQ3(n) + τxQ1(n)xQ2(n)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)

Similarly the discrete Lorenz system isx(n + 1) minus x(n)

τ σ(y(n) minus x(n))

y(n + 1) minus y(n)

τ rx(n) minus y(n) minus x(n)z(n)

z(n + 1) minus z(n)

τ x(n)y(n) minus βz(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Correspondingly the iteration equations of Lorenz systemare

x3

ndash15 ndash1 ndash05 0 05 1 15 2times104

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

x

(e)

ndash20 ndash10 0 10 20x

0

10

20

30

40

50

z

(f )

Figure 1 Bifurcation diagram along with b variation and partial phase portraits for different initial values (blue line for the first initial valueset and red line for the second initial value set)

Source interface

ZYNQ

HDMI JTAG

Figure 2 FPGA hardware diagram

Semitensor

Video data m (n)

c (n)

p (n)

lfloorze (n + 1)rfloor

lfloorye (n + 1)rfloor

lfloorxe (n + 1)rfloor

lfloorxeQ1 (n + 1)rfloor

lfloorxeQ2 (n + 1)rfloor

lfloorxeQ3 (n + 1)rfloor

lfloorxeQ4 (n + 1)rfloor

lfloorxi (n)rfloor i = 1 2 8

mod (mod (lfloorxi (n)rfloor 2tndash1) 2tndashq)

Figure 3 Block diagram of encryption algorithm

Complexity 7

x(n + 1) στy(n) +(1 minus στ)x(n)

y(n + 1) rτx(n) +(1 minus τ)y(n) minus τx(n)z(n)

z(n + 1) τx(n)y(n) +(1 minus βτ)z(n)

⎧⎪⎪⎨

⎪⎪⎩(18)

In general FPGA can store float data and fixed-pointdata Since fixed-point data require less computing resourcesthan that of float data this paper uses 64-bit fixed-point

number to represent the data+e detailed data format of 64-bit fixed-point numbers is shown in Figure 4

In Figure 4 I represents the integer part of 64-bit fixed-point numbers and f is the fractional part

As mentioned before because of the limitation of bitwidth in FPGA all data are truncated numbers in hardwareimplementation +erefore the Qi and Lorenz systembecomes

lfloorxQ1(n + 1)rfloor aτlfloorxQ2(n)rfloor +(1 minus aτ)lfloorxQ1(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor bτlfloorxQ1(n)rfloor +(1 + bτ)lfloorxQ2(n)rfloor minus τlfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor (1 minus cτ)lfloorxQ3(n)rfloor minus eτlfloorxQ4(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor (1 minus dτ)lfloorxQ4(n)rfloor + fτlfloorxQ3(n)rfloor + τlfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

lfloorx(n + 1)rfloor στlfloory(n)rfloor +(1 minus στ)lfloorx(n)rfloor

lfloory(n + 1)rfloor rτlfloorx(n)rfloor +(1 minus τ)lfloory(n)rfloor minus τlfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor τlfloorx(n)rfloorlfloory(n)rfloor +(1 minus βτ)lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(20)

Let the iteration step be τ 000001 and use the sameparameters in system (12) +en substitute them into (19)

and (20) respectively +erefore Qi system and Lorenzsystem are changed as follows

lfloorxQ1(n + 1)rfloor 09995lfloorxQ2(n)rfloor + 00005lfloorxQ1(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor 00002lfloorxQ1(n)rfloor + 10002lfloorxQ2(n)rfloor minus 000001lfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor 099987lfloorxQ3(n)rfloor minus 000033lfloorxQ4(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor 000008lfloorxQ4(n)rfloor + 09997lfloorxQ3(n)rfloor + 000001lfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)

lfloorx(n + 1)rfloor 00001lfloory(n)rfloor + 09999lfloorx(n)rfloor

lfloory(n + 1)rfloor 000028lfloorx(n)rfloor + 099999lfloory(n)rfloor minus 000001lfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor 000001lfloorx(n)rfloorlfloory(n)rfloor + 09999733lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(22)

To iterate Qi system and Lorenz system and makesemitensor product operation on these two systems aftereach iteration respectively the discretized first 8 statevariables of the new system are obtained

x1(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x2(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x3(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x4(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

x5(n + 1) lfloory(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x6(n + 1) lfloory(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x7(n + 1) lfloory(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x8(n + 1) lfloory(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

where xi(n+ 1) and xi (n) are system state variables lfloory(n +

1)rfloor is the approximate value of y (n+ 1) using fixed-pointnumber

32 Implementation and Analysis of Encryption Algorithm ofthe New Chaotic System In order to ensure the randomnessof the random sequence therefore select the low bits from tto q as shown in Figure 5 +e positions of these bits are notclose to those of sign and exponent bits +en the chosenrandom encryption sequence ci (n) is shown in equation(17) +is paper selects t 1 and q 6

ci (n) modlfloorxi(n)rfloor2tminus 11113888 1113889 2qminust+1

1113888 1113889 i 1 2 8

(24)Random sequences which are selected from eight states

based on the method mentioned previously are combined togenerate the random sequence c (n)

c(n) c1(n) c2(v) c8(n)( 1113857 (25)

In order to resist the differential attack and decrease thecorrelation between adjacent random sequences the paperselects the very first iteration sequence among every N it-erations and stacks these selected sequences to construct a

8 Complexity

random sequence c (n) as shown in equation (18) +is canimprove the randomness of the random sequence

Next the random sequence c (n) conducts XOR op-eration with the divided video data Since a frame video dataincludes tricolor integer sequences R (n) G (n) and B (n)these three sequences will be encrypted simultaneously afterchanging the random sequence c (n) into three columnsevenly c1 (n) c2 (n) and c3 (n)

p1(n) c1(n)oplusR(n)

p2(n) c2(n)oplusG(n)

p3(n) c3(n)oplusB(n)

(26)

where p1 (n) p2 (n) and p3 (n) are encrypted sequences andoplus is an XOR operation One has

1113954R(n) modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplusp1(n)

modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusR(n)

1113954G(n) modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp2(n)rfloor

modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusG(n)

1113954B(n) modlfloor1113954x3(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp3(n)rfloor

mod mod lfloor1113954x3(n)rfloor 2t1113872 1113873 2tminus q

1113872 1113873oplus c3(n)oplusB(n)

(27)

where lfloor1113954xpprime(n)rfloor pprime 1 2 3 are receiver terminal sequences

33 Analysis for NIST Test NIST test is provided by NationalInstitute of Standards and Technology and it is a standard to testthe randomness of a random sequence According to the en-cryption algorithm in this paper c (n) in equation (25) should betested by NIST standard +e comparisons of the random se-quence among the new system Qi system and Lorenz system c(n) cL (n) and cQ (n) are conducted which are obtained fromserial interfaces +e results of the tests are shown in Table 1

As shown in Table 1 all the test results for the randomsequences of the new system meet the NIST test index

standards Partial test results are larger than 08 which meansthese random indexes are quite close to those of the realrandom sequences +e randomness indexes and some othertest results are better than those generated from Lorenz systemand Qi system such as frequency block frequency cumulativesums nonoverlapping template approximate entropy randomexcursions random excursions variant and linear complexity

34 Statistical Analyses Vivado IDE is used to conduct thehardware simulation +e paper also performs the statisticsanalysis for the encrypted video data generated by hardwareFigure 6(a) is one picture of a video before encryptionFigure 6(b) is the encrypted picture of a video

Figure 7 demonstrates the comparisons of statisticshistogram between the original and encrypted pictures

Figure 7 demonstrates the comparisons of statisticshistograms between the original and encrypted pictures Asillustrated in Figure 7(a) the difference of the pixels dis-tribution is obvious However distribution of differentpixels for the encrypted picture shown in Figure 7(b) is theapproximately uniform distribution It can be concludedthat the proposed encryption algorithm for the new systemcan better resist statistic attack effectively

35 Differential Analysis Differential attack is used tomeasure the sensitivity of plaintext change for the encryp-tion algorithm and commonly uses NPCR (Number of PixelsChange Rate) and UACI (Unified Average Changing In-tensity) as indexes defined as follows

NPRC 1113936efD(e f)

W times Htimes 100

UACI 1

W times Htimes 1113944

ef

C(e f) minus Cprime(e f)1113868111386811138681113868

1113868111386811138681113868

255times 100

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

NPRC 1113944m

NPRC (m)

UACI 1113944m

UACI (m)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(29)

where C (e f) is the pixel value before encryption and Cprime (ef ) is the pixel value after encryption If C (e f )Cprime (e f ) D(e f ) 0 else 1 NPRC and UACI calculated by (29) and theproposed system and encryption algorithm are 9960 and1228 for the first-time encryption respectively +ereforethe ability to resist differential attack improves to someextent In video encryption application the requirement forencryption speed is more concerned

36 Correlation Analysis Correlation analysis is used tocheck whether the neighbor pixels are close or not +ispaper analyzes the correlation for Figure 6 +e paper selects5000 random pixels from the original and the encryptedimages and analyzes the correlation among these random-pixel pairs as shown in Figure 8 As Figure 8 illustrates the

ffffffff ffffffff ffffffff ffffffff ffffffff4024

IIIIIIII IIIIIIII IIIIIIII

Figure 4 Data format for 64-bit fixed-point numbers

q t 164 63

hellip hellip hellip

Figure 5 +e schematics of numbered data bits

Complexity 9

Table 1 NIST test results for the random sequences of the new system Lorenz system and Qi system (the number of sequences is 100 andtheir lengths are 1000000)

Statistical test P value of cL (n) P value of cQ (n) P value of c (n) Test results of c (n) ProportionFrequency 0000000 0236810 0474986 radic 100100Block frequency (m 128) 0289667 0699313 0946308 radic 99100Cumulative sums 0000000 0108791 0779188 radic 100100Runs 0000000 0554420 0075719 radic 100100Longest run 0419021 0383827 0289667 radic 100100Rank 0851383 0996335 0289667 radic 100100FFT 0911413 0911413 0213309 radic 99100Nonoverlapping template (m 9) 0000003 0181557 0983453 radic 100100Overlapping template (m 9) 0181557 0935716 0924076 radic 100100Universal 0616305 0289667 0014550 radic 97100Approximate entropy (m 10) 0000000 0798139 0816537 radic 99100Random excursions 0008879 0319084 0739918 radic 6464Random excursions variant 0213309 0289667 0949602 radic 6464Linear complexity (M 500) 0010988 0013569 0108791 radic 99100Serial (m 16) 0616305 0028817 0262249 radic 100100

(a) (b)

(c)

Figure 6+e original and encrypted pictures of a video and the corresponding encrypted video data through FPGA (a)+e original pictureof a video (b) +e encrypted picture of a video (c) Encrypted video data shown on a monitor

0

05

1

15

2

0 50 100 150 200 250

104

(a)

0

05

1

15

2

0 50 100 150 200 250

104

(b)

Figure 7 Histogram between the original and encrypted pictures (a) Histogram of the original picture (b) Histogram of the encrypted picture

10 Complexity

correlation dramatically decreases when comparing twofigures before and after encryption as shown in Figure 6

4 Conclusions

+is paper proposed a new chaotic system generated byusing semitensor product on two chaotic systems and therelated dynamic characteristics are analyzed +e new sys-tem is employed in video encryption as well and the pro-posed method can generate 8 or even 12 state variables whencompared to Qi system and Lorenz system which onlygenerate 7 state variables at most in one iteration period+eproposed method can improve the speed of random se-quence generation +e NIST test results demonstrate thatthe pseudorandomness of new system is better than that ofsingle Qi system and single Lorenz system

+e proposed encryption algorithm based on semitensorproduct can be used in other chaotic systems +e synchro-nization of the new system can be implemented by synchro-nizing two original chaotic systems separately In this paperFPGA is used to implement the generation of the new chaoticsystem and to encrypt video data +e corresponding statisticsand differential and correlation analyses were also conductedwhich demonstrates that the new system has obvious advan-tages such as better random features better resistance todifferential attacks and lower pixel correlation for encryptedimages +e future work will focus on the decryption of videoinformation by the proposed chaotic system generated by thesemitensor product method in hardware

Data Availability

+e figuresrsquo zip data used to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities-Civil Aviation Universityof China (3122019044)

References

[1] J Li J Li X Chen C Jia and W Lou ldquoIdentity-based en-cryption with outsourced revocation in cloud computingrdquoIEEE Transactions on Computers vol 64 no 2 pp 425ndash4372015

[2] J Li Y K Li X Chen P P C Lee and W Lou ldquoA hybridcloud approach for secure authorized deduplicationrdquo IEEETransactions on Parallel and Distributed Systems vol 26no 5 pp 1206ndash1216 2015

[3] M Feki B Robert G Gelle and M Colas ldquoSecure digitalcommunication using discrete-time chaos synchronizationrdquoChaos Solitons amp Fractals vol 18 no 4 pp 881ndash890 2003

[4] V T Pham S Vaidyanathan C Volos S Jafari andT Kapitaniak ldquoA new multi-stable chaotic hyperjerk systemits special features circuit realization control and synchro-nizationrdquo Archives of Control Sciences vol 30 no 1pp 23ndash45 2020

[5] A Ouannas X Wang V T Pham G Grassi andV V Huynh ldquoSynchronization results for a class of frac-tional-order spatiotemporal partial differential systems basedon fractional lyapunov approachrdquo Boundary Value Problemsvol 2019 no 1 2019

[6] F Yu H Shen L Liu et al ldquoCCII and FPGA realization amultistable modified fourth-order autonomous Chuarsquos

Value of pixel (x y)

0

50

100

150

200

250Va

lue o

f pix

el (x

+ 1

y)

0 50 100 150 200 250

(a)

0 50 100 150 200 250Value of pixel (x y)

0

50

100

150

200

250

Valu

e of p

ixel

(x +

1 y

)(b)

Figure 8 Correlation analyses for the original and encrypted figures in Figure 6 (a) Correlation analysis for the original figure as shown inFigure 6(a) (b) Correlation analysis for the encrypted figure as shown in Figure 6(b)

Complexity 11

chaotic system with coexisting multiple attractorsrdquo Com-plexity vol 2020 Article ID 5212601 17 pages 2020

[7] Q Lai P D K Kuate F Liu and H H Lu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs vol 67 no 6 2019

[8] Q Lai A Akgul C B Li G H Xu and U Cavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and s-box designrdquo Entropy vol 20 no 12018

[9] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[10] R Wang M Li Z Gao and H Sun ldquoA new memristor-based5D chaotic system and circuit implementationrdquo Complexityvol 2018 Article ID 6069401 12 pages 2018

[11] Q Hong Y Li X Wang and Z Zeng ldquoA versatile pulsecontrol method to generate arbitrary multidirection multi-butterfly chaotic attractorsrdquo IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol 38 no 8pp 1480ndash1492 2019

[12] P S Shabestari Z Rostami V T Pham F E Alsaadi andT Hayat ldquoModeling of neurodegenerative diseases usingdiscrete chaotic systemsrdquo Communications in gteoreticalPhysics vol 71 no 10 pp 1241ndash1245 2019

[13] Q Jiang J Jiang and J Li ldquoAn image encryption algorithmbased on high-dimensional chaotic systemsrdquo in Proceedings ofthe IEEE International Conference on Signal ProcessingCommunications and Computing pp 1ndash4 Hong Kong ChinaAugust 2016

[14] S Singh M Ahmad and D Malik ldquoBreaking an imageencryption scheme based on chaotic synchronization phe-nomenonrdquo in Proceedings of the 2016 Ninth InternationalConference on Contemporary Computing (IC3) Noida IndiaAugust 2016

[15] P Chen S Yu B Chen L Xiao and J Lu ldquoDesign and SOPC-based realization of a video chaotic secure communicationschemerdquo International Journal of Bifurcation and Chaosvol 28 no 13 Article ID 1850160 24 pages 2018

[16] P R Sankpal and P A Vijaya ldquoImage encryption Usingchaotic maps a surveyrdquo in Proceedings of the 2014 Fifth In-ternational Conference on Signal and Image Processingpp 102ndash107 IEEE Bangalore India January 2014

[17] A Rani and B Raman ldquoAn image copyright protectionsystem using chaotic mapsrdquo Multimedia Tools and Applica-tions vol 76 no 2 pp 3121ndash3138 2016

[18] N Shyamala and K Anusudha ldquoReversible chaotic encryp-tion techniques for imagesrdquo in Proceedings of the 2017 FourthInternational Conference on Signal Processing Communica-tion and Networking (ICSCN) Chennai India March 2017

[19] S Chen S Yu J Lu G Chen and J He ldquoDesign and FPGA-based realization of a chaotic secure video communicationsystemrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 28 no 9 pp 2359ndash2371 2018

[20] Z Lin S Yu J Lu S Cai and G Chen ldquoDesign and ARM-embedded implementation of a chaotic map-based real-timesecure video communication systemrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 25 no 7pp 1203ndash1216 2015

[21] D Chang Z Li M Wang and Y Zeng ldquoA novel digital pro-grammable multi-scroll chaotic system and its application inFPGA-based audio secure communicationrdquo AEUmdashInternational

Journal of Electronics and Communications vol 88 pp 20ndash292018

[22] F J Farsana and K Gopakumar ldquoPrivate key encryption ofspeech signal based on three dimensional chaotic maprdquo inProceedings of the International Conference on Communica-tion amp Signal Processing pp 2197ndash2201 Chennai India April2017

[23] H Jia Z Guo S Wang and Z Chen ldquoMechanics analysis andhardware implementation of a new 3D chaotic systemrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 13Article ID 1850161 14 pages 2018

[24] E Tlelo-Cuautle L G De La Fraga V-T Pham C VolosS Jafari and A D J Quintas-Valles ldquoDynamics FPGA re-alization and application of a chaotic system with an infinitenumber of equilibrium pointsrdquo Nonlinear Dynamics vol 89no 2 pp 1129ndash1139 2017

[25] A Akgul H Calgan I Koyuncu I Pehlivan andA Istanbullu ldquoChaos-based engineering applications with a3D chaotic system without equilibrium pointsrdquo NonlinearDynamics vol 84 no 2 pp 481ndash495 2016

[26] Q Lai X Zhao K Rajagopal G Xu A Akgul andE Guleryuz ldquoDynamic analyses FPGA implementation andengineering applications of multi-butterfly chaotic attractorsgenerated from generalised sprott c systemrdquo Pramana vol 90no 1 p 12 2018

[27] W S Sayed M F Tolba A G Radwan and S K Abd-El-Hafiz ldquoFPGA realization of a speech encryption system basedon a generalized modified chaotic transition map and bitpermutationrdquo Multimedia Tools and Applications vol 78no 12 pp 16097ndash16127 2019

[28] Q Wang S Yu C Li et al ldquo+eoretical design and FPGA-based implementation of higher-dimensional digital chaoticsystemsrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 63 no 3 pp 401ndash412 2016

[29] E Dong Z Wang Z Chen and Z Wang ldquoTopologicalhorseshoe analysis and field-programmable gate arrayimplementation of a fractional-order four-wing chaoticattractorrdquo Chinese Physics B vol 27 no 1 pp 300ndash306 2018

[30] Z Hua B Zhou and Y Zhou ldquoSine chaotification model forenhancing chaos and its hardware implementationrdquo IEEETransactions on Industrial Electronics vol 66 no 2 pp 1273ndash1284 2019

[31] I Ozturk and R Kılıccedil ldquoA novel method for producing pseudorandom numbers from differential equation-based chaoticsystemsrdquo Nonlinear Dynamics vol 80 no 3 pp 1147ndash11572015

[32] R Wang Q Xie Y Huang H Sun and Y Sun ldquoDesign of aswitched hyperchaotic system and its applicationrdquo Interna-tional Journal of Computer Applications in Technology vol 57no 3 pp 207ndash218 2018

[33] R Wang H Sun J Wang L Wang and Y Wang ldquoAppli-cations of modularized circuit designs in a new hyper-chaoticsystem circuit implementationrdquo Chinese Physics B vol 24no 4 Article ID 020501 2015

[34] Z Lin S Yu and J Li ldquoChosen ciphertext attack on a chaoticstream cipherrdquo in Proceedings of the 30th Chinese Control andDecision Conference pp 1238ndash1242 Shenyang China June2018

[35] G Qi G Chen M A Van Wyk and B J Van Wyk ldquoOn anew hyperchaotic systemrdquo Physics Letters A vol 372 no 2pp 124ndash136 2008

[36] Z Hu and C-K Chan ldquoA real-valued chaotic orthogonalmatrix transform-based encryption for OFDM-PONrdquo IEEE

12 Complexity

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13

_x1 σ x5 minus x1( 1113857 + a x2 minus x1( 1113857 +x2x3

x

_x2 σ x6 minus x2( 1113857 + b x2 + x1( 1113857 minusx1x3

x

_x3 σ x7minusx3( 1113857 minus cx3 minus ex4 +x1x2

x

_x4 σ x8minusx4( 1113857 minus dx4 + fx3 +x1x2

x

_x5 rx1 minus x5 minus x1z + a x6 minus x5( 1113857 +x6x7

y

_x6 rx2 minus x6 minus x2z + b x5 + x6( 1113857 minusx5x7

y

_x7 rx3 minus x7 minus x3z minus cx7 minus ex8 +x5x6

y

_x8 rx4 minus x8 minus x4z minus dx8 + fx7 +x5x6

y

_x σ(y minus x)

_y rx minus y minus xz

_z xy minus βz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(8)

As seen from equation (8) if one substitutes x⟶ minusxy⟶ minusy and z⟶ z _z (minusx)(minusy) minus βz xy minus βz itproves that it is symmetric with respect to z variable for xand y

23 Numerical Analysis of the New System +e paper ana-lyzes some dynamics characteristics of the new system in-cluding symmetry dissipativity equilibrium pointequilibria bifurcation diagram Lyapunov diagram andphase portraits

231 Symmetry As described in system (8) the system issymmetry with respect to z-axis since the system is invariantunder the coordinate transformations (x1 x2 x3 x4 x5 x6

x7 x8 x y z)⟶ (minusx1 minusx2 minusx3 minusx4 minusx5 minusx6 minusx7 minusx8

minus x minusy z)

232 Dissipativity +e divergence of system (12) is given by

nabla middot f zf1

zx1+

zf2

zx2+ middot middot middot +

zf8

zx8+

zf9

zx+

zf10

zy+

zf11

zz

minus5σ minus 2a + 2b minus 2c minus 2 d minus 5 minus β

(9)

and when minus5σ minus 2a + 2b minus 2c minus 2d minus 5 minus βlt 0 the systemundergoes dissipation

233 Equilibria As shown in system (8) x y and z couldnot be zero when calculating equilibria +en the equilibriaof system (8) are (0 0 0 0 0 0 0 0 plusmn

β(r minus 1)

1113968

plusmnβ(r minus 1)

1113968 r minus 1) One has

J12

minusσ minus a a 0 0 σ 0 0 0 0 0 0

b b minus σ 0 0 0 σ 0 0 0 0 0

0 0 minusσ minus c minuse 0 0 σ 0 0 0 0

0 0 f minusσ minus d 0 0 0 σ 0 0 0

1 0 0 0 minus1 minus a a 0 0 0 0 0

0 1 0 0 b b minus 1 0 0 0 0 0

0 0 1 0 0 0 minus1 minus c minuse 0 0 0

0 0 0 1 0 0 f minus1 minus d 0 0 0

0 0 0 0 0 0 0 0 minusσ 0 0

0 0 0 0 0 0 0 0 1 minus1 0

0 0 0 0 0 0 0 0 plusmnβ(r minus 1)

1113968plusmn

β(r minus 1)

1113968β

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

equilibria

(10)

+e corresponding polynomial is

f(λ) λ(λ + β)(λ minus 1)f1(λ) (11)where f1(λ) is an eighth-order polynomial It is obvious thatat least 0 1 and minusβ are eigenvalues of system (8) for the theseequilibrium points therefore not each real part of the

4 Complexity

eigenvalues is negative +en it can be concluded that theseare not stable equilibrium points

234 Bifurcation Diagram Lyapunov Diagram and PhasePortraits It is known that when 49le ale 55 20le ble 24c 13 d 8 e 33 and f 30 Qi system is a hyperchaoticsystem When σ 10 r 28 and β 83 Lorenz system is achaotic system +erefore the paper selects the parametersa 50 c 13 d 8 e 33 f 30 σ 10 r 28 and β 83and varies b to analyze the bifurcation of system (12) asshown in Figure 1(a) As the bifurcation diagram shows thesystem demonstrates the chaotic characteristics whenb isin [minus5 26] +e corresponding Lyapunov diagram is il-lustrated in Figure 1(b) Furthermore partial phase portraitsof system (7) for different initials when b 24 are shown inFigures 1(c) One has

_x1 10 x5 minus x1( 1113857 + 50 x2 minus x1( 1113857 +x2x3

x

_x2 10 x6 minus x2( 1113857 + b x2 + x1( 1113857 minusx1x3

x

_x3 10 x7minusx3( 1113857 minus 13x3 minus 33x4 +x1x2

x

_x4 10 x8minusx4( 1113857 minus 8x4 + 30x3 +x1x2

x

_x5 28x1 minus x5 minus x1z + 50 x6 minus x5( 1113857 +x6x7

y

_x6 28x2 minus x6 minus x2z + b x5 + x6( 1113857 minusx5x7

y

_x7 28x3 minus x7 minus x3z minus 13x7 minus 33x8 +x5x6

y

_x8 28x4 minus x8 minus x4z minus 8x8 + 30x7 +x5x6

y

_x 10(y minus x)

_y 28x minus y minus xz

_z xy minus83z

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

Figures 1(c)ndash1(f) illustrate different phase portraits in-cluding x1 versus x2 x2 versus x4 x3 versus x and x versuszwhen b 24 for two initial value sets the initial values for theblue line phase portraits are 001418 004217 009157007922 009594 006557 000357 008491 009339006787 and 007577 and those for the red line phase

portraits are 001417 004218 009156 007921 009593006558 000356 008492 009338 006788 and 007576+ese portraits demonstrate that system (12) has obviouschaotic attractors and approaches periodic characteristics asinitial values changes

3 Encryption Implementation with the NewChaotic System Based on FPGA

+is paper employs the random sequence of system (12) asthe random sequence to encrypt video data and realize thehardware implement on FPGA Figure 2 is the FPGAhardware diagram used for the encryption +e maincomponents are HDMI ZYNQ JTAG and source interface+e video is collected from JTAG then the encryption al-gorithm is performed in ZYNQ powered by 5V DC and theoutputs will be shown in the monitor through HDMI

+e encryption algorithm is described in the followingand the corresponding block diagram is demonstrated inFigure 3

Step 1 to generate the random sequences for each statevariable for both discretized Qi system and Lorenzsystem respectivelyStep 2 to generate the random sequence for the newsystem (12) constructed by semitensor product oper-ation on (xQ1 xQ2 xQ3 xQ4)T and (x y)TStep 3 to generate the sequence xi(xi1 xi2 xi3 xi32)by the new system (i 1 2 8 j 1 2 32) wherexij is a binary number i represents the number of statevariables and j is the bit number for each state variableChoose a sequence xi with fixed bits from t to q that is

ci(n) xi mod 2q( 1113857mod 2tminusq

1113872 1113873 (i 1 2 amp 8 1le tlt qle 32)

(13)

Make an XOR operation on ci (n) and divide video databased on pixels that is

m(n) xi mod 2tminus11113872 1113873mod 2tminusq

1113872 1113873oplus (m) (14)

where oplus is the XOR operation

31 Discretization for the New System and Its ImplementationBasedonFPGA In the hardware experiment it is impossibleto implement the continuous Lorenz and Qi chaotic systemsbecause of limitation of the bit width in FPGA +erefore itis necessary to discretize continuous system first Multiplemethods can be used to discretize a differential equationsuch as Euler method improved Euler method and Runge-Kutta method To meet the requirement of real-time per-formance and the limitation of hardware implementationEuler method is used to discretize the differential equationsdue to its low computation complexity First Euler methodis used to discretize Qi and Lorenz systems respectively +ecorresponding process of Qi system is proposed as follows

Complexity 5

8000

6000

4000

2000

0

ndash2000

ndash4000ndash5 0 5 10 15 20 25

b

(a)

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

Lyap

unov

expo

nent

t

ndash5 0 5 10 15 20 25b

(b)

ndash8000 ndash6000 ndash4000 ndash2000 0 2000 4000 6000ndash6000

ndash4000

ndash2000

0

2000

4000

6000

x 2

x1

(c)

x 4

ndash6000 ndash4000 ndash2000 0 2000 4000 6000x2

ndash4

ndash3

ndash2

ndash1

0

1

2

3 times104

(d)

Figure 1 Continued

6 Complexity

xQ1(n + 1) minus xQ1(n)1113872 1113873

τ a xQ2(n) minus xQ1(n)) + xQ2(n)xQ3(n)1113872 1113873

xQ2(n + 1) minus xQ2(n)1113872 1113873

τ b xQ1(n) + xQ2(n)) minus xQ1(n)xQ3(n)1113872 1113873

xQ3(n + 1) minus xQ3(n)1113872 1113873

τ minuscxQ3 (n) minus exQ4(n) + xQ1(n)xQ2(n)1113872 1113873

xQ4(n + 1) minus xQ4(n)1113872 1113873

τ minusdxQ4 (n) + fxQ3(n) + xQ1(n)xQ2(n)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

+en the iteration equations of Qi system are shown inxQ1(n + 1) aτxQ2(n) +(1 minus aτ)xQ1(n) + τxQ2(n)xQ3(n)

xQ2(n + 1) bτxQ1(n) +(1 + bτ)xQ2(n) minus τxQ1(n)xQ3(n)

xQ3(n + 1) (1 minus cτ)xQ3(n) minus eτxQ4(n) + τxQ1(n)xQ2(n)

xQ4(n + 1) (1 minus dτ)xQ4(n) + fτxQ3(n) + τxQ1(n)xQ2(n)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)

Similarly the discrete Lorenz system isx(n + 1) minus x(n)

τ σ(y(n) minus x(n))

y(n + 1) minus y(n)

τ rx(n) minus y(n) minus x(n)z(n)

z(n + 1) minus z(n)

τ x(n)y(n) minus βz(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Correspondingly the iteration equations of Lorenz systemare

x3

ndash15 ndash1 ndash05 0 05 1 15 2times104

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

x

(e)

ndash20 ndash10 0 10 20x

0

10

20

30

40

50

z

(f )

Figure 1 Bifurcation diagram along with b variation and partial phase portraits for different initial values (blue line for the first initial valueset and red line for the second initial value set)

Source interface

ZYNQ

HDMI JTAG

Figure 2 FPGA hardware diagram

Semitensor

Video data m (n)

c (n)

p (n)

lfloorze (n + 1)rfloor

lfloorye (n + 1)rfloor

lfloorxe (n + 1)rfloor

lfloorxeQ1 (n + 1)rfloor

lfloorxeQ2 (n + 1)rfloor

lfloorxeQ3 (n + 1)rfloor

lfloorxeQ4 (n + 1)rfloor

lfloorxi (n)rfloor i = 1 2 8

mod (mod (lfloorxi (n)rfloor 2tndash1) 2tndashq)

Figure 3 Block diagram of encryption algorithm

Complexity 7

x(n + 1) στy(n) +(1 minus στ)x(n)

y(n + 1) rτx(n) +(1 minus τ)y(n) minus τx(n)z(n)

z(n + 1) τx(n)y(n) +(1 minus βτ)z(n)

⎧⎪⎪⎨

⎪⎪⎩(18)

In general FPGA can store float data and fixed-pointdata Since fixed-point data require less computing resourcesthan that of float data this paper uses 64-bit fixed-point

number to represent the data+e detailed data format of 64-bit fixed-point numbers is shown in Figure 4

In Figure 4 I represents the integer part of 64-bit fixed-point numbers and f is the fractional part

As mentioned before because of the limitation of bitwidth in FPGA all data are truncated numbers in hardwareimplementation +erefore the Qi and Lorenz systembecomes

lfloorxQ1(n + 1)rfloor aτlfloorxQ2(n)rfloor +(1 minus aτ)lfloorxQ1(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor bτlfloorxQ1(n)rfloor +(1 + bτ)lfloorxQ2(n)rfloor minus τlfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor (1 minus cτ)lfloorxQ3(n)rfloor minus eτlfloorxQ4(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor (1 minus dτ)lfloorxQ4(n)rfloor + fτlfloorxQ3(n)rfloor + τlfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

lfloorx(n + 1)rfloor στlfloory(n)rfloor +(1 minus στ)lfloorx(n)rfloor

lfloory(n + 1)rfloor rτlfloorx(n)rfloor +(1 minus τ)lfloory(n)rfloor minus τlfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor τlfloorx(n)rfloorlfloory(n)rfloor +(1 minus βτ)lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(20)

Let the iteration step be τ 000001 and use the sameparameters in system (12) +en substitute them into (19)

and (20) respectively +erefore Qi system and Lorenzsystem are changed as follows

lfloorxQ1(n + 1)rfloor 09995lfloorxQ2(n)rfloor + 00005lfloorxQ1(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor 00002lfloorxQ1(n)rfloor + 10002lfloorxQ2(n)rfloor minus 000001lfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor 099987lfloorxQ3(n)rfloor minus 000033lfloorxQ4(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor 000008lfloorxQ4(n)rfloor + 09997lfloorxQ3(n)rfloor + 000001lfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)

lfloorx(n + 1)rfloor 00001lfloory(n)rfloor + 09999lfloorx(n)rfloor

lfloory(n + 1)rfloor 000028lfloorx(n)rfloor + 099999lfloory(n)rfloor minus 000001lfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor 000001lfloorx(n)rfloorlfloory(n)rfloor + 09999733lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(22)

To iterate Qi system and Lorenz system and makesemitensor product operation on these two systems aftereach iteration respectively the discretized first 8 statevariables of the new system are obtained

x1(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x2(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x3(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x4(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

x5(n + 1) lfloory(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x6(n + 1) lfloory(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x7(n + 1) lfloory(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x8(n + 1) lfloory(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

where xi(n+ 1) and xi (n) are system state variables lfloory(n +

1)rfloor is the approximate value of y (n+ 1) using fixed-pointnumber

32 Implementation and Analysis of Encryption Algorithm ofthe New Chaotic System In order to ensure the randomnessof the random sequence therefore select the low bits from tto q as shown in Figure 5 +e positions of these bits are notclose to those of sign and exponent bits +en the chosenrandom encryption sequence ci (n) is shown in equation(17) +is paper selects t 1 and q 6

ci (n) modlfloorxi(n)rfloor2tminus 11113888 1113889 2qminust+1

1113888 1113889 i 1 2 8

(24)Random sequences which are selected from eight states

based on the method mentioned previously are combined togenerate the random sequence c (n)

c(n) c1(n) c2(v) c8(n)( 1113857 (25)

In order to resist the differential attack and decrease thecorrelation between adjacent random sequences the paperselects the very first iteration sequence among every N it-erations and stacks these selected sequences to construct a

8 Complexity

random sequence c (n) as shown in equation (18) +is canimprove the randomness of the random sequence

Next the random sequence c (n) conducts XOR op-eration with the divided video data Since a frame video dataincludes tricolor integer sequences R (n) G (n) and B (n)these three sequences will be encrypted simultaneously afterchanging the random sequence c (n) into three columnsevenly c1 (n) c2 (n) and c3 (n)

p1(n) c1(n)oplusR(n)

p2(n) c2(n)oplusG(n)

p3(n) c3(n)oplusB(n)

(26)

where p1 (n) p2 (n) and p3 (n) are encrypted sequences andoplus is an XOR operation One has

1113954R(n) modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplusp1(n)

modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusR(n)

1113954G(n) modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp2(n)rfloor

modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusG(n)

1113954B(n) modlfloor1113954x3(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp3(n)rfloor

mod mod lfloor1113954x3(n)rfloor 2t1113872 1113873 2tminus q

1113872 1113873oplus c3(n)oplusB(n)

(27)

where lfloor1113954xpprime(n)rfloor pprime 1 2 3 are receiver terminal sequences

33 Analysis for NIST Test NIST test is provided by NationalInstitute of Standards and Technology and it is a standard to testthe randomness of a random sequence According to the en-cryption algorithm in this paper c (n) in equation (25) should betested by NIST standard +e comparisons of the random se-quence among the new system Qi system and Lorenz system c(n) cL (n) and cQ (n) are conducted which are obtained fromserial interfaces +e results of the tests are shown in Table 1

As shown in Table 1 all the test results for the randomsequences of the new system meet the NIST test index

standards Partial test results are larger than 08 which meansthese random indexes are quite close to those of the realrandom sequences +e randomness indexes and some othertest results are better than those generated from Lorenz systemand Qi system such as frequency block frequency cumulativesums nonoverlapping template approximate entropy randomexcursions random excursions variant and linear complexity

34 Statistical Analyses Vivado IDE is used to conduct thehardware simulation +e paper also performs the statisticsanalysis for the encrypted video data generated by hardwareFigure 6(a) is one picture of a video before encryptionFigure 6(b) is the encrypted picture of a video

Figure 7 demonstrates the comparisons of statisticshistogram between the original and encrypted pictures

Figure 7 demonstrates the comparisons of statisticshistograms between the original and encrypted pictures Asillustrated in Figure 7(a) the difference of the pixels dis-tribution is obvious However distribution of differentpixels for the encrypted picture shown in Figure 7(b) is theapproximately uniform distribution It can be concludedthat the proposed encryption algorithm for the new systemcan better resist statistic attack effectively

35 Differential Analysis Differential attack is used tomeasure the sensitivity of plaintext change for the encryp-tion algorithm and commonly uses NPCR (Number of PixelsChange Rate) and UACI (Unified Average Changing In-tensity) as indexes defined as follows

NPRC 1113936efD(e f)

W times Htimes 100

UACI 1

W times Htimes 1113944

ef

C(e f) minus Cprime(e f)1113868111386811138681113868

1113868111386811138681113868

255times 100

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

NPRC 1113944m

NPRC (m)

UACI 1113944m

UACI (m)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(29)

where C (e f) is the pixel value before encryption and Cprime (ef ) is the pixel value after encryption If C (e f )Cprime (e f ) D(e f ) 0 else 1 NPRC and UACI calculated by (29) and theproposed system and encryption algorithm are 9960 and1228 for the first-time encryption respectively +ereforethe ability to resist differential attack improves to someextent In video encryption application the requirement forencryption speed is more concerned

36 Correlation Analysis Correlation analysis is used tocheck whether the neighbor pixels are close or not +ispaper analyzes the correlation for Figure 6 +e paper selects5000 random pixels from the original and the encryptedimages and analyzes the correlation among these random-pixel pairs as shown in Figure 8 As Figure 8 illustrates the

ffffffff ffffffff ffffffff ffffffff ffffffff4024

IIIIIIII IIIIIIII IIIIIIII

Figure 4 Data format for 64-bit fixed-point numbers

q t 164 63

hellip hellip hellip

Figure 5 +e schematics of numbered data bits

Complexity 9

Table 1 NIST test results for the random sequences of the new system Lorenz system and Qi system (the number of sequences is 100 andtheir lengths are 1000000)

Statistical test P value of cL (n) P value of cQ (n) P value of c (n) Test results of c (n) ProportionFrequency 0000000 0236810 0474986 radic 100100Block frequency (m 128) 0289667 0699313 0946308 radic 99100Cumulative sums 0000000 0108791 0779188 radic 100100Runs 0000000 0554420 0075719 radic 100100Longest run 0419021 0383827 0289667 radic 100100Rank 0851383 0996335 0289667 radic 100100FFT 0911413 0911413 0213309 radic 99100Nonoverlapping template (m 9) 0000003 0181557 0983453 radic 100100Overlapping template (m 9) 0181557 0935716 0924076 radic 100100Universal 0616305 0289667 0014550 radic 97100Approximate entropy (m 10) 0000000 0798139 0816537 radic 99100Random excursions 0008879 0319084 0739918 radic 6464Random excursions variant 0213309 0289667 0949602 radic 6464Linear complexity (M 500) 0010988 0013569 0108791 radic 99100Serial (m 16) 0616305 0028817 0262249 radic 100100

(a) (b)

(c)

Figure 6+e original and encrypted pictures of a video and the corresponding encrypted video data through FPGA (a)+e original pictureof a video (b) +e encrypted picture of a video (c) Encrypted video data shown on a monitor

0

05

1

15

2

0 50 100 150 200 250

104

(a)

0

05

1

15

2

0 50 100 150 200 250

104

(b)

Figure 7 Histogram between the original and encrypted pictures (a) Histogram of the original picture (b) Histogram of the encrypted picture

10 Complexity

correlation dramatically decreases when comparing twofigures before and after encryption as shown in Figure 6

4 Conclusions

+is paper proposed a new chaotic system generated byusing semitensor product on two chaotic systems and therelated dynamic characteristics are analyzed +e new sys-tem is employed in video encryption as well and the pro-posed method can generate 8 or even 12 state variables whencompared to Qi system and Lorenz system which onlygenerate 7 state variables at most in one iteration period+eproposed method can improve the speed of random se-quence generation +e NIST test results demonstrate thatthe pseudorandomness of new system is better than that ofsingle Qi system and single Lorenz system

+e proposed encryption algorithm based on semitensorproduct can be used in other chaotic systems +e synchro-nization of the new system can be implemented by synchro-nizing two original chaotic systems separately In this paperFPGA is used to implement the generation of the new chaoticsystem and to encrypt video data +e corresponding statisticsand differential and correlation analyses were also conductedwhich demonstrates that the new system has obvious advan-tages such as better random features better resistance todifferential attacks and lower pixel correlation for encryptedimages +e future work will focus on the decryption of videoinformation by the proposed chaotic system generated by thesemitensor product method in hardware

Data Availability

+e figuresrsquo zip data used to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities-Civil Aviation Universityof China (3122019044)

References

[1] J Li J Li X Chen C Jia and W Lou ldquoIdentity-based en-cryption with outsourced revocation in cloud computingrdquoIEEE Transactions on Computers vol 64 no 2 pp 425ndash4372015

[2] J Li Y K Li X Chen P P C Lee and W Lou ldquoA hybridcloud approach for secure authorized deduplicationrdquo IEEETransactions on Parallel and Distributed Systems vol 26no 5 pp 1206ndash1216 2015

[3] M Feki B Robert G Gelle and M Colas ldquoSecure digitalcommunication using discrete-time chaos synchronizationrdquoChaos Solitons amp Fractals vol 18 no 4 pp 881ndash890 2003

[4] V T Pham S Vaidyanathan C Volos S Jafari andT Kapitaniak ldquoA new multi-stable chaotic hyperjerk systemits special features circuit realization control and synchro-nizationrdquo Archives of Control Sciences vol 30 no 1pp 23ndash45 2020

[5] A Ouannas X Wang V T Pham G Grassi andV V Huynh ldquoSynchronization results for a class of frac-tional-order spatiotemporal partial differential systems basedon fractional lyapunov approachrdquo Boundary Value Problemsvol 2019 no 1 2019

[6] F Yu H Shen L Liu et al ldquoCCII and FPGA realization amultistable modified fourth-order autonomous Chuarsquos

Value of pixel (x y)

0

50

100

150

200

250Va

lue o

f pix

el (x

+ 1

y)

0 50 100 150 200 250

(a)

0 50 100 150 200 250Value of pixel (x y)

0

50

100

150

200

250

Valu

e of p

ixel

(x +

1 y

)(b)

Figure 8 Correlation analyses for the original and encrypted figures in Figure 6 (a) Correlation analysis for the original figure as shown inFigure 6(a) (b) Correlation analysis for the encrypted figure as shown in Figure 6(b)

Complexity 11

chaotic system with coexisting multiple attractorsrdquo Com-plexity vol 2020 Article ID 5212601 17 pages 2020

[7] Q Lai P D K Kuate F Liu and H H Lu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs vol 67 no 6 2019

[8] Q Lai A Akgul C B Li G H Xu and U Cavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and s-box designrdquo Entropy vol 20 no 12018

[9] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[10] R Wang M Li Z Gao and H Sun ldquoA new memristor-based5D chaotic system and circuit implementationrdquo Complexityvol 2018 Article ID 6069401 12 pages 2018

[11] Q Hong Y Li X Wang and Z Zeng ldquoA versatile pulsecontrol method to generate arbitrary multidirection multi-butterfly chaotic attractorsrdquo IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol 38 no 8pp 1480ndash1492 2019

[12] P S Shabestari Z Rostami V T Pham F E Alsaadi andT Hayat ldquoModeling of neurodegenerative diseases usingdiscrete chaotic systemsrdquo Communications in gteoreticalPhysics vol 71 no 10 pp 1241ndash1245 2019

[13] Q Jiang J Jiang and J Li ldquoAn image encryption algorithmbased on high-dimensional chaotic systemsrdquo in Proceedings ofthe IEEE International Conference on Signal ProcessingCommunications and Computing pp 1ndash4 Hong Kong ChinaAugust 2016

[14] S Singh M Ahmad and D Malik ldquoBreaking an imageencryption scheme based on chaotic synchronization phe-nomenonrdquo in Proceedings of the 2016 Ninth InternationalConference on Contemporary Computing (IC3) Noida IndiaAugust 2016

[15] P Chen S Yu B Chen L Xiao and J Lu ldquoDesign and SOPC-based realization of a video chaotic secure communicationschemerdquo International Journal of Bifurcation and Chaosvol 28 no 13 Article ID 1850160 24 pages 2018

[16] P R Sankpal and P A Vijaya ldquoImage encryption Usingchaotic maps a surveyrdquo in Proceedings of the 2014 Fifth In-ternational Conference on Signal and Image Processingpp 102ndash107 IEEE Bangalore India January 2014

[17] A Rani and B Raman ldquoAn image copyright protectionsystem using chaotic mapsrdquo Multimedia Tools and Applica-tions vol 76 no 2 pp 3121ndash3138 2016

[18] N Shyamala and K Anusudha ldquoReversible chaotic encryp-tion techniques for imagesrdquo in Proceedings of the 2017 FourthInternational Conference on Signal Processing Communica-tion and Networking (ICSCN) Chennai India March 2017

[19] S Chen S Yu J Lu G Chen and J He ldquoDesign and FPGA-based realization of a chaotic secure video communicationsystemrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 28 no 9 pp 2359ndash2371 2018

[20] Z Lin S Yu J Lu S Cai and G Chen ldquoDesign and ARM-embedded implementation of a chaotic map-based real-timesecure video communication systemrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 25 no 7pp 1203ndash1216 2015

[21] D Chang Z Li M Wang and Y Zeng ldquoA novel digital pro-grammable multi-scroll chaotic system and its application inFPGA-based audio secure communicationrdquo AEUmdashInternational

Journal of Electronics and Communications vol 88 pp 20ndash292018

[22] F J Farsana and K Gopakumar ldquoPrivate key encryption ofspeech signal based on three dimensional chaotic maprdquo inProceedings of the International Conference on Communica-tion amp Signal Processing pp 2197ndash2201 Chennai India April2017

[23] H Jia Z Guo S Wang and Z Chen ldquoMechanics analysis andhardware implementation of a new 3D chaotic systemrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 13Article ID 1850161 14 pages 2018

[24] E Tlelo-Cuautle L G De La Fraga V-T Pham C VolosS Jafari and A D J Quintas-Valles ldquoDynamics FPGA re-alization and application of a chaotic system with an infinitenumber of equilibrium pointsrdquo Nonlinear Dynamics vol 89no 2 pp 1129ndash1139 2017

[25] A Akgul H Calgan I Koyuncu I Pehlivan andA Istanbullu ldquoChaos-based engineering applications with a3D chaotic system without equilibrium pointsrdquo NonlinearDynamics vol 84 no 2 pp 481ndash495 2016

[26] Q Lai X Zhao K Rajagopal G Xu A Akgul andE Guleryuz ldquoDynamic analyses FPGA implementation andengineering applications of multi-butterfly chaotic attractorsgenerated from generalised sprott c systemrdquo Pramana vol 90no 1 p 12 2018

[27] W S Sayed M F Tolba A G Radwan and S K Abd-El-Hafiz ldquoFPGA realization of a speech encryption system basedon a generalized modified chaotic transition map and bitpermutationrdquo Multimedia Tools and Applications vol 78no 12 pp 16097ndash16127 2019

[28] Q Wang S Yu C Li et al ldquo+eoretical design and FPGA-based implementation of higher-dimensional digital chaoticsystemsrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 63 no 3 pp 401ndash412 2016

[29] E Dong Z Wang Z Chen and Z Wang ldquoTopologicalhorseshoe analysis and field-programmable gate arrayimplementation of a fractional-order four-wing chaoticattractorrdquo Chinese Physics B vol 27 no 1 pp 300ndash306 2018

[30] Z Hua B Zhou and Y Zhou ldquoSine chaotification model forenhancing chaos and its hardware implementationrdquo IEEETransactions on Industrial Electronics vol 66 no 2 pp 1273ndash1284 2019

[31] I Ozturk and R Kılıccedil ldquoA novel method for producing pseudorandom numbers from differential equation-based chaoticsystemsrdquo Nonlinear Dynamics vol 80 no 3 pp 1147ndash11572015

[32] R Wang Q Xie Y Huang H Sun and Y Sun ldquoDesign of aswitched hyperchaotic system and its applicationrdquo Interna-tional Journal of Computer Applications in Technology vol 57no 3 pp 207ndash218 2018

[33] R Wang H Sun J Wang L Wang and Y Wang ldquoAppli-cations of modularized circuit designs in a new hyper-chaoticsystem circuit implementationrdquo Chinese Physics B vol 24no 4 Article ID 020501 2015

[34] Z Lin S Yu and J Li ldquoChosen ciphertext attack on a chaoticstream cipherrdquo in Proceedings of the 30th Chinese Control andDecision Conference pp 1238ndash1242 Shenyang China June2018

[35] G Qi G Chen M A Van Wyk and B J Van Wyk ldquoOn anew hyperchaotic systemrdquo Physics Letters A vol 372 no 2pp 124ndash136 2008

[36] Z Hu and C-K Chan ldquoA real-valued chaotic orthogonalmatrix transform-based encryption for OFDM-PONrdquo IEEE

12 Complexity

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13

eigenvalues is negative +en it can be concluded that theseare not stable equilibrium points

234 Bifurcation Diagram Lyapunov Diagram and PhasePortraits It is known that when 49le ale 55 20le ble 24c 13 d 8 e 33 and f 30 Qi system is a hyperchaoticsystem When σ 10 r 28 and β 83 Lorenz system is achaotic system +erefore the paper selects the parametersa 50 c 13 d 8 e 33 f 30 σ 10 r 28 and β 83and varies b to analyze the bifurcation of system (12) asshown in Figure 1(a) As the bifurcation diagram shows thesystem demonstrates the chaotic characteristics whenb isin [minus5 26] +e corresponding Lyapunov diagram is il-lustrated in Figure 1(b) Furthermore partial phase portraitsof system (7) for different initials when b 24 are shown inFigures 1(c) One has

_x1 10 x5 minus x1( 1113857 + 50 x2 minus x1( 1113857 +x2x3

x

_x2 10 x6 minus x2( 1113857 + b x2 + x1( 1113857 minusx1x3

x

_x3 10 x7minusx3( 1113857 minus 13x3 minus 33x4 +x1x2

x

_x4 10 x8minusx4( 1113857 minus 8x4 + 30x3 +x1x2

x

_x5 28x1 minus x5 minus x1z + 50 x6 minus x5( 1113857 +x6x7

y

_x6 28x2 minus x6 minus x2z + b x5 + x6( 1113857 minusx5x7

y

_x7 28x3 minus x7 minus x3z minus 13x7 minus 33x8 +x5x6

y

_x8 28x4 minus x8 minus x4z minus 8x8 + 30x7 +x5x6

y

_x 10(y minus x)

_y 28x minus y minus xz

_z xy minus83z

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

Figures 1(c)ndash1(f) illustrate different phase portraits in-cluding x1 versus x2 x2 versus x4 x3 versus x and x versuszwhen b 24 for two initial value sets the initial values for theblue line phase portraits are 001418 004217 009157007922 009594 006557 000357 008491 009339006787 and 007577 and those for the red line phase

portraits are 001417 004218 009156 007921 009593006558 000356 008492 009338 006788 and 007576+ese portraits demonstrate that system (12) has obviouschaotic attractors and approaches periodic characteristics asinitial values changes

3 Encryption Implementation with the NewChaotic System Based on FPGA

+is paper employs the random sequence of system (12) asthe random sequence to encrypt video data and realize thehardware implement on FPGA Figure 2 is the FPGAhardware diagram used for the encryption +e maincomponents are HDMI ZYNQ JTAG and source interface+e video is collected from JTAG then the encryption al-gorithm is performed in ZYNQ powered by 5V DC and theoutputs will be shown in the monitor through HDMI

+e encryption algorithm is described in the followingand the corresponding block diagram is demonstrated inFigure 3

Step 1 to generate the random sequences for each statevariable for both discretized Qi system and Lorenzsystem respectivelyStep 2 to generate the random sequence for the newsystem (12) constructed by semitensor product oper-ation on (xQ1 xQ2 xQ3 xQ4)T and (x y)TStep 3 to generate the sequence xi(xi1 xi2 xi3 xi32)by the new system (i 1 2 8 j 1 2 32) wherexij is a binary number i represents the number of statevariables and j is the bit number for each state variableChoose a sequence xi with fixed bits from t to q that is

ci(n) xi mod 2q( 1113857mod 2tminusq

1113872 1113873 (i 1 2 amp 8 1le tlt qle 32)

(13)

Make an XOR operation on ci (n) and divide video databased on pixels that is

m(n) xi mod 2tminus11113872 1113873mod 2tminusq

1113872 1113873oplus (m) (14)

where oplus is the XOR operation

31 Discretization for the New System and Its ImplementationBasedonFPGA In the hardware experiment it is impossibleto implement the continuous Lorenz and Qi chaotic systemsbecause of limitation of the bit width in FPGA +erefore itis necessary to discretize continuous system first Multiplemethods can be used to discretize a differential equationsuch as Euler method improved Euler method and Runge-Kutta method To meet the requirement of real-time per-formance and the limitation of hardware implementationEuler method is used to discretize the differential equationsdue to its low computation complexity First Euler methodis used to discretize Qi and Lorenz systems respectively +ecorresponding process of Qi system is proposed as follows

Complexity 5

8000

6000

4000

2000

0

ndash2000

ndash4000ndash5 0 5 10 15 20 25

b

(a)

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

Lyap

unov

expo

nent

t

ndash5 0 5 10 15 20 25b

(b)

ndash8000 ndash6000 ndash4000 ndash2000 0 2000 4000 6000ndash6000

ndash4000

ndash2000

0

2000

4000

6000

x 2

x1

(c)

x 4

ndash6000 ndash4000 ndash2000 0 2000 4000 6000x2

ndash4

ndash3

ndash2

ndash1

0

1

2

3 times104

(d)

Figure 1 Continued

6 Complexity

xQ1(n + 1) minus xQ1(n)1113872 1113873

τ a xQ2(n) minus xQ1(n)) + xQ2(n)xQ3(n)1113872 1113873

xQ2(n + 1) minus xQ2(n)1113872 1113873

τ b xQ1(n) + xQ2(n)) minus xQ1(n)xQ3(n)1113872 1113873

xQ3(n + 1) minus xQ3(n)1113872 1113873

τ minuscxQ3 (n) minus exQ4(n) + xQ1(n)xQ2(n)1113872 1113873

xQ4(n + 1) minus xQ4(n)1113872 1113873

τ minusdxQ4 (n) + fxQ3(n) + xQ1(n)xQ2(n)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

+en the iteration equations of Qi system are shown inxQ1(n + 1) aτxQ2(n) +(1 minus aτ)xQ1(n) + τxQ2(n)xQ3(n)

xQ2(n + 1) bτxQ1(n) +(1 + bτ)xQ2(n) minus τxQ1(n)xQ3(n)

xQ3(n + 1) (1 minus cτ)xQ3(n) minus eτxQ4(n) + τxQ1(n)xQ2(n)

xQ4(n + 1) (1 minus dτ)xQ4(n) + fτxQ3(n) + τxQ1(n)xQ2(n)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)

Similarly the discrete Lorenz system isx(n + 1) minus x(n)

τ σ(y(n) minus x(n))

y(n + 1) minus y(n)

τ rx(n) minus y(n) minus x(n)z(n)

z(n + 1) minus z(n)

τ x(n)y(n) minus βz(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Correspondingly the iteration equations of Lorenz systemare

x3

ndash15 ndash1 ndash05 0 05 1 15 2times104

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

x

(e)

ndash20 ndash10 0 10 20x

0

10

20

30

40

50

z

(f )

Figure 1 Bifurcation diagram along with b variation and partial phase portraits for different initial values (blue line for the first initial valueset and red line for the second initial value set)

Source interface

ZYNQ

HDMI JTAG

Figure 2 FPGA hardware diagram

Semitensor

Video data m (n)

c (n)

p (n)

lfloorze (n + 1)rfloor

lfloorye (n + 1)rfloor

lfloorxe (n + 1)rfloor

lfloorxeQ1 (n + 1)rfloor

lfloorxeQ2 (n + 1)rfloor

lfloorxeQ3 (n + 1)rfloor

lfloorxeQ4 (n + 1)rfloor

lfloorxi (n)rfloor i = 1 2 8

mod (mod (lfloorxi (n)rfloor 2tndash1) 2tndashq)

Figure 3 Block diagram of encryption algorithm

Complexity 7

x(n + 1) στy(n) +(1 minus στ)x(n)

y(n + 1) rτx(n) +(1 minus τ)y(n) minus τx(n)z(n)

z(n + 1) τx(n)y(n) +(1 minus βτ)z(n)

⎧⎪⎪⎨

⎪⎪⎩(18)

In general FPGA can store float data and fixed-pointdata Since fixed-point data require less computing resourcesthan that of float data this paper uses 64-bit fixed-point

number to represent the data+e detailed data format of 64-bit fixed-point numbers is shown in Figure 4

In Figure 4 I represents the integer part of 64-bit fixed-point numbers and f is the fractional part

As mentioned before because of the limitation of bitwidth in FPGA all data are truncated numbers in hardwareimplementation +erefore the Qi and Lorenz systembecomes

lfloorxQ1(n + 1)rfloor aτlfloorxQ2(n)rfloor +(1 minus aτ)lfloorxQ1(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor bτlfloorxQ1(n)rfloor +(1 + bτ)lfloorxQ2(n)rfloor minus τlfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor (1 minus cτ)lfloorxQ3(n)rfloor minus eτlfloorxQ4(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor (1 minus dτ)lfloorxQ4(n)rfloor + fτlfloorxQ3(n)rfloor + τlfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

lfloorx(n + 1)rfloor στlfloory(n)rfloor +(1 minus στ)lfloorx(n)rfloor

lfloory(n + 1)rfloor rτlfloorx(n)rfloor +(1 minus τ)lfloory(n)rfloor minus τlfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor τlfloorx(n)rfloorlfloory(n)rfloor +(1 minus βτ)lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(20)

Let the iteration step be τ 000001 and use the sameparameters in system (12) +en substitute them into (19)

and (20) respectively +erefore Qi system and Lorenzsystem are changed as follows

lfloorxQ1(n + 1)rfloor 09995lfloorxQ2(n)rfloor + 00005lfloorxQ1(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor 00002lfloorxQ1(n)rfloor + 10002lfloorxQ2(n)rfloor minus 000001lfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor 099987lfloorxQ3(n)rfloor minus 000033lfloorxQ4(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor 000008lfloorxQ4(n)rfloor + 09997lfloorxQ3(n)rfloor + 000001lfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)

lfloorx(n + 1)rfloor 00001lfloory(n)rfloor + 09999lfloorx(n)rfloor

lfloory(n + 1)rfloor 000028lfloorx(n)rfloor + 099999lfloory(n)rfloor minus 000001lfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor 000001lfloorx(n)rfloorlfloory(n)rfloor + 09999733lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(22)

To iterate Qi system and Lorenz system and makesemitensor product operation on these two systems aftereach iteration respectively the discretized first 8 statevariables of the new system are obtained

x1(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x2(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x3(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x4(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

x5(n + 1) lfloory(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x6(n + 1) lfloory(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x7(n + 1) lfloory(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x8(n + 1) lfloory(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

where xi(n+ 1) and xi (n) are system state variables lfloory(n +

1)rfloor is the approximate value of y (n+ 1) using fixed-pointnumber

32 Implementation and Analysis of Encryption Algorithm ofthe New Chaotic System In order to ensure the randomnessof the random sequence therefore select the low bits from tto q as shown in Figure 5 +e positions of these bits are notclose to those of sign and exponent bits +en the chosenrandom encryption sequence ci (n) is shown in equation(17) +is paper selects t 1 and q 6

ci (n) modlfloorxi(n)rfloor2tminus 11113888 1113889 2qminust+1

1113888 1113889 i 1 2 8

(24)Random sequences which are selected from eight states

based on the method mentioned previously are combined togenerate the random sequence c (n)

c(n) c1(n) c2(v) c8(n)( 1113857 (25)

In order to resist the differential attack and decrease thecorrelation between adjacent random sequences the paperselects the very first iteration sequence among every N it-erations and stacks these selected sequences to construct a

8 Complexity

random sequence c (n) as shown in equation (18) +is canimprove the randomness of the random sequence

Next the random sequence c (n) conducts XOR op-eration with the divided video data Since a frame video dataincludes tricolor integer sequences R (n) G (n) and B (n)these three sequences will be encrypted simultaneously afterchanging the random sequence c (n) into three columnsevenly c1 (n) c2 (n) and c3 (n)

p1(n) c1(n)oplusR(n)

p2(n) c2(n)oplusG(n)

p3(n) c3(n)oplusB(n)

(26)

where p1 (n) p2 (n) and p3 (n) are encrypted sequences andoplus is an XOR operation One has

1113954R(n) modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplusp1(n)

modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusR(n)

1113954G(n) modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp2(n)rfloor

modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusG(n)

1113954B(n) modlfloor1113954x3(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp3(n)rfloor

mod mod lfloor1113954x3(n)rfloor 2t1113872 1113873 2tminus q

1113872 1113873oplus c3(n)oplusB(n)

(27)

where lfloor1113954xpprime(n)rfloor pprime 1 2 3 are receiver terminal sequences

33 Analysis for NIST Test NIST test is provided by NationalInstitute of Standards and Technology and it is a standard to testthe randomness of a random sequence According to the en-cryption algorithm in this paper c (n) in equation (25) should betested by NIST standard +e comparisons of the random se-quence among the new system Qi system and Lorenz system c(n) cL (n) and cQ (n) are conducted which are obtained fromserial interfaces +e results of the tests are shown in Table 1

As shown in Table 1 all the test results for the randomsequences of the new system meet the NIST test index

standards Partial test results are larger than 08 which meansthese random indexes are quite close to those of the realrandom sequences +e randomness indexes and some othertest results are better than those generated from Lorenz systemand Qi system such as frequency block frequency cumulativesums nonoverlapping template approximate entropy randomexcursions random excursions variant and linear complexity

34 Statistical Analyses Vivado IDE is used to conduct thehardware simulation +e paper also performs the statisticsanalysis for the encrypted video data generated by hardwareFigure 6(a) is one picture of a video before encryptionFigure 6(b) is the encrypted picture of a video

Figure 7 demonstrates the comparisons of statisticshistogram between the original and encrypted pictures

Figure 7 demonstrates the comparisons of statisticshistograms between the original and encrypted pictures Asillustrated in Figure 7(a) the difference of the pixels dis-tribution is obvious However distribution of differentpixels for the encrypted picture shown in Figure 7(b) is theapproximately uniform distribution It can be concludedthat the proposed encryption algorithm for the new systemcan better resist statistic attack effectively

35 Differential Analysis Differential attack is used tomeasure the sensitivity of plaintext change for the encryp-tion algorithm and commonly uses NPCR (Number of PixelsChange Rate) and UACI (Unified Average Changing In-tensity) as indexes defined as follows

NPRC 1113936efD(e f)

W times Htimes 100

UACI 1

W times Htimes 1113944

ef

C(e f) minus Cprime(e f)1113868111386811138681113868

1113868111386811138681113868

255times 100

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

NPRC 1113944m

NPRC (m)

UACI 1113944m

UACI (m)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(29)

where C (e f) is the pixel value before encryption and Cprime (ef ) is the pixel value after encryption If C (e f )Cprime (e f ) D(e f ) 0 else 1 NPRC and UACI calculated by (29) and theproposed system and encryption algorithm are 9960 and1228 for the first-time encryption respectively +ereforethe ability to resist differential attack improves to someextent In video encryption application the requirement forencryption speed is more concerned

36 Correlation Analysis Correlation analysis is used tocheck whether the neighbor pixels are close or not +ispaper analyzes the correlation for Figure 6 +e paper selects5000 random pixels from the original and the encryptedimages and analyzes the correlation among these random-pixel pairs as shown in Figure 8 As Figure 8 illustrates the

ffffffff ffffffff ffffffff ffffffff ffffffff4024

IIIIIIII IIIIIIII IIIIIIII

Figure 4 Data format for 64-bit fixed-point numbers

q t 164 63

hellip hellip hellip

Figure 5 +e schematics of numbered data bits

Complexity 9

Table 1 NIST test results for the random sequences of the new system Lorenz system and Qi system (the number of sequences is 100 andtheir lengths are 1000000)

Statistical test P value of cL (n) P value of cQ (n) P value of c (n) Test results of c (n) ProportionFrequency 0000000 0236810 0474986 radic 100100Block frequency (m 128) 0289667 0699313 0946308 radic 99100Cumulative sums 0000000 0108791 0779188 radic 100100Runs 0000000 0554420 0075719 radic 100100Longest run 0419021 0383827 0289667 radic 100100Rank 0851383 0996335 0289667 radic 100100FFT 0911413 0911413 0213309 radic 99100Nonoverlapping template (m 9) 0000003 0181557 0983453 radic 100100Overlapping template (m 9) 0181557 0935716 0924076 radic 100100Universal 0616305 0289667 0014550 radic 97100Approximate entropy (m 10) 0000000 0798139 0816537 radic 99100Random excursions 0008879 0319084 0739918 radic 6464Random excursions variant 0213309 0289667 0949602 radic 6464Linear complexity (M 500) 0010988 0013569 0108791 radic 99100Serial (m 16) 0616305 0028817 0262249 radic 100100

(a) (b)

(c)

Figure 6+e original and encrypted pictures of a video and the corresponding encrypted video data through FPGA (a)+e original pictureof a video (b) +e encrypted picture of a video (c) Encrypted video data shown on a monitor

0

05

1

15

2

0 50 100 150 200 250

104

(a)

0

05

1

15

2

0 50 100 150 200 250

104

(b)

Figure 7 Histogram between the original and encrypted pictures (a) Histogram of the original picture (b) Histogram of the encrypted picture

10 Complexity

correlation dramatically decreases when comparing twofigures before and after encryption as shown in Figure 6

4 Conclusions

+is paper proposed a new chaotic system generated byusing semitensor product on two chaotic systems and therelated dynamic characteristics are analyzed +e new sys-tem is employed in video encryption as well and the pro-posed method can generate 8 or even 12 state variables whencompared to Qi system and Lorenz system which onlygenerate 7 state variables at most in one iteration period+eproposed method can improve the speed of random se-quence generation +e NIST test results demonstrate thatthe pseudorandomness of new system is better than that ofsingle Qi system and single Lorenz system

+e proposed encryption algorithm based on semitensorproduct can be used in other chaotic systems +e synchro-nization of the new system can be implemented by synchro-nizing two original chaotic systems separately In this paperFPGA is used to implement the generation of the new chaoticsystem and to encrypt video data +e corresponding statisticsand differential and correlation analyses were also conductedwhich demonstrates that the new system has obvious advan-tages such as better random features better resistance todifferential attacks and lower pixel correlation for encryptedimages +e future work will focus on the decryption of videoinformation by the proposed chaotic system generated by thesemitensor product method in hardware

Data Availability

+e figuresrsquo zip data used to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities-Civil Aviation Universityof China (3122019044)

References

[1] J Li J Li X Chen C Jia and W Lou ldquoIdentity-based en-cryption with outsourced revocation in cloud computingrdquoIEEE Transactions on Computers vol 64 no 2 pp 425ndash4372015

[2] J Li Y K Li X Chen P P C Lee and W Lou ldquoA hybridcloud approach for secure authorized deduplicationrdquo IEEETransactions on Parallel and Distributed Systems vol 26no 5 pp 1206ndash1216 2015

[3] M Feki B Robert G Gelle and M Colas ldquoSecure digitalcommunication using discrete-time chaos synchronizationrdquoChaos Solitons amp Fractals vol 18 no 4 pp 881ndash890 2003

[4] V T Pham S Vaidyanathan C Volos S Jafari andT Kapitaniak ldquoA new multi-stable chaotic hyperjerk systemits special features circuit realization control and synchro-nizationrdquo Archives of Control Sciences vol 30 no 1pp 23ndash45 2020

[5] A Ouannas X Wang V T Pham G Grassi andV V Huynh ldquoSynchronization results for a class of frac-tional-order spatiotemporal partial differential systems basedon fractional lyapunov approachrdquo Boundary Value Problemsvol 2019 no 1 2019

[6] F Yu H Shen L Liu et al ldquoCCII and FPGA realization amultistable modified fourth-order autonomous Chuarsquos

Value of pixel (x y)

0

50

100

150

200

250Va

lue o

f pix

el (x

+ 1

y)

0 50 100 150 200 250

(a)

0 50 100 150 200 250Value of pixel (x y)

0

50

100

150

200

250

Valu

e of p

ixel

(x +

1 y

)(b)

Figure 8 Correlation analyses for the original and encrypted figures in Figure 6 (a) Correlation analysis for the original figure as shown inFigure 6(a) (b) Correlation analysis for the encrypted figure as shown in Figure 6(b)

Complexity 11

chaotic system with coexisting multiple attractorsrdquo Com-plexity vol 2020 Article ID 5212601 17 pages 2020

[7] Q Lai P D K Kuate F Liu and H H Lu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs vol 67 no 6 2019

[8] Q Lai A Akgul C B Li G H Xu and U Cavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and s-box designrdquo Entropy vol 20 no 12018

[9] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[10] R Wang M Li Z Gao and H Sun ldquoA new memristor-based5D chaotic system and circuit implementationrdquo Complexityvol 2018 Article ID 6069401 12 pages 2018

[11] Q Hong Y Li X Wang and Z Zeng ldquoA versatile pulsecontrol method to generate arbitrary multidirection multi-butterfly chaotic attractorsrdquo IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol 38 no 8pp 1480ndash1492 2019

[12] P S Shabestari Z Rostami V T Pham F E Alsaadi andT Hayat ldquoModeling of neurodegenerative diseases usingdiscrete chaotic systemsrdquo Communications in gteoreticalPhysics vol 71 no 10 pp 1241ndash1245 2019

[13] Q Jiang J Jiang and J Li ldquoAn image encryption algorithmbased on high-dimensional chaotic systemsrdquo in Proceedings ofthe IEEE International Conference on Signal ProcessingCommunications and Computing pp 1ndash4 Hong Kong ChinaAugust 2016

[14] S Singh M Ahmad and D Malik ldquoBreaking an imageencryption scheme based on chaotic synchronization phe-nomenonrdquo in Proceedings of the 2016 Ninth InternationalConference on Contemporary Computing (IC3) Noida IndiaAugust 2016

[15] P Chen S Yu B Chen L Xiao and J Lu ldquoDesign and SOPC-based realization of a video chaotic secure communicationschemerdquo International Journal of Bifurcation and Chaosvol 28 no 13 Article ID 1850160 24 pages 2018

[16] P R Sankpal and P A Vijaya ldquoImage encryption Usingchaotic maps a surveyrdquo in Proceedings of the 2014 Fifth In-ternational Conference on Signal and Image Processingpp 102ndash107 IEEE Bangalore India January 2014

[17] A Rani and B Raman ldquoAn image copyright protectionsystem using chaotic mapsrdquo Multimedia Tools and Applica-tions vol 76 no 2 pp 3121ndash3138 2016

[18] N Shyamala and K Anusudha ldquoReversible chaotic encryp-tion techniques for imagesrdquo in Proceedings of the 2017 FourthInternational Conference on Signal Processing Communica-tion and Networking (ICSCN) Chennai India March 2017

[19] S Chen S Yu J Lu G Chen and J He ldquoDesign and FPGA-based realization of a chaotic secure video communicationsystemrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 28 no 9 pp 2359ndash2371 2018

[20] Z Lin S Yu J Lu S Cai and G Chen ldquoDesign and ARM-embedded implementation of a chaotic map-based real-timesecure video communication systemrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 25 no 7pp 1203ndash1216 2015

[21] D Chang Z Li M Wang and Y Zeng ldquoA novel digital pro-grammable multi-scroll chaotic system and its application inFPGA-based audio secure communicationrdquo AEUmdashInternational

Journal of Electronics and Communications vol 88 pp 20ndash292018

[22] F J Farsana and K Gopakumar ldquoPrivate key encryption ofspeech signal based on three dimensional chaotic maprdquo inProceedings of the International Conference on Communica-tion amp Signal Processing pp 2197ndash2201 Chennai India April2017

[23] H Jia Z Guo S Wang and Z Chen ldquoMechanics analysis andhardware implementation of a new 3D chaotic systemrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 13Article ID 1850161 14 pages 2018

[24] E Tlelo-Cuautle L G De La Fraga V-T Pham C VolosS Jafari and A D J Quintas-Valles ldquoDynamics FPGA re-alization and application of a chaotic system with an infinitenumber of equilibrium pointsrdquo Nonlinear Dynamics vol 89no 2 pp 1129ndash1139 2017

[25] A Akgul H Calgan I Koyuncu I Pehlivan andA Istanbullu ldquoChaos-based engineering applications with a3D chaotic system without equilibrium pointsrdquo NonlinearDynamics vol 84 no 2 pp 481ndash495 2016

[26] Q Lai X Zhao K Rajagopal G Xu A Akgul andE Guleryuz ldquoDynamic analyses FPGA implementation andengineering applications of multi-butterfly chaotic attractorsgenerated from generalised sprott c systemrdquo Pramana vol 90no 1 p 12 2018

[27] W S Sayed M F Tolba A G Radwan and S K Abd-El-Hafiz ldquoFPGA realization of a speech encryption system basedon a generalized modified chaotic transition map and bitpermutationrdquo Multimedia Tools and Applications vol 78no 12 pp 16097ndash16127 2019

[28] Q Wang S Yu C Li et al ldquo+eoretical design and FPGA-based implementation of higher-dimensional digital chaoticsystemsrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 63 no 3 pp 401ndash412 2016

[29] E Dong Z Wang Z Chen and Z Wang ldquoTopologicalhorseshoe analysis and field-programmable gate arrayimplementation of a fractional-order four-wing chaoticattractorrdquo Chinese Physics B vol 27 no 1 pp 300ndash306 2018

[30] Z Hua B Zhou and Y Zhou ldquoSine chaotification model forenhancing chaos and its hardware implementationrdquo IEEETransactions on Industrial Electronics vol 66 no 2 pp 1273ndash1284 2019

[31] I Ozturk and R Kılıccedil ldquoA novel method for producing pseudorandom numbers from differential equation-based chaoticsystemsrdquo Nonlinear Dynamics vol 80 no 3 pp 1147ndash11572015

[32] R Wang Q Xie Y Huang H Sun and Y Sun ldquoDesign of aswitched hyperchaotic system and its applicationrdquo Interna-tional Journal of Computer Applications in Technology vol 57no 3 pp 207ndash218 2018

[33] R Wang H Sun J Wang L Wang and Y Wang ldquoAppli-cations of modularized circuit designs in a new hyper-chaoticsystem circuit implementationrdquo Chinese Physics B vol 24no 4 Article ID 020501 2015

[34] Z Lin S Yu and J Li ldquoChosen ciphertext attack on a chaoticstream cipherrdquo in Proceedings of the 30th Chinese Control andDecision Conference pp 1238ndash1242 Shenyang China June2018

[35] G Qi G Chen M A Van Wyk and B J Van Wyk ldquoOn anew hyperchaotic systemrdquo Physics Letters A vol 372 no 2pp 124ndash136 2008

[36] Z Hu and C-K Chan ldquoA real-valued chaotic orthogonalmatrix transform-based encryption for OFDM-PONrdquo IEEE

12 Complexity

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13

8000

6000

4000

2000

0

ndash2000

ndash4000ndash5 0 5 10 15 20 25

b

(a)

ndash80

ndash70

ndash60

ndash50

ndash40

ndash30

ndash20

ndash10

0

10

20

Lyap

unov

expo

nent

t

ndash5 0 5 10 15 20 25b

(b)

ndash8000 ndash6000 ndash4000 ndash2000 0 2000 4000 6000ndash6000

ndash4000

ndash2000

0

2000

4000

6000

x 2

x1

(c)

x 4

ndash6000 ndash4000 ndash2000 0 2000 4000 6000x2

ndash4

ndash3

ndash2

ndash1

0

1

2

3 times104

(d)

Figure 1 Continued

6 Complexity

xQ1(n + 1) minus xQ1(n)1113872 1113873

τ a xQ2(n) minus xQ1(n)) + xQ2(n)xQ3(n)1113872 1113873

xQ2(n + 1) minus xQ2(n)1113872 1113873

τ b xQ1(n) + xQ2(n)) minus xQ1(n)xQ3(n)1113872 1113873

xQ3(n + 1) minus xQ3(n)1113872 1113873

τ minuscxQ3 (n) minus exQ4(n) + xQ1(n)xQ2(n)1113872 1113873

xQ4(n + 1) minus xQ4(n)1113872 1113873

τ minusdxQ4 (n) + fxQ3(n) + xQ1(n)xQ2(n)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

+en the iteration equations of Qi system are shown inxQ1(n + 1) aτxQ2(n) +(1 minus aτ)xQ1(n) + τxQ2(n)xQ3(n)

xQ2(n + 1) bτxQ1(n) +(1 + bτ)xQ2(n) minus τxQ1(n)xQ3(n)

xQ3(n + 1) (1 minus cτ)xQ3(n) minus eτxQ4(n) + τxQ1(n)xQ2(n)

xQ4(n + 1) (1 minus dτ)xQ4(n) + fτxQ3(n) + τxQ1(n)xQ2(n)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)

Similarly the discrete Lorenz system isx(n + 1) minus x(n)

τ σ(y(n) minus x(n))

y(n + 1) minus y(n)

τ rx(n) minus y(n) minus x(n)z(n)

z(n + 1) minus z(n)

τ x(n)y(n) minus βz(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Correspondingly the iteration equations of Lorenz systemare

x3

ndash15 ndash1 ndash05 0 05 1 15 2times104

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

x

(e)

ndash20 ndash10 0 10 20x

0

10

20

30

40

50

z

(f )

Figure 1 Bifurcation diagram along with b variation and partial phase portraits for different initial values (blue line for the first initial valueset and red line for the second initial value set)

Source interface

ZYNQ

HDMI JTAG

Figure 2 FPGA hardware diagram

Semitensor

Video data m (n)

c (n)

p (n)

lfloorze (n + 1)rfloor

lfloorye (n + 1)rfloor

lfloorxe (n + 1)rfloor

lfloorxeQ1 (n + 1)rfloor

lfloorxeQ2 (n + 1)rfloor

lfloorxeQ3 (n + 1)rfloor

lfloorxeQ4 (n + 1)rfloor

lfloorxi (n)rfloor i = 1 2 8

mod (mod (lfloorxi (n)rfloor 2tndash1) 2tndashq)

Figure 3 Block diagram of encryption algorithm

Complexity 7

x(n + 1) στy(n) +(1 minus στ)x(n)

y(n + 1) rτx(n) +(1 minus τ)y(n) minus τx(n)z(n)

z(n + 1) τx(n)y(n) +(1 minus βτ)z(n)

⎧⎪⎪⎨

⎪⎪⎩(18)

In general FPGA can store float data and fixed-pointdata Since fixed-point data require less computing resourcesthan that of float data this paper uses 64-bit fixed-point

number to represent the data+e detailed data format of 64-bit fixed-point numbers is shown in Figure 4

In Figure 4 I represents the integer part of 64-bit fixed-point numbers and f is the fractional part

As mentioned before because of the limitation of bitwidth in FPGA all data are truncated numbers in hardwareimplementation +erefore the Qi and Lorenz systembecomes

lfloorxQ1(n + 1)rfloor aτlfloorxQ2(n)rfloor +(1 minus aτ)lfloorxQ1(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor bτlfloorxQ1(n)rfloor +(1 + bτ)lfloorxQ2(n)rfloor minus τlfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor (1 minus cτ)lfloorxQ3(n)rfloor minus eτlfloorxQ4(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor (1 minus dτ)lfloorxQ4(n)rfloor + fτlfloorxQ3(n)rfloor + τlfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

lfloorx(n + 1)rfloor στlfloory(n)rfloor +(1 minus στ)lfloorx(n)rfloor

lfloory(n + 1)rfloor rτlfloorx(n)rfloor +(1 minus τ)lfloory(n)rfloor minus τlfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor τlfloorx(n)rfloorlfloory(n)rfloor +(1 minus βτ)lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(20)

Let the iteration step be τ 000001 and use the sameparameters in system (12) +en substitute them into (19)

and (20) respectively +erefore Qi system and Lorenzsystem are changed as follows

lfloorxQ1(n + 1)rfloor 09995lfloorxQ2(n)rfloor + 00005lfloorxQ1(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor 00002lfloorxQ1(n)rfloor + 10002lfloorxQ2(n)rfloor minus 000001lfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor 099987lfloorxQ3(n)rfloor minus 000033lfloorxQ4(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor 000008lfloorxQ4(n)rfloor + 09997lfloorxQ3(n)rfloor + 000001lfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)

lfloorx(n + 1)rfloor 00001lfloory(n)rfloor + 09999lfloorx(n)rfloor

lfloory(n + 1)rfloor 000028lfloorx(n)rfloor + 099999lfloory(n)rfloor minus 000001lfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor 000001lfloorx(n)rfloorlfloory(n)rfloor + 09999733lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(22)

To iterate Qi system and Lorenz system and makesemitensor product operation on these two systems aftereach iteration respectively the discretized first 8 statevariables of the new system are obtained

x1(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x2(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x3(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x4(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

x5(n + 1) lfloory(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x6(n + 1) lfloory(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x7(n + 1) lfloory(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x8(n + 1) lfloory(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

where xi(n+ 1) and xi (n) are system state variables lfloory(n +

1)rfloor is the approximate value of y (n+ 1) using fixed-pointnumber

32 Implementation and Analysis of Encryption Algorithm ofthe New Chaotic System In order to ensure the randomnessof the random sequence therefore select the low bits from tto q as shown in Figure 5 +e positions of these bits are notclose to those of sign and exponent bits +en the chosenrandom encryption sequence ci (n) is shown in equation(17) +is paper selects t 1 and q 6

ci (n) modlfloorxi(n)rfloor2tminus 11113888 1113889 2qminust+1

1113888 1113889 i 1 2 8

(24)Random sequences which are selected from eight states

based on the method mentioned previously are combined togenerate the random sequence c (n)

c(n) c1(n) c2(v) c8(n)( 1113857 (25)

In order to resist the differential attack and decrease thecorrelation between adjacent random sequences the paperselects the very first iteration sequence among every N it-erations and stacks these selected sequences to construct a

8 Complexity

random sequence c (n) as shown in equation (18) +is canimprove the randomness of the random sequence

Next the random sequence c (n) conducts XOR op-eration with the divided video data Since a frame video dataincludes tricolor integer sequences R (n) G (n) and B (n)these three sequences will be encrypted simultaneously afterchanging the random sequence c (n) into three columnsevenly c1 (n) c2 (n) and c3 (n)

p1(n) c1(n)oplusR(n)

p2(n) c2(n)oplusG(n)

p3(n) c3(n)oplusB(n)

(26)

where p1 (n) p2 (n) and p3 (n) are encrypted sequences andoplus is an XOR operation One has

1113954R(n) modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplusp1(n)

modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusR(n)

1113954G(n) modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp2(n)rfloor

modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusG(n)

1113954B(n) modlfloor1113954x3(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp3(n)rfloor

mod mod lfloor1113954x3(n)rfloor 2t1113872 1113873 2tminus q

1113872 1113873oplus c3(n)oplusB(n)

(27)

where lfloor1113954xpprime(n)rfloor pprime 1 2 3 are receiver terminal sequences

33 Analysis for NIST Test NIST test is provided by NationalInstitute of Standards and Technology and it is a standard to testthe randomness of a random sequence According to the en-cryption algorithm in this paper c (n) in equation (25) should betested by NIST standard +e comparisons of the random se-quence among the new system Qi system and Lorenz system c(n) cL (n) and cQ (n) are conducted which are obtained fromserial interfaces +e results of the tests are shown in Table 1

As shown in Table 1 all the test results for the randomsequences of the new system meet the NIST test index

standards Partial test results are larger than 08 which meansthese random indexes are quite close to those of the realrandom sequences +e randomness indexes and some othertest results are better than those generated from Lorenz systemand Qi system such as frequency block frequency cumulativesums nonoverlapping template approximate entropy randomexcursions random excursions variant and linear complexity

34 Statistical Analyses Vivado IDE is used to conduct thehardware simulation +e paper also performs the statisticsanalysis for the encrypted video data generated by hardwareFigure 6(a) is one picture of a video before encryptionFigure 6(b) is the encrypted picture of a video

Figure 7 demonstrates the comparisons of statisticshistogram between the original and encrypted pictures

Figure 7 demonstrates the comparisons of statisticshistograms between the original and encrypted pictures Asillustrated in Figure 7(a) the difference of the pixels dis-tribution is obvious However distribution of differentpixels for the encrypted picture shown in Figure 7(b) is theapproximately uniform distribution It can be concludedthat the proposed encryption algorithm for the new systemcan better resist statistic attack effectively

35 Differential Analysis Differential attack is used tomeasure the sensitivity of plaintext change for the encryp-tion algorithm and commonly uses NPCR (Number of PixelsChange Rate) and UACI (Unified Average Changing In-tensity) as indexes defined as follows

NPRC 1113936efD(e f)

W times Htimes 100

UACI 1

W times Htimes 1113944

ef

C(e f) minus Cprime(e f)1113868111386811138681113868

1113868111386811138681113868

255times 100

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

NPRC 1113944m

NPRC (m)

UACI 1113944m

UACI (m)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(29)

where C (e f) is the pixel value before encryption and Cprime (ef ) is the pixel value after encryption If C (e f )Cprime (e f ) D(e f ) 0 else 1 NPRC and UACI calculated by (29) and theproposed system and encryption algorithm are 9960 and1228 for the first-time encryption respectively +ereforethe ability to resist differential attack improves to someextent In video encryption application the requirement forencryption speed is more concerned

36 Correlation Analysis Correlation analysis is used tocheck whether the neighbor pixels are close or not +ispaper analyzes the correlation for Figure 6 +e paper selects5000 random pixels from the original and the encryptedimages and analyzes the correlation among these random-pixel pairs as shown in Figure 8 As Figure 8 illustrates the

ffffffff ffffffff ffffffff ffffffff ffffffff4024

IIIIIIII IIIIIIII IIIIIIII

Figure 4 Data format for 64-bit fixed-point numbers

q t 164 63

hellip hellip hellip

Figure 5 +e schematics of numbered data bits

Complexity 9

Table 1 NIST test results for the random sequences of the new system Lorenz system and Qi system (the number of sequences is 100 andtheir lengths are 1000000)

Statistical test P value of cL (n) P value of cQ (n) P value of c (n) Test results of c (n) ProportionFrequency 0000000 0236810 0474986 radic 100100Block frequency (m 128) 0289667 0699313 0946308 radic 99100Cumulative sums 0000000 0108791 0779188 radic 100100Runs 0000000 0554420 0075719 radic 100100Longest run 0419021 0383827 0289667 radic 100100Rank 0851383 0996335 0289667 radic 100100FFT 0911413 0911413 0213309 radic 99100Nonoverlapping template (m 9) 0000003 0181557 0983453 radic 100100Overlapping template (m 9) 0181557 0935716 0924076 radic 100100Universal 0616305 0289667 0014550 radic 97100Approximate entropy (m 10) 0000000 0798139 0816537 radic 99100Random excursions 0008879 0319084 0739918 radic 6464Random excursions variant 0213309 0289667 0949602 radic 6464Linear complexity (M 500) 0010988 0013569 0108791 radic 99100Serial (m 16) 0616305 0028817 0262249 radic 100100

(a) (b)

(c)

Figure 6+e original and encrypted pictures of a video and the corresponding encrypted video data through FPGA (a)+e original pictureof a video (b) +e encrypted picture of a video (c) Encrypted video data shown on a monitor

0

05

1

15

2

0 50 100 150 200 250

104

(a)

0

05

1

15

2

0 50 100 150 200 250

104

(b)

Figure 7 Histogram between the original and encrypted pictures (a) Histogram of the original picture (b) Histogram of the encrypted picture

10 Complexity

correlation dramatically decreases when comparing twofigures before and after encryption as shown in Figure 6

4 Conclusions

+is paper proposed a new chaotic system generated byusing semitensor product on two chaotic systems and therelated dynamic characteristics are analyzed +e new sys-tem is employed in video encryption as well and the pro-posed method can generate 8 or even 12 state variables whencompared to Qi system and Lorenz system which onlygenerate 7 state variables at most in one iteration period+eproposed method can improve the speed of random se-quence generation +e NIST test results demonstrate thatthe pseudorandomness of new system is better than that ofsingle Qi system and single Lorenz system

+e proposed encryption algorithm based on semitensorproduct can be used in other chaotic systems +e synchro-nization of the new system can be implemented by synchro-nizing two original chaotic systems separately In this paperFPGA is used to implement the generation of the new chaoticsystem and to encrypt video data +e corresponding statisticsand differential and correlation analyses were also conductedwhich demonstrates that the new system has obvious advan-tages such as better random features better resistance todifferential attacks and lower pixel correlation for encryptedimages +e future work will focus on the decryption of videoinformation by the proposed chaotic system generated by thesemitensor product method in hardware

Data Availability

+e figuresrsquo zip data used to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities-Civil Aviation Universityof China (3122019044)

References

[1] J Li J Li X Chen C Jia and W Lou ldquoIdentity-based en-cryption with outsourced revocation in cloud computingrdquoIEEE Transactions on Computers vol 64 no 2 pp 425ndash4372015

[2] J Li Y K Li X Chen P P C Lee and W Lou ldquoA hybridcloud approach for secure authorized deduplicationrdquo IEEETransactions on Parallel and Distributed Systems vol 26no 5 pp 1206ndash1216 2015

[3] M Feki B Robert G Gelle and M Colas ldquoSecure digitalcommunication using discrete-time chaos synchronizationrdquoChaos Solitons amp Fractals vol 18 no 4 pp 881ndash890 2003

[4] V T Pham S Vaidyanathan C Volos S Jafari andT Kapitaniak ldquoA new multi-stable chaotic hyperjerk systemits special features circuit realization control and synchro-nizationrdquo Archives of Control Sciences vol 30 no 1pp 23ndash45 2020

[5] A Ouannas X Wang V T Pham G Grassi andV V Huynh ldquoSynchronization results for a class of frac-tional-order spatiotemporal partial differential systems basedon fractional lyapunov approachrdquo Boundary Value Problemsvol 2019 no 1 2019

[6] F Yu H Shen L Liu et al ldquoCCII and FPGA realization amultistable modified fourth-order autonomous Chuarsquos

Value of pixel (x y)

0

50

100

150

200

250Va

lue o

f pix

el (x

+ 1

y)

0 50 100 150 200 250

(a)

0 50 100 150 200 250Value of pixel (x y)

0

50

100

150

200

250

Valu

e of p

ixel

(x +

1 y

)(b)

Figure 8 Correlation analyses for the original and encrypted figures in Figure 6 (a) Correlation analysis for the original figure as shown inFigure 6(a) (b) Correlation analysis for the encrypted figure as shown in Figure 6(b)

Complexity 11

chaotic system with coexisting multiple attractorsrdquo Com-plexity vol 2020 Article ID 5212601 17 pages 2020

[7] Q Lai P D K Kuate F Liu and H H Lu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs vol 67 no 6 2019

[8] Q Lai A Akgul C B Li G H Xu and U Cavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and s-box designrdquo Entropy vol 20 no 12018

[9] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[10] R Wang M Li Z Gao and H Sun ldquoA new memristor-based5D chaotic system and circuit implementationrdquo Complexityvol 2018 Article ID 6069401 12 pages 2018

[11] Q Hong Y Li X Wang and Z Zeng ldquoA versatile pulsecontrol method to generate arbitrary multidirection multi-butterfly chaotic attractorsrdquo IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol 38 no 8pp 1480ndash1492 2019

[12] P S Shabestari Z Rostami V T Pham F E Alsaadi andT Hayat ldquoModeling of neurodegenerative diseases usingdiscrete chaotic systemsrdquo Communications in gteoreticalPhysics vol 71 no 10 pp 1241ndash1245 2019

[13] Q Jiang J Jiang and J Li ldquoAn image encryption algorithmbased on high-dimensional chaotic systemsrdquo in Proceedings ofthe IEEE International Conference on Signal ProcessingCommunications and Computing pp 1ndash4 Hong Kong ChinaAugust 2016

[14] S Singh M Ahmad and D Malik ldquoBreaking an imageencryption scheme based on chaotic synchronization phe-nomenonrdquo in Proceedings of the 2016 Ninth InternationalConference on Contemporary Computing (IC3) Noida IndiaAugust 2016

[15] P Chen S Yu B Chen L Xiao and J Lu ldquoDesign and SOPC-based realization of a video chaotic secure communicationschemerdquo International Journal of Bifurcation and Chaosvol 28 no 13 Article ID 1850160 24 pages 2018

[16] P R Sankpal and P A Vijaya ldquoImage encryption Usingchaotic maps a surveyrdquo in Proceedings of the 2014 Fifth In-ternational Conference on Signal and Image Processingpp 102ndash107 IEEE Bangalore India January 2014

[17] A Rani and B Raman ldquoAn image copyright protectionsystem using chaotic mapsrdquo Multimedia Tools and Applica-tions vol 76 no 2 pp 3121ndash3138 2016

[18] N Shyamala and K Anusudha ldquoReversible chaotic encryp-tion techniques for imagesrdquo in Proceedings of the 2017 FourthInternational Conference on Signal Processing Communica-tion and Networking (ICSCN) Chennai India March 2017

[19] S Chen S Yu J Lu G Chen and J He ldquoDesign and FPGA-based realization of a chaotic secure video communicationsystemrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 28 no 9 pp 2359ndash2371 2018

[20] Z Lin S Yu J Lu S Cai and G Chen ldquoDesign and ARM-embedded implementation of a chaotic map-based real-timesecure video communication systemrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 25 no 7pp 1203ndash1216 2015

[21] D Chang Z Li M Wang and Y Zeng ldquoA novel digital pro-grammable multi-scroll chaotic system and its application inFPGA-based audio secure communicationrdquo AEUmdashInternational

Journal of Electronics and Communications vol 88 pp 20ndash292018

[22] F J Farsana and K Gopakumar ldquoPrivate key encryption ofspeech signal based on three dimensional chaotic maprdquo inProceedings of the International Conference on Communica-tion amp Signal Processing pp 2197ndash2201 Chennai India April2017

[23] H Jia Z Guo S Wang and Z Chen ldquoMechanics analysis andhardware implementation of a new 3D chaotic systemrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 13Article ID 1850161 14 pages 2018

[24] E Tlelo-Cuautle L G De La Fraga V-T Pham C VolosS Jafari and A D J Quintas-Valles ldquoDynamics FPGA re-alization and application of a chaotic system with an infinitenumber of equilibrium pointsrdquo Nonlinear Dynamics vol 89no 2 pp 1129ndash1139 2017

[25] A Akgul H Calgan I Koyuncu I Pehlivan andA Istanbullu ldquoChaos-based engineering applications with a3D chaotic system without equilibrium pointsrdquo NonlinearDynamics vol 84 no 2 pp 481ndash495 2016

[26] Q Lai X Zhao K Rajagopal G Xu A Akgul andE Guleryuz ldquoDynamic analyses FPGA implementation andengineering applications of multi-butterfly chaotic attractorsgenerated from generalised sprott c systemrdquo Pramana vol 90no 1 p 12 2018

[27] W S Sayed M F Tolba A G Radwan and S K Abd-El-Hafiz ldquoFPGA realization of a speech encryption system basedon a generalized modified chaotic transition map and bitpermutationrdquo Multimedia Tools and Applications vol 78no 12 pp 16097ndash16127 2019

[28] Q Wang S Yu C Li et al ldquo+eoretical design and FPGA-based implementation of higher-dimensional digital chaoticsystemsrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 63 no 3 pp 401ndash412 2016

[29] E Dong Z Wang Z Chen and Z Wang ldquoTopologicalhorseshoe analysis and field-programmable gate arrayimplementation of a fractional-order four-wing chaoticattractorrdquo Chinese Physics B vol 27 no 1 pp 300ndash306 2018

[30] Z Hua B Zhou and Y Zhou ldquoSine chaotification model forenhancing chaos and its hardware implementationrdquo IEEETransactions on Industrial Electronics vol 66 no 2 pp 1273ndash1284 2019

[31] I Ozturk and R Kılıccedil ldquoA novel method for producing pseudorandom numbers from differential equation-based chaoticsystemsrdquo Nonlinear Dynamics vol 80 no 3 pp 1147ndash11572015

[32] R Wang Q Xie Y Huang H Sun and Y Sun ldquoDesign of aswitched hyperchaotic system and its applicationrdquo Interna-tional Journal of Computer Applications in Technology vol 57no 3 pp 207ndash218 2018

[33] R Wang H Sun J Wang L Wang and Y Wang ldquoAppli-cations of modularized circuit designs in a new hyper-chaoticsystem circuit implementationrdquo Chinese Physics B vol 24no 4 Article ID 020501 2015

[34] Z Lin S Yu and J Li ldquoChosen ciphertext attack on a chaoticstream cipherrdquo in Proceedings of the 30th Chinese Control andDecision Conference pp 1238ndash1242 Shenyang China June2018

[35] G Qi G Chen M A Van Wyk and B J Van Wyk ldquoOn anew hyperchaotic systemrdquo Physics Letters A vol 372 no 2pp 124ndash136 2008

[36] Z Hu and C-K Chan ldquoA real-valued chaotic orthogonalmatrix transform-based encryption for OFDM-PONrdquo IEEE

12 Complexity

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13

xQ1(n + 1) minus xQ1(n)1113872 1113873

τ a xQ2(n) minus xQ1(n)) + xQ2(n)xQ3(n)1113872 1113873

xQ2(n + 1) minus xQ2(n)1113872 1113873

τ b xQ1(n) + xQ2(n)) minus xQ1(n)xQ3(n)1113872 1113873

xQ3(n + 1) minus xQ3(n)1113872 1113873

τ minuscxQ3 (n) minus exQ4(n) + xQ1(n)xQ2(n)1113872 1113873

xQ4(n + 1) minus xQ4(n)1113872 1113873

τ minusdxQ4 (n) + fxQ3(n) + xQ1(n)xQ2(n)1113872 1113873

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

+en the iteration equations of Qi system are shown inxQ1(n + 1) aτxQ2(n) +(1 minus aτ)xQ1(n) + τxQ2(n)xQ3(n)

xQ2(n + 1) bτxQ1(n) +(1 + bτ)xQ2(n) minus τxQ1(n)xQ3(n)

xQ3(n + 1) (1 minus cτ)xQ3(n) minus eτxQ4(n) + τxQ1(n)xQ2(n)

xQ4(n + 1) (1 minus dτ)xQ4(n) + fτxQ3(n) + τxQ1(n)xQ2(n)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)

Similarly the discrete Lorenz system isx(n + 1) minus x(n)

τ σ(y(n) minus x(n))

y(n + 1) minus y(n)

τ rx(n) minus y(n) minus x(n)z(n)

z(n + 1) minus z(n)

τ x(n)y(n) minus βz(n)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

Correspondingly the iteration equations of Lorenz systemare

x3

ndash15 ndash1 ndash05 0 05 1 15 2times104

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

x

(e)

ndash20 ndash10 0 10 20x

0

10

20

30

40

50

z

(f )

Figure 1 Bifurcation diagram along with b variation and partial phase portraits for different initial values (blue line for the first initial valueset and red line for the second initial value set)

Source interface

ZYNQ

HDMI JTAG

Figure 2 FPGA hardware diagram

Semitensor

Video data m (n)

c (n)

p (n)

lfloorze (n + 1)rfloor

lfloorye (n + 1)rfloor

lfloorxe (n + 1)rfloor

lfloorxeQ1 (n + 1)rfloor

lfloorxeQ2 (n + 1)rfloor

lfloorxeQ3 (n + 1)rfloor

lfloorxeQ4 (n + 1)rfloor

lfloorxi (n)rfloor i = 1 2 8

mod (mod (lfloorxi (n)rfloor 2tndash1) 2tndashq)

Figure 3 Block diagram of encryption algorithm

Complexity 7

x(n + 1) στy(n) +(1 minus στ)x(n)

y(n + 1) rτx(n) +(1 minus τ)y(n) minus τx(n)z(n)

z(n + 1) τx(n)y(n) +(1 minus βτ)z(n)

⎧⎪⎪⎨

⎪⎪⎩(18)

In general FPGA can store float data and fixed-pointdata Since fixed-point data require less computing resourcesthan that of float data this paper uses 64-bit fixed-point

number to represent the data+e detailed data format of 64-bit fixed-point numbers is shown in Figure 4

In Figure 4 I represents the integer part of 64-bit fixed-point numbers and f is the fractional part

As mentioned before because of the limitation of bitwidth in FPGA all data are truncated numbers in hardwareimplementation +erefore the Qi and Lorenz systembecomes

lfloorxQ1(n + 1)rfloor aτlfloorxQ2(n)rfloor +(1 minus aτ)lfloorxQ1(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor bτlfloorxQ1(n)rfloor +(1 + bτ)lfloorxQ2(n)rfloor minus τlfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor (1 minus cτ)lfloorxQ3(n)rfloor minus eτlfloorxQ4(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor (1 minus dτ)lfloorxQ4(n)rfloor + fτlfloorxQ3(n)rfloor + τlfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

lfloorx(n + 1)rfloor στlfloory(n)rfloor +(1 minus στ)lfloorx(n)rfloor

lfloory(n + 1)rfloor rτlfloorx(n)rfloor +(1 minus τ)lfloory(n)rfloor minus τlfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor τlfloorx(n)rfloorlfloory(n)rfloor +(1 minus βτ)lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(20)

Let the iteration step be τ 000001 and use the sameparameters in system (12) +en substitute them into (19)

and (20) respectively +erefore Qi system and Lorenzsystem are changed as follows

lfloorxQ1(n + 1)rfloor 09995lfloorxQ2(n)rfloor + 00005lfloorxQ1(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor 00002lfloorxQ1(n)rfloor + 10002lfloorxQ2(n)rfloor minus 000001lfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor 099987lfloorxQ3(n)rfloor minus 000033lfloorxQ4(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor 000008lfloorxQ4(n)rfloor + 09997lfloorxQ3(n)rfloor + 000001lfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)

lfloorx(n + 1)rfloor 00001lfloory(n)rfloor + 09999lfloorx(n)rfloor

lfloory(n + 1)rfloor 000028lfloorx(n)rfloor + 099999lfloory(n)rfloor minus 000001lfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor 000001lfloorx(n)rfloorlfloory(n)rfloor + 09999733lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(22)

To iterate Qi system and Lorenz system and makesemitensor product operation on these two systems aftereach iteration respectively the discretized first 8 statevariables of the new system are obtained

x1(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x2(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x3(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x4(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

x5(n + 1) lfloory(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x6(n + 1) lfloory(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x7(n + 1) lfloory(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x8(n + 1) lfloory(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

where xi(n+ 1) and xi (n) are system state variables lfloory(n +

1)rfloor is the approximate value of y (n+ 1) using fixed-pointnumber

32 Implementation and Analysis of Encryption Algorithm ofthe New Chaotic System In order to ensure the randomnessof the random sequence therefore select the low bits from tto q as shown in Figure 5 +e positions of these bits are notclose to those of sign and exponent bits +en the chosenrandom encryption sequence ci (n) is shown in equation(17) +is paper selects t 1 and q 6

ci (n) modlfloorxi(n)rfloor2tminus 11113888 1113889 2qminust+1

1113888 1113889 i 1 2 8

(24)Random sequences which are selected from eight states

based on the method mentioned previously are combined togenerate the random sequence c (n)

c(n) c1(n) c2(v) c8(n)( 1113857 (25)

In order to resist the differential attack and decrease thecorrelation between adjacent random sequences the paperselects the very first iteration sequence among every N it-erations and stacks these selected sequences to construct a

8 Complexity

random sequence c (n) as shown in equation (18) +is canimprove the randomness of the random sequence

Next the random sequence c (n) conducts XOR op-eration with the divided video data Since a frame video dataincludes tricolor integer sequences R (n) G (n) and B (n)these three sequences will be encrypted simultaneously afterchanging the random sequence c (n) into three columnsevenly c1 (n) c2 (n) and c3 (n)

p1(n) c1(n)oplusR(n)

p2(n) c2(n)oplusG(n)

p3(n) c3(n)oplusB(n)

(26)

where p1 (n) p2 (n) and p3 (n) are encrypted sequences andoplus is an XOR operation One has

1113954R(n) modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplusp1(n)

modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusR(n)

1113954G(n) modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp2(n)rfloor

modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusG(n)

1113954B(n) modlfloor1113954x3(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp3(n)rfloor

mod mod lfloor1113954x3(n)rfloor 2t1113872 1113873 2tminus q

1113872 1113873oplus c3(n)oplusB(n)

(27)

where lfloor1113954xpprime(n)rfloor pprime 1 2 3 are receiver terminal sequences

33 Analysis for NIST Test NIST test is provided by NationalInstitute of Standards and Technology and it is a standard to testthe randomness of a random sequence According to the en-cryption algorithm in this paper c (n) in equation (25) should betested by NIST standard +e comparisons of the random se-quence among the new system Qi system and Lorenz system c(n) cL (n) and cQ (n) are conducted which are obtained fromserial interfaces +e results of the tests are shown in Table 1

As shown in Table 1 all the test results for the randomsequences of the new system meet the NIST test index

standards Partial test results are larger than 08 which meansthese random indexes are quite close to those of the realrandom sequences +e randomness indexes and some othertest results are better than those generated from Lorenz systemand Qi system such as frequency block frequency cumulativesums nonoverlapping template approximate entropy randomexcursions random excursions variant and linear complexity

34 Statistical Analyses Vivado IDE is used to conduct thehardware simulation +e paper also performs the statisticsanalysis for the encrypted video data generated by hardwareFigure 6(a) is one picture of a video before encryptionFigure 6(b) is the encrypted picture of a video

Figure 7 demonstrates the comparisons of statisticshistogram between the original and encrypted pictures

Figure 7 demonstrates the comparisons of statisticshistograms between the original and encrypted pictures Asillustrated in Figure 7(a) the difference of the pixels dis-tribution is obvious However distribution of differentpixels for the encrypted picture shown in Figure 7(b) is theapproximately uniform distribution It can be concludedthat the proposed encryption algorithm for the new systemcan better resist statistic attack effectively

35 Differential Analysis Differential attack is used tomeasure the sensitivity of plaintext change for the encryp-tion algorithm and commonly uses NPCR (Number of PixelsChange Rate) and UACI (Unified Average Changing In-tensity) as indexes defined as follows

NPRC 1113936efD(e f)

W times Htimes 100

UACI 1

W times Htimes 1113944

ef

C(e f) minus Cprime(e f)1113868111386811138681113868

1113868111386811138681113868

255times 100

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

NPRC 1113944m

NPRC (m)

UACI 1113944m

UACI (m)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(29)

where C (e f) is the pixel value before encryption and Cprime (ef ) is the pixel value after encryption If C (e f )Cprime (e f ) D(e f ) 0 else 1 NPRC and UACI calculated by (29) and theproposed system and encryption algorithm are 9960 and1228 for the first-time encryption respectively +ereforethe ability to resist differential attack improves to someextent In video encryption application the requirement forencryption speed is more concerned

36 Correlation Analysis Correlation analysis is used tocheck whether the neighbor pixels are close or not +ispaper analyzes the correlation for Figure 6 +e paper selects5000 random pixels from the original and the encryptedimages and analyzes the correlation among these random-pixel pairs as shown in Figure 8 As Figure 8 illustrates the

ffffffff ffffffff ffffffff ffffffff ffffffff4024

IIIIIIII IIIIIIII IIIIIIII

Figure 4 Data format for 64-bit fixed-point numbers

q t 164 63

hellip hellip hellip

Figure 5 +e schematics of numbered data bits

Complexity 9

Table 1 NIST test results for the random sequences of the new system Lorenz system and Qi system (the number of sequences is 100 andtheir lengths are 1000000)

Statistical test P value of cL (n) P value of cQ (n) P value of c (n) Test results of c (n) ProportionFrequency 0000000 0236810 0474986 radic 100100Block frequency (m 128) 0289667 0699313 0946308 radic 99100Cumulative sums 0000000 0108791 0779188 radic 100100Runs 0000000 0554420 0075719 radic 100100Longest run 0419021 0383827 0289667 radic 100100Rank 0851383 0996335 0289667 radic 100100FFT 0911413 0911413 0213309 radic 99100Nonoverlapping template (m 9) 0000003 0181557 0983453 radic 100100Overlapping template (m 9) 0181557 0935716 0924076 radic 100100Universal 0616305 0289667 0014550 radic 97100Approximate entropy (m 10) 0000000 0798139 0816537 radic 99100Random excursions 0008879 0319084 0739918 radic 6464Random excursions variant 0213309 0289667 0949602 radic 6464Linear complexity (M 500) 0010988 0013569 0108791 radic 99100Serial (m 16) 0616305 0028817 0262249 radic 100100

(a) (b)

(c)

Figure 6+e original and encrypted pictures of a video and the corresponding encrypted video data through FPGA (a)+e original pictureof a video (b) +e encrypted picture of a video (c) Encrypted video data shown on a monitor

0

05

1

15

2

0 50 100 150 200 250

104

(a)

0

05

1

15

2

0 50 100 150 200 250

104

(b)

Figure 7 Histogram between the original and encrypted pictures (a) Histogram of the original picture (b) Histogram of the encrypted picture

10 Complexity

correlation dramatically decreases when comparing twofigures before and after encryption as shown in Figure 6

4 Conclusions

+is paper proposed a new chaotic system generated byusing semitensor product on two chaotic systems and therelated dynamic characteristics are analyzed +e new sys-tem is employed in video encryption as well and the pro-posed method can generate 8 or even 12 state variables whencompared to Qi system and Lorenz system which onlygenerate 7 state variables at most in one iteration period+eproposed method can improve the speed of random se-quence generation +e NIST test results demonstrate thatthe pseudorandomness of new system is better than that ofsingle Qi system and single Lorenz system

+e proposed encryption algorithm based on semitensorproduct can be used in other chaotic systems +e synchro-nization of the new system can be implemented by synchro-nizing two original chaotic systems separately In this paperFPGA is used to implement the generation of the new chaoticsystem and to encrypt video data +e corresponding statisticsand differential and correlation analyses were also conductedwhich demonstrates that the new system has obvious advan-tages such as better random features better resistance todifferential attacks and lower pixel correlation for encryptedimages +e future work will focus on the decryption of videoinformation by the proposed chaotic system generated by thesemitensor product method in hardware

Data Availability

+e figuresrsquo zip data used to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities-Civil Aviation Universityof China (3122019044)

References

[1] J Li J Li X Chen C Jia and W Lou ldquoIdentity-based en-cryption with outsourced revocation in cloud computingrdquoIEEE Transactions on Computers vol 64 no 2 pp 425ndash4372015

[2] J Li Y K Li X Chen P P C Lee and W Lou ldquoA hybridcloud approach for secure authorized deduplicationrdquo IEEETransactions on Parallel and Distributed Systems vol 26no 5 pp 1206ndash1216 2015

[3] M Feki B Robert G Gelle and M Colas ldquoSecure digitalcommunication using discrete-time chaos synchronizationrdquoChaos Solitons amp Fractals vol 18 no 4 pp 881ndash890 2003

[4] V T Pham S Vaidyanathan C Volos S Jafari andT Kapitaniak ldquoA new multi-stable chaotic hyperjerk systemits special features circuit realization control and synchro-nizationrdquo Archives of Control Sciences vol 30 no 1pp 23ndash45 2020

[5] A Ouannas X Wang V T Pham G Grassi andV V Huynh ldquoSynchronization results for a class of frac-tional-order spatiotemporal partial differential systems basedon fractional lyapunov approachrdquo Boundary Value Problemsvol 2019 no 1 2019

[6] F Yu H Shen L Liu et al ldquoCCII and FPGA realization amultistable modified fourth-order autonomous Chuarsquos

Value of pixel (x y)

0

50

100

150

200

250Va

lue o

f pix

el (x

+ 1

y)

0 50 100 150 200 250

(a)

0 50 100 150 200 250Value of pixel (x y)

0

50

100

150

200

250

Valu

e of p

ixel

(x +

1 y

)(b)

Figure 8 Correlation analyses for the original and encrypted figures in Figure 6 (a) Correlation analysis for the original figure as shown inFigure 6(a) (b) Correlation analysis for the encrypted figure as shown in Figure 6(b)

Complexity 11

chaotic system with coexisting multiple attractorsrdquo Com-plexity vol 2020 Article ID 5212601 17 pages 2020

[7] Q Lai P D K Kuate F Liu and H H Lu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs vol 67 no 6 2019

[8] Q Lai A Akgul C B Li G H Xu and U Cavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and s-box designrdquo Entropy vol 20 no 12018

[9] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[10] R Wang M Li Z Gao and H Sun ldquoA new memristor-based5D chaotic system and circuit implementationrdquo Complexityvol 2018 Article ID 6069401 12 pages 2018

[11] Q Hong Y Li X Wang and Z Zeng ldquoA versatile pulsecontrol method to generate arbitrary multidirection multi-butterfly chaotic attractorsrdquo IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol 38 no 8pp 1480ndash1492 2019

[12] P S Shabestari Z Rostami V T Pham F E Alsaadi andT Hayat ldquoModeling of neurodegenerative diseases usingdiscrete chaotic systemsrdquo Communications in gteoreticalPhysics vol 71 no 10 pp 1241ndash1245 2019

[13] Q Jiang J Jiang and J Li ldquoAn image encryption algorithmbased on high-dimensional chaotic systemsrdquo in Proceedings ofthe IEEE International Conference on Signal ProcessingCommunications and Computing pp 1ndash4 Hong Kong ChinaAugust 2016

[14] S Singh M Ahmad and D Malik ldquoBreaking an imageencryption scheme based on chaotic synchronization phe-nomenonrdquo in Proceedings of the 2016 Ninth InternationalConference on Contemporary Computing (IC3) Noida IndiaAugust 2016

[15] P Chen S Yu B Chen L Xiao and J Lu ldquoDesign and SOPC-based realization of a video chaotic secure communicationschemerdquo International Journal of Bifurcation and Chaosvol 28 no 13 Article ID 1850160 24 pages 2018

[16] P R Sankpal and P A Vijaya ldquoImage encryption Usingchaotic maps a surveyrdquo in Proceedings of the 2014 Fifth In-ternational Conference on Signal and Image Processingpp 102ndash107 IEEE Bangalore India January 2014

[17] A Rani and B Raman ldquoAn image copyright protectionsystem using chaotic mapsrdquo Multimedia Tools and Applica-tions vol 76 no 2 pp 3121ndash3138 2016

[18] N Shyamala and K Anusudha ldquoReversible chaotic encryp-tion techniques for imagesrdquo in Proceedings of the 2017 FourthInternational Conference on Signal Processing Communica-tion and Networking (ICSCN) Chennai India March 2017

[19] S Chen S Yu J Lu G Chen and J He ldquoDesign and FPGA-based realization of a chaotic secure video communicationsystemrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 28 no 9 pp 2359ndash2371 2018

[20] Z Lin S Yu J Lu S Cai and G Chen ldquoDesign and ARM-embedded implementation of a chaotic map-based real-timesecure video communication systemrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 25 no 7pp 1203ndash1216 2015

[21] D Chang Z Li M Wang and Y Zeng ldquoA novel digital pro-grammable multi-scroll chaotic system and its application inFPGA-based audio secure communicationrdquo AEUmdashInternational

Journal of Electronics and Communications vol 88 pp 20ndash292018

[22] F J Farsana and K Gopakumar ldquoPrivate key encryption ofspeech signal based on three dimensional chaotic maprdquo inProceedings of the International Conference on Communica-tion amp Signal Processing pp 2197ndash2201 Chennai India April2017

[23] H Jia Z Guo S Wang and Z Chen ldquoMechanics analysis andhardware implementation of a new 3D chaotic systemrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 13Article ID 1850161 14 pages 2018

[24] E Tlelo-Cuautle L G De La Fraga V-T Pham C VolosS Jafari and A D J Quintas-Valles ldquoDynamics FPGA re-alization and application of a chaotic system with an infinitenumber of equilibrium pointsrdquo Nonlinear Dynamics vol 89no 2 pp 1129ndash1139 2017

[25] A Akgul H Calgan I Koyuncu I Pehlivan andA Istanbullu ldquoChaos-based engineering applications with a3D chaotic system without equilibrium pointsrdquo NonlinearDynamics vol 84 no 2 pp 481ndash495 2016

[26] Q Lai X Zhao K Rajagopal G Xu A Akgul andE Guleryuz ldquoDynamic analyses FPGA implementation andengineering applications of multi-butterfly chaotic attractorsgenerated from generalised sprott c systemrdquo Pramana vol 90no 1 p 12 2018

[27] W S Sayed M F Tolba A G Radwan and S K Abd-El-Hafiz ldquoFPGA realization of a speech encryption system basedon a generalized modified chaotic transition map and bitpermutationrdquo Multimedia Tools and Applications vol 78no 12 pp 16097ndash16127 2019

[28] Q Wang S Yu C Li et al ldquo+eoretical design and FPGA-based implementation of higher-dimensional digital chaoticsystemsrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 63 no 3 pp 401ndash412 2016

[29] E Dong Z Wang Z Chen and Z Wang ldquoTopologicalhorseshoe analysis and field-programmable gate arrayimplementation of a fractional-order four-wing chaoticattractorrdquo Chinese Physics B vol 27 no 1 pp 300ndash306 2018

[30] Z Hua B Zhou and Y Zhou ldquoSine chaotification model forenhancing chaos and its hardware implementationrdquo IEEETransactions on Industrial Electronics vol 66 no 2 pp 1273ndash1284 2019

[31] I Ozturk and R Kılıccedil ldquoA novel method for producing pseudorandom numbers from differential equation-based chaoticsystemsrdquo Nonlinear Dynamics vol 80 no 3 pp 1147ndash11572015

[32] R Wang Q Xie Y Huang H Sun and Y Sun ldquoDesign of aswitched hyperchaotic system and its applicationrdquo Interna-tional Journal of Computer Applications in Technology vol 57no 3 pp 207ndash218 2018

[33] R Wang H Sun J Wang L Wang and Y Wang ldquoAppli-cations of modularized circuit designs in a new hyper-chaoticsystem circuit implementationrdquo Chinese Physics B vol 24no 4 Article ID 020501 2015

[34] Z Lin S Yu and J Li ldquoChosen ciphertext attack on a chaoticstream cipherrdquo in Proceedings of the 30th Chinese Control andDecision Conference pp 1238ndash1242 Shenyang China June2018

[35] G Qi G Chen M A Van Wyk and B J Van Wyk ldquoOn anew hyperchaotic systemrdquo Physics Letters A vol 372 no 2pp 124ndash136 2008

[36] Z Hu and C-K Chan ldquoA real-valued chaotic orthogonalmatrix transform-based encryption for OFDM-PONrdquo IEEE

12 Complexity

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13

x(n + 1) στy(n) +(1 minus στ)x(n)

y(n + 1) rτx(n) +(1 minus τ)y(n) minus τx(n)z(n)

z(n + 1) τx(n)y(n) +(1 minus βτ)z(n)

⎧⎪⎪⎨

⎪⎪⎩(18)

In general FPGA can store float data and fixed-pointdata Since fixed-point data require less computing resourcesthan that of float data this paper uses 64-bit fixed-point

number to represent the data+e detailed data format of 64-bit fixed-point numbers is shown in Figure 4

In Figure 4 I represents the integer part of 64-bit fixed-point numbers and f is the fractional part

As mentioned before because of the limitation of bitwidth in FPGA all data are truncated numbers in hardwareimplementation +erefore the Qi and Lorenz systembecomes

lfloorxQ1(n + 1)rfloor aτlfloorxQ2(n)rfloor +(1 minus aτ)lfloorxQ1(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor bτlfloorxQ1(n)rfloor +(1 + bτ)lfloorxQ2(n)rfloor minus τlfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor (1 minus cτ)lfloorxQ3(n)rfloor minus eτlfloorxQ4(n)rfloor + τlfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor (1 minus dτ)lfloorxQ4(n)rfloor + fτlfloorxQ3(n)rfloor + τlfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(19)

lfloorx(n + 1)rfloor στlfloory(n)rfloor +(1 minus στ)lfloorx(n)rfloor

lfloory(n + 1)rfloor rτlfloorx(n)rfloor +(1 minus τ)lfloory(n)rfloor minus τlfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor τlfloorx(n)rfloorlfloory(n)rfloor +(1 minus βτ)lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(20)

Let the iteration step be τ 000001 and use the sameparameters in system (12) +en substitute them into (19)

and (20) respectively +erefore Qi system and Lorenzsystem are changed as follows

lfloorxQ1(n + 1)rfloor 09995lfloorxQ2(n)rfloor + 00005lfloorxQ1(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ2(n + 1)rfloor 00002lfloorxQ1(n)rfloor + 10002lfloorxQ2(n)rfloor minus 000001lfloorxQ1(n)rfloorlfloorxQ3(n)rfloor

lfloorxQ3(n + 1)rfloor 099987lfloorxQ3(n)rfloor minus 000033lfloorxQ4(n)rfloor + 000001lfloorxQ2(n)rfloorlfloorxQ1(n)rfloor

lfloorxQ4(n + 1)rfloor 000008lfloorxQ4(n)rfloor + 09997lfloorxQ3(n)rfloor + 000001lfloorxQ1(n)rfloorlfloorxQ2(n)rfloor

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(21)

lfloorx(n + 1)rfloor 00001lfloory(n)rfloor + 09999lfloorx(n)rfloor

lfloory(n + 1)rfloor 000028lfloorx(n)rfloor + 099999lfloory(n)rfloor minus 000001lfloorx(n)rfloorlfloorz(n)rfloor

lfloorz(n + 1)rfloor 000001lfloorx(n)rfloorlfloory(n)rfloor + 09999733lfloorz(n)rfloor

⎧⎪⎪⎨

⎪⎪⎩(22)

To iterate Qi system and Lorenz system and makesemitensor product operation on these two systems aftereach iteration respectively the discretized first 8 statevariables of the new system are obtained

x1(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x2(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x3(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x4(n + 1) lfloorx(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

x5(n + 1) lfloory(n + 1)rfloor timeslfloorxQ1(n + 1)rfloor

x6(n + 1) lfloory(n + 1)rfloor timeslfloorxQ2(n + 1)rfloor

x7(n + 1) lfloory(n + 1)rfloor timeslfloorxQ3(n + 1)rfloor

x8(n + 1) lfloory(n + 1)rfloor timeslfloorxQ4(n + 1)rfloor

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

where xi(n+ 1) and xi (n) are system state variables lfloory(n +

1)rfloor is the approximate value of y (n+ 1) using fixed-pointnumber

32 Implementation and Analysis of Encryption Algorithm ofthe New Chaotic System In order to ensure the randomnessof the random sequence therefore select the low bits from tto q as shown in Figure 5 +e positions of these bits are notclose to those of sign and exponent bits +en the chosenrandom encryption sequence ci (n) is shown in equation(17) +is paper selects t 1 and q 6

ci (n) modlfloorxi(n)rfloor2tminus 11113888 1113889 2qminust+1

1113888 1113889 i 1 2 8

(24)Random sequences which are selected from eight states

based on the method mentioned previously are combined togenerate the random sequence c (n)

c(n) c1(n) c2(v) c8(n)( 1113857 (25)

In order to resist the differential attack and decrease thecorrelation between adjacent random sequences the paperselects the very first iteration sequence among every N it-erations and stacks these selected sequences to construct a

8 Complexity

random sequence c (n) as shown in equation (18) +is canimprove the randomness of the random sequence

Next the random sequence c (n) conducts XOR op-eration with the divided video data Since a frame video dataincludes tricolor integer sequences R (n) G (n) and B (n)these three sequences will be encrypted simultaneously afterchanging the random sequence c (n) into three columnsevenly c1 (n) c2 (n) and c3 (n)

p1(n) c1(n)oplusR(n)

p2(n) c2(n)oplusG(n)

p3(n) c3(n)oplusB(n)

(26)

where p1 (n) p2 (n) and p3 (n) are encrypted sequences andoplus is an XOR operation One has

1113954R(n) modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplusp1(n)

modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusR(n)

1113954G(n) modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp2(n)rfloor

modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusG(n)

1113954B(n) modlfloor1113954x3(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp3(n)rfloor

mod mod lfloor1113954x3(n)rfloor 2t1113872 1113873 2tminus q

1113872 1113873oplus c3(n)oplusB(n)

(27)

where lfloor1113954xpprime(n)rfloor pprime 1 2 3 are receiver terminal sequences

33 Analysis for NIST Test NIST test is provided by NationalInstitute of Standards and Technology and it is a standard to testthe randomness of a random sequence According to the en-cryption algorithm in this paper c (n) in equation (25) should betested by NIST standard +e comparisons of the random se-quence among the new system Qi system and Lorenz system c(n) cL (n) and cQ (n) are conducted which are obtained fromserial interfaces +e results of the tests are shown in Table 1

As shown in Table 1 all the test results for the randomsequences of the new system meet the NIST test index

standards Partial test results are larger than 08 which meansthese random indexes are quite close to those of the realrandom sequences +e randomness indexes and some othertest results are better than those generated from Lorenz systemand Qi system such as frequency block frequency cumulativesums nonoverlapping template approximate entropy randomexcursions random excursions variant and linear complexity

34 Statistical Analyses Vivado IDE is used to conduct thehardware simulation +e paper also performs the statisticsanalysis for the encrypted video data generated by hardwareFigure 6(a) is one picture of a video before encryptionFigure 6(b) is the encrypted picture of a video

Figure 7 demonstrates the comparisons of statisticshistogram between the original and encrypted pictures

Figure 7 demonstrates the comparisons of statisticshistograms between the original and encrypted pictures Asillustrated in Figure 7(a) the difference of the pixels dis-tribution is obvious However distribution of differentpixels for the encrypted picture shown in Figure 7(b) is theapproximately uniform distribution It can be concludedthat the proposed encryption algorithm for the new systemcan better resist statistic attack effectively

35 Differential Analysis Differential attack is used tomeasure the sensitivity of plaintext change for the encryp-tion algorithm and commonly uses NPCR (Number of PixelsChange Rate) and UACI (Unified Average Changing In-tensity) as indexes defined as follows

NPRC 1113936efD(e f)

W times Htimes 100

UACI 1

W times Htimes 1113944

ef

C(e f) minus Cprime(e f)1113868111386811138681113868

1113868111386811138681113868

255times 100

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

NPRC 1113944m

NPRC (m)

UACI 1113944m

UACI (m)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(29)

where C (e f) is the pixel value before encryption and Cprime (ef ) is the pixel value after encryption If C (e f )Cprime (e f ) D(e f ) 0 else 1 NPRC and UACI calculated by (29) and theproposed system and encryption algorithm are 9960 and1228 for the first-time encryption respectively +ereforethe ability to resist differential attack improves to someextent In video encryption application the requirement forencryption speed is more concerned

36 Correlation Analysis Correlation analysis is used tocheck whether the neighbor pixels are close or not +ispaper analyzes the correlation for Figure 6 +e paper selects5000 random pixels from the original and the encryptedimages and analyzes the correlation among these random-pixel pairs as shown in Figure 8 As Figure 8 illustrates the

ffffffff ffffffff ffffffff ffffffff ffffffff4024

IIIIIIII IIIIIIII IIIIIIII

Figure 4 Data format for 64-bit fixed-point numbers

q t 164 63

hellip hellip hellip

Figure 5 +e schematics of numbered data bits

Complexity 9

Table 1 NIST test results for the random sequences of the new system Lorenz system and Qi system (the number of sequences is 100 andtheir lengths are 1000000)

Statistical test P value of cL (n) P value of cQ (n) P value of c (n) Test results of c (n) ProportionFrequency 0000000 0236810 0474986 radic 100100Block frequency (m 128) 0289667 0699313 0946308 radic 99100Cumulative sums 0000000 0108791 0779188 radic 100100Runs 0000000 0554420 0075719 radic 100100Longest run 0419021 0383827 0289667 radic 100100Rank 0851383 0996335 0289667 radic 100100FFT 0911413 0911413 0213309 radic 99100Nonoverlapping template (m 9) 0000003 0181557 0983453 radic 100100Overlapping template (m 9) 0181557 0935716 0924076 radic 100100Universal 0616305 0289667 0014550 radic 97100Approximate entropy (m 10) 0000000 0798139 0816537 radic 99100Random excursions 0008879 0319084 0739918 radic 6464Random excursions variant 0213309 0289667 0949602 radic 6464Linear complexity (M 500) 0010988 0013569 0108791 radic 99100Serial (m 16) 0616305 0028817 0262249 radic 100100

(a) (b)

(c)

Figure 6+e original and encrypted pictures of a video and the corresponding encrypted video data through FPGA (a)+e original pictureof a video (b) +e encrypted picture of a video (c) Encrypted video data shown on a monitor

0

05

1

15

2

0 50 100 150 200 250

104

(a)

0

05

1

15

2

0 50 100 150 200 250

104

(b)

Figure 7 Histogram between the original and encrypted pictures (a) Histogram of the original picture (b) Histogram of the encrypted picture

10 Complexity

correlation dramatically decreases when comparing twofigures before and after encryption as shown in Figure 6

4 Conclusions

+is paper proposed a new chaotic system generated byusing semitensor product on two chaotic systems and therelated dynamic characteristics are analyzed +e new sys-tem is employed in video encryption as well and the pro-posed method can generate 8 or even 12 state variables whencompared to Qi system and Lorenz system which onlygenerate 7 state variables at most in one iteration period+eproposed method can improve the speed of random se-quence generation +e NIST test results demonstrate thatthe pseudorandomness of new system is better than that ofsingle Qi system and single Lorenz system

+e proposed encryption algorithm based on semitensorproduct can be used in other chaotic systems +e synchro-nization of the new system can be implemented by synchro-nizing two original chaotic systems separately In this paperFPGA is used to implement the generation of the new chaoticsystem and to encrypt video data +e corresponding statisticsand differential and correlation analyses were also conductedwhich demonstrates that the new system has obvious advan-tages such as better random features better resistance todifferential attacks and lower pixel correlation for encryptedimages +e future work will focus on the decryption of videoinformation by the proposed chaotic system generated by thesemitensor product method in hardware

Data Availability

+e figuresrsquo zip data used to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities-Civil Aviation Universityof China (3122019044)

References

[1] J Li J Li X Chen C Jia and W Lou ldquoIdentity-based en-cryption with outsourced revocation in cloud computingrdquoIEEE Transactions on Computers vol 64 no 2 pp 425ndash4372015

[2] J Li Y K Li X Chen P P C Lee and W Lou ldquoA hybridcloud approach for secure authorized deduplicationrdquo IEEETransactions on Parallel and Distributed Systems vol 26no 5 pp 1206ndash1216 2015

[3] M Feki B Robert G Gelle and M Colas ldquoSecure digitalcommunication using discrete-time chaos synchronizationrdquoChaos Solitons amp Fractals vol 18 no 4 pp 881ndash890 2003

[4] V T Pham S Vaidyanathan C Volos S Jafari andT Kapitaniak ldquoA new multi-stable chaotic hyperjerk systemits special features circuit realization control and synchro-nizationrdquo Archives of Control Sciences vol 30 no 1pp 23ndash45 2020

[5] A Ouannas X Wang V T Pham G Grassi andV V Huynh ldquoSynchronization results for a class of frac-tional-order spatiotemporal partial differential systems basedon fractional lyapunov approachrdquo Boundary Value Problemsvol 2019 no 1 2019

[6] F Yu H Shen L Liu et al ldquoCCII and FPGA realization amultistable modified fourth-order autonomous Chuarsquos

Value of pixel (x y)

0

50

100

150

200

250Va

lue o

f pix

el (x

+ 1

y)

0 50 100 150 200 250

(a)

0 50 100 150 200 250Value of pixel (x y)

0

50

100

150

200

250

Valu

e of p

ixel

(x +

1 y

)(b)

Figure 8 Correlation analyses for the original and encrypted figures in Figure 6 (a) Correlation analysis for the original figure as shown inFigure 6(a) (b) Correlation analysis for the encrypted figure as shown in Figure 6(b)

Complexity 11

chaotic system with coexisting multiple attractorsrdquo Com-plexity vol 2020 Article ID 5212601 17 pages 2020

[7] Q Lai P D K Kuate F Liu and H H Lu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs vol 67 no 6 2019

[8] Q Lai A Akgul C B Li G H Xu and U Cavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and s-box designrdquo Entropy vol 20 no 12018

[9] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[10] R Wang M Li Z Gao and H Sun ldquoA new memristor-based5D chaotic system and circuit implementationrdquo Complexityvol 2018 Article ID 6069401 12 pages 2018

[11] Q Hong Y Li X Wang and Z Zeng ldquoA versatile pulsecontrol method to generate arbitrary multidirection multi-butterfly chaotic attractorsrdquo IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol 38 no 8pp 1480ndash1492 2019

[12] P S Shabestari Z Rostami V T Pham F E Alsaadi andT Hayat ldquoModeling of neurodegenerative diseases usingdiscrete chaotic systemsrdquo Communications in gteoreticalPhysics vol 71 no 10 pp 1241ndash1245 2019

[13] Q Jiang J Jiang and J Li ldquoAn image encryption algorithmbased on high-dimensional chaotic systemsrdquo in Proceedings ofthe IEEE International Conference on Signal ProcessingCommunications and Computing pp 1ndash4 Hong Kong ChinaAugust 2016

[14] S Singh M Ahmad and D Malik ldquoBreaking an imageencryption scheme based on chaotic synchronization phe-nomenonrdquo in Proceedings of the 2016 Ninth InternationalConference on Contemporary Computing (IC3) Noida IndiaAugust 2016

[15] P Chen S Yu B Chen L Xiao and J Lu ldquoDesign and SOPC-based realization of a video chaotic secure communicationschemerdquo International Journal of Bifurcation and Chaosvol 28 no 13 Article ID 1850160 24 pages 2018

[16] P R Sankpal and P A Vijaya ldquoImage encryption Usingchaotic maps a surveyrdquo in Proceedings of the 2014 Fifth In-ternational Conference on Signal and Image Processingpp 102ndash107 IEEE Bangalore India January 2014

[17] A Rani and B Raman ldquoAn image copyright protectionsystem using chaotic mapsrdquo Multimedia Tools and Applica-tions vol 76 no 2 pp 3121ndash3138 2016

[18] N Shyamala and K Anusudha ldquoReversible chaotic encryp-tion techniques for imagesrdquo in Proceedings of the 2017 FourthInternational Conference on Signal Processing Communica-tion and Networking (ICSCN) Chennai India March 2017

[19] S Chen S Yu J Lu G Chen and J He ldquoDesign and FPGA-based realization of a chaotic secure video communicationsystemrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 28 no 9 pp 2359ndash2371 2018

[20] Z Lin S Yu J Lu S Cai and G Chen ldquoDesign and ARM-embedded implementation of a chaotic map-based real-timesecure video communication systemrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 25 no 7pp 1203ndash1216 2015

[21] D Chang Z Li M Wang and Y Zeng ldquoA novel digital pro-grammable multi-scroll chaotic system and its application inFPGA-based audio secure communicationrdquo AEUmdashInternational

Journal of Electronics and Communications vol 88 pp 20ndash292018

[22] F J Farsana and K Gopakumar ldquoPrivate key encryption ofspeech signal based on three dimensional chaotic maprdquo inProceedings of the International Conference on Communica-tion amp Signal Processing pp 2197ndash2201 Chennai India April2017

[23] H Jia Z Guo S Wang and Z Chen ldquoMechanics analysis andhardware implementation of a new 3D chaotic systemrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 13Article ID 1850161 14 pages 2018

[24] E Tlelo-Cuautle L G De La Fraga V-T Pham C VolosS Jafari and A D J Quintas-Valles ldquoDynamics FPGA re-alization and application of a chaotic system with an infinitenumber of equilibrium pointsrdquo Nonlinear Dynamics vol 89no 2 pp 1129ndash1139 2017

[25] A Akgul H Calgan I Koyuncu I Pehlivan andA Istanbullu ldquoChaos-based engineering applications with a3D chaotic system without equilibrium pointsrdquo NonlinearDynamics vol 84 no 2 pp 481ndash495 2016

[26] Q Lai X Zhao K Rajagopal G Xu A Akgul andE Guleryuz ldquoDynamic analyses FPGA implementation andengineering applications of multi-butterfly chaotic attractorsgenerated from generalised sprott c systemrdquo Pramana vol 90no 1 p 12 2018

[27] W S Sayed M F Tolba A G Radwan and S K Abd-El-Hafiz ldquoFPGA realization of a speech encryption system basedon a generalized modified chaotic transition map and bitpermutationrdquo Multimedia Tools and Applications vol 78no 12 pp 16097ndash16127 2019

[28] Q Wang S Yu C Li et al ldquo+eoretical design and FPGA-based implementation of higher-dimensional digital chaoticsystemsrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 63 no 3 pp 401ndash412 2016

[29] E Dong Z Wang Z Chen and Z Wang ldquoTopologicalhorseshoe analysis and field-programmable gate arrayimplementation of a fractional-order four-wing chaoticattractorrdquo Chinese Physics B vol 27 no 1 pp 300ndash306 2018

[30] Z Hua B Zhou and Y Zhou ldquoSine chaotification model forenhancing chaos and its hardware implementationrdquo IEEETransactions on Industrial Electronics vol 66 no 2 pp 1273ndash1284 2019

[31] I Ozturk and R Kılıccedil ldquoA novel method for producing pseudorandom numbers from differential equation-based chaoticsystemsrdquo Nonlinear Dynamics vol 80 no 3 pp 1147ndash11572015

[32] R Wang Q Xie Y Huang H Sun and Y Sun ldquoDesign of aswitched hyperchaotic system and its applicationrdquo Interna-tional Journal of Computer Applications in Technology vol 57no 3 pp 207ndash218 2018

[33] R Wang H Sun J Wang L Wang and Y Wang ldquoAppli-cations of modularized circuit designs in a new hyper-chaoticsystem circuit implementationrdquo Chinese Physics B vol 24no 4 Article ID 020501 2015

[34] Z Lin S Yu and J Li ldquoChosen ciphertext attack on a chaoticstream cipherrdquo in Proceedings of the 30th Chinese Control andDecision Conference pp 1238ndash1242 Shenyang China June2018

[35] G Qi G Chen M A Van Wyk and B J Van Wyk ldquoOn anew hyperchaotic systemrdquo Physics Letters A vol 372 no 2pp 124ndash136 2008

[36] Z Hu and C-K Chan ldquoA real-valued chaotic orthogonalmatrix transform-based encryption for OFDM-PONrdquo IEEE

12 Complexity

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13

random sequence c (n) as shown in equation (18) +is canimprove the randomness of the random sequence

Next the random sequence c (n) conducts XOR op-eration with the divided video data Since a frame video dataincludes tricolor integer sequences R (n) G (n) and B (n)these three sequences will be encrypted simultaneously afterchanging the random sequence c (n) into three columnsevenly c1 (n) c2 (n) and c3 (n)

p1(n) c1(n)oplusR(n)

p2(n) c2(n)oplusG(n)

p3(n) c3(n)oplusB(n)

(26)

where p1 (n) p2 (n) and p3 (n) are encrypted sequences andoplus is an XOR operation One has

1113954R(n) modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplusp1(n)

modlfloor1113954x1(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusR(n)

1113954G(n) modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp2(n)rfloor

modlfloor1113954x2(n)rfloor

2t 2qminus t+1

1113888 1113889oplus c1(n)oplusG(n)

1113954B(n) modlfloor1113954x3(n)rfloor

2t 2qminus t+1

1113888 1113889oplus lfloorp3(n)rfloor

mod mod lfloor1113954x3(n)rfloor 2t1113872 1113873 2tminus q

1113872 1113873oplus c3(n)oplusB(n)

(27)

where lfloor1113954xpprime(n)rfloor pprime 1 2 3 are receiver terminal sequences

33 Analysis for NIST Test NIST test is provided by NationalInstitute of Standards and Technology and it is a standard to testthe randomness of a random sequence According to the en-cryption algorithm in this paper c (n) in equation (25) should betested by NIST standard +e comparisons of the random se-quence among the new system Qi system and Lorenz system c(n) cL (n) and cQ (n) are conducted which are obtained fromserial interfaces +e results of the tests are shown in Table 1

As shown in Table 1 all the test results for the randomsequences of the new system meet the NIST test index

standards Partial test results are larger than 08 which meansthese random indexes are quite close to those of the realrandom sequences +e randomness indexes and some othertest results are better than those generated from Lorenz systemand Qi system such as frequency block frequency cumulativesums nonoverlapping template approximate entropy randomexcursions random excursions variant and linear complexity

34 Statistical Analyses Vivado IDE is used to conduct thehardware simulation +e paper also performs the statisticsanalysis for the encrypted video data generated by hardwareFigure 6(a) is one picture of a video before encryptionFigure 6(b) is the encrypted picture of a video

Figure 7 demonstrates the comparisons of statisticshistogram between the original and encrypted pictures

Figure 7 demonstrates the comparisons of statisticshistograms between the original and encrypted pictures Asillustrated in Figure 7(a) the difference of the pixels dis-tribution is obvious However distribution of differentpixels for the encrypted picture shown in Figure 7(b) is theapproximately uniform distribution It can be concludedthat the proposed encryption algorithm for the new systemcan better resist statistic attack effectively

35 Differential Analysis Differential attack is used tomeasure the sensitivity of plaintext change for the encryp-tion algorithm and commonly uses NPCR (Number of PixelsChange Rate) and UACI (Unified Average Changing In-tensity) as indexes defined as follows

NPRC 1113936efD(e f)

W times Htimes 100

UACI 1

W times Htimes 1113944

ef

C(e f) minus Cprime(e f)1113868111386811138681113868

1113868111386811138681113868

255times 100

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(28)

NPRC 1113944m

NPRC (m)

UACI 1113944m

UACI (m)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(29)

where C (e f) is the pixel value before encryption and Cprime (ef ) is the pixel value after encryption If C (e f )Cprime (e f ) D(e f ) 0 else 1 NPRC and UACI calculated by (29) and theproposed system and encryption algorithm are 9960 and1228 for the first-time encryption respectively +ereforethe ability to resist differential attack improves to someextent In video encryption application the requirement forencryption speed is more concerned

36 Correlation Analysis Correlation analysis is used tocheck whether the neighbor pixels are close or not +ispaper analyzes the correlation for Figure 6 +e paper selects5000 random pixels from the original and the encryptedimages and analyzes the correlation among these random-pixel pairs as shown in Figure 8 As Figure 8 illustrates the

ffffffff ffffffff ffffffff ffffffff ffffffff4024

IIIIIIII IIIIIIII IIIIIIII

Figure 4 Data format for 64-bit fixed-point numbers

q t 164 63

hellip hellip hellip

Figure 5 +e schematics of numbered data bits

Complexity 9

Table 1 NIST test results for the random sequences of the new system Lorenz system and Qi system (the number of sequences is 100 andtheir lengths are 1000000)

Statistical test P value of cL (n) P value of cQ (n) P value of c (n) Test results of c (n) ProportionFrequency 0000000 0236810 0474986 radic 100100Block frequency (m 128) 0289667 0699313 0946308 radic 99100Cumulative sums 0000000 0108791 0779188 radic 100100Runs 0000000 0554420 0075719 radic 100100Longest run 0419021 0383827 0289667 radic 100100Rank 0851383 0996335 0289667 radic 100100FFT 0911413 0911413 0213309 radic 99100Nonoverlapping template (m 9) 0000003 0181557 0983453 radic 100100Overlapping template (m 9) 0181557 0935716 0924076 radic 100100Universal 0616305 0289667 0014550 radic 97100Approximate entropy (m 10) 0000000 0798139 0816537 radic 99100Random excursions 0008879 0319084 0739918 radic 6464Random excursions variant 0213309 0289667 0949602 radic 6464Linear complexity (M 500) 0010988 0013569 0108791 radic 99100Serial (m 16) 0616305 0028817 0262249 radic 100100

(a) (b)

(c)

Figure 6+e original and encrypted pictures of a video and the corresponding encrypted video data through FPGA (a)+e original pictureof a video (b) +e encrypted picture of a video (c) Encrypted video data shown on a monitor

0

05

1

15

2

0 50 100 150 200 250

104

(a)

0

05

1

15

2

0 50 100 150 200 250

104

(b)

Figure 7 Histogram between the original and encrypted pictures (a) Histogram of the original picture (b) Histogram of the encrypted picture

10 Complexity

correlation dramatically decreases when comparing twofigures before and after encryption as shown in Figure 6

4 Conclusions

+is paper proposed a new chaotic system generated byusing semitensor product on two chaotic systems and therelated dynamic characteristics are analyzed +e new sys-tem is employed in video encryption as well and the pro-posed method can generate 8 or even 12 state variables whencompared to Qi system and Lorenz system which onlygenerate 7 state variables at most in one iteration period+eproposed method can improve the speed of random se-quence generation +e NIST test results demonstrate thatthe pseudorandomness of new system is better than that ofsingle Qi system and single Lorenz system

+e proposed encryption algorithm based on semitensorproduct can be used in other chaotic systems +e synchro-nization of the new system can be implemented by synchro-nizing two original chaotic systems separately In this paperFPGA is used to implement the generation of the new chaoticsystem and to encrypt video data +e corresponding statisticsand differential and correlation analyses were also conductedwhich demonstrates that the new system has obvious advan-tages such as better random features better resistance todifferential attacks and lower pixel correlation for encryptedimages +e future work will focus on the decryption of videoinformation by the proposed chaotic system generated by thesemitensor product method in hardware

Data Availability

+e figuresrsquo zip data used to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities-Civil Aviation Universityof China (3122019044)

References

[1] J Li J Li X Chen C Jia and W Lou ldquoIdentity-based en-cryption with outsourced revocation in cloud computingrdquoIEEE Transactions on Computers vol 64 no 2 pp 425ndash4372015

[2] J Li Y K Li X Chen P P C Lee and W Lou ldquoA hybridcloud approach for secure authorized deduplicationrdquo IEEETransactions on Parallel and Distributed Systems vol 26no 5 pp 1206ndash1216 2015

[3] M Feki B Robert G Gelle and M Colas ldquoSecure digitalcommunication using discrete-time chaos synchronizationrdquoChaos Solitons amp Fractals vol 18 no 4 pp 881ndash890 2003

[4] V T Pham S Vaidyanathan C Volos S Jafari andT Kapitaniak ldquoA new multi-stable chaotic hyperjerk systemits special features circuit realization control and synchro-nizationrdquo Archives of Control Sciences vol 30 no 1pp 23ndash45 2020

[5] A Ouannas X Wang V T Pham G Grassi andV V Huynh ldquoSynchronization results for a class of frac-tional-order spatiotemporal partial differential systems basedon fractional lyapunov approachrdquo Boundary Value Problemsvol 2019 no 1 2019

[6] F Yu H Shen L Liu et al ldquoCCII and FPGA realization amultistable modified fourth-order autonomous Chuarsquos

Value of pixel (x y)

0

50

100

150

200

250Va

lue o

f pix

el (x

+ 1

y)

0 50 100 150 200 250

(a)

0 50 100 150 200 250Value of pixel (x y)

0

50

100

150

200

250

Valu

e of p

ixel

(x +

1 y

)(b)

Figure 8 Correlation analyses for the original and encrypted figures in Figure 6 (a) Correlation analysis for the original figure as shown inFigure 6(a) (b) Correlation analysis for the encrypted figure as shown in Figure 6(b)

Complexity 11

chaotic system with coexisting multiple attractorsrdquo Com-plexity vol 2020 Article ID 5212601 17 pages 2020

[7] Q Lai P D K Kuate F Liu and H H Lu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs vol 67 no 6 2019

[8] Q Lai A Akgul C B Li G H Xu and U Cavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and s-box designrdquo Entropy vol 20 no 12018

[9] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[10] R Wang M Li Z Gao and H Sun ldquoA new memristor-based5D chaotic system and circuit implementationrdquo Complexityvol 2018 Article ID 6069401 12 pages 2018

[11] Q Hong Y Li X Wang and Z Zeng ldquoA versatile pulsecontrol method to generate arbitrary multidirection multi-butterfly chaotic attractorsrdquo IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol 38 no 8pp 1480ndash1492 2019

[12] P S Shabestari Z Rostami V T Pham F E Alsaadi andT Hayat ldquoModeling of neurodegenerative diseases usingdiscrete chaotic systemsrdquo Communications in gteoreticalPhysics vol 71 no 10 pp 1241ndash1245 2019

[13] Q Jiang J Jiang and J Li ldquoAn image encryption algorithmbased on high-dimensional chaotic systemsrdquo in Proceedings ofthe IEEE International Conference on Signal ProcessingCommunications and Computing pp 1ndash4 Hong Kong ChinaAugust 2016

[14] S Singh M Ahmad and D Malik ldquoBreaking an imageencryption scheme based on chaotic synchronization phe-nomenonrdquo in Proceedings of the 2016 Ninth InternationalConference on Contemporary Computing (IC3) Noida IndiaAugust 2016

[15] P Chen S Yu B Chen L Xiao and J Lu ldquoDesign and SOPC-based realization of a video chaotic secure communicationschemerdquo International Journal of Bifurcation and Chaosvol 28 no 13 Article ID 1850160 24 pages 2018

[16] P R Sankpal and P A Vijaya ldquoImage encryption Usingchaotic maps a surveyrdquo in Proceedings of the 2014 Fifth In-ternational Conference on Signal and Image Processingpp 102ndash107 IEEE Bangalore India January 2014

[17] A Rani and B Raman ldquoAn image copyright protectionsystem using chaotic mapsrdquo Multimedia Tools and Applica-tions vol 76 no 2 pp 3121ndash3138 2016

[18] N Shyamala and K Anusudha ldquoReversible chaotic encryp-tion techniques for imagesrdquo in Proceedings of the 2017 FourthInternational Conference on Signal Processing Communica-tion and Networking (ICSCN) Chennai India March 2017

[19] S Chen S Yu J Lu G Chen and J He ldquoDesign and FPGA-based realization of a chaotic secure video communicationsystemrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 28 no 9 pp 2359ndash2371 2018

[20] Z Lin S Yu J Lu S Cai and G Chen ldquoDesign and ARM-embedded implementation of a chaotic map-based real-timesecure video communication systemrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 25 no 7pp 1203ndash1216 2015

[21] D Chang Z Li M Wang and Y Zeng ldquoA novel digital pro-grammable multi-scroll chaotic system and its application inFPGA-based audio secure communicationrdquo AEUmdashInternational

Journal of Electronics and Communications vol 88 pp 20ndash292018

[22] F J Farsana and K Gopakumar ldquoPrivate key encryption ofspeech signal based on three dimensional chaotic maprdquo inProceedings of the International Conference on Communica-tion amp Signal Processing pp 2197ndash2201 Chennai India April2017

[23] H Jia Z Guo S Wang and Z Chen ldquoMechanics analysis andhardware implementation of a new 3D chaotic systemrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 13Article ID 1850161 14 pages 2018

[24] E Tlelo-Cuautle L G De La Fraga V-T Pham C VolosS Jafari and A D J Quintas-Valles ldquoDynamics FPGA re-alization and application of a chaotic system with an infinitenumber of equilibrium pointsrdquo Nonlinear Dynamics vol 89no 2 pp 1129ndash1139 2017

[25] A Akgul H Calgan I Koyuncu I Pehlivan andA Istanbullu ldquoChaos-based engineering applications with a3D chaotic system without equilibrium pointsrdquo NonlinearDynamics vol 84 no 2 pp 481ndash495 2016

[26] Q Lai X Zhao K Rajagopal G Xu A Akgul andE Guleryuz ldquoDynamic analyses FPGA implementation andengineering applications of multi-butterfly chaotic attractorsgenerated from generalised sprott c systemrdquo Pramana vol 90no 1 p 12 2018

[27] W S Sayed M F Tolba A G Radwan and S K Abd-El-Hafiz ldquoFPGA realization of a speech encryption system basedon a generalized modified chaotic transition map and bitpermutationrdquo Multimedia Tools and Applications vol 78no 12 pp 16097ndash16127 2019

[28] Q Wang S Yu C Li et al ldquo+eoretical design and FPGA-based implementation of higher-dimensional digital chaoticsystemsrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 63 no 3 pp 401ndash412 2016

[29] E Dong Z Wang Z Chen and Z Wang ldquoTopologicalhorseshoe analysis and field-programmable gate arrayimplementation of a fractional-order four-wing chaoticattractorrdquo Chinese Physics B vol 27 no 1 pp 300ndash306 2018

[30] Z Hua B Zhou and Y Zhou ldquoSine chaotification model forenhancing chaos and its hardware implementationrdquo IEEETransactions on Industrial Electronics vol 66 no 2 pp 1273ndash1284 2019

[31] I Ozturk and R Kılıccedil ldquoA novel method for producing pseudorandom numbers from differential equation-based chaoticsystemsrdquo Nonlinear Dynamics vol 80 no 3 pp 1147ndash11572015

[32] R Wang Q Xie Y Huang H Sun and Y Sun ldquoDesign of aswitched hyperchaotic system and its applicationrdquo Interna-tional Journal of Computer Applications in Technology vol 57no 3 pp 207ndash218 2018

[33] R Wang H Sun J Wang L Wang and Y Wang ldquoAppli-cations of modularized circuit designs in a new hyper-chaoticsystem circuit implementationrdquo Chinese Physics B vol 24no 4 Article ID 020501 2015

[34] Z Lin S Yu and J Li ldquoChosen ciphertext attack on a chaoticstream cipherrdquo in Proceedings of the 30th Chinese Control andDecision Conference pp 1238ndash1242 Shenyang China June2018

[35] G Qi G Chen M A Van Wyk and B J Van Wyk ldquoOn anew hyperchaotic systemrdquo Physics Letters A vol 372 no 2pp 124ndash136 2008

[36] Z Hu and C-K Chan ldquoA real-valued chaotic orthogonalmatrix transform-based encryption for OFDM-PONrdquo IEEE

12 Complexity

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13

Table 1 NIST test results for the random sequences of the new system Lorenz system and Qi system (the number of sequences is 100 andtheir lengths are 1000000)

Statistical test P value of cL (n) P value of cQ (n) P value of c (n) Test results of c (n) ProportionFrequency 0000000 0236810 0474986 radic 100100Block frequency (m 128) 0289667 0699313 0946308 radic 99100Cumulative sums 0000000 0108791 0779188 radic 100100Runs 0000000 0554420 0075719 radic 100100Longest run 0419021 0383827 0289667 radic 100100Rank 0851383 0996335 0289667 radic 100100FFT 0911413 0911413 0213309 radic 99100Nonoverlapping template (m 9) 0000003 0181557 0983453 radic 100100Overlapping template (m 9) 0181557 0935716 0924076 radic 100100Universal 0616305 0289667 0014550 radic 97100Approximate entropy (m 10) 0000000 0798139 0816537 radic 99100Random excursions 0008879 0319084 0739918 radic 6464Random excursions variant 0213309 0289667 0949602 radic 6464Linear complexity (M 500) 0010988 0013569 0108791 radic 99100Serial (m 16) 0616305 0028817 0262249 radic 100100

(a) (b)

(c)

Figure 6+e original and encrypted pictures of a video and the corresponding encrypted video data through FPGA (a)+e original pictureof a video (b) +e encrypted picture of a video (c) Encrypted video data shown on a monitor

0

05

1

15

2

0 50 100 150 200 250

104

(a)

0

05

1

15

2

0 50 100 150 200 250

104

(b)

Figure 7 Histogram between the original and encrypted pictures (a) Histogram of the original picture (b) Histogram of the encrypted picture

10 Complexity

correlation dramatically decreases when comparing twofigures before and after encryption as shown in Figure 6

4 Conclusions

+is paper proposed a new chaotic system generated byusing semitensor product on two chaotic systems and therelated dynamic characteristics are analyzed +e new sys-tem is employed in video encryption as well and the pro-posed method can generate 8 or even 12 state variables whencompared to Qi system and Lorenz system which onlygenerate 7 state variables at most in one iteration period+eproposed method can improve the speed of random se-quence generation +e NIST test results demonstrate thatthe pseudorandomness of new system is better than that ofsingle Qi system and single Lorenz system

+e proposed encryption algorithm based on semitensorproduct can be used in other chaotic systems +e synchro-nization of the new system can be implemented by synchro-nizing two original chaotic systems separately In this paperFPGA is used to implement the generation of the new chaoticsystem and to encrypt video data +e corresponding statisticsand differential and correlation analyses were also conductedwhich demonstrates that the new system has obvious advan-tages such as better random features better resistance todifferential attacks and lower pixel correlation for encryptedimages +e future work will focus on the decryption of videoinformation by the proposed chaotic system generated by thesemitensor product method in hardware

Data Availability

+e figuresrsquo zip data used to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities-Civil Aviation Universityof China (3122019044)

References

[1] J Li J Li X Chen C Jia and W Lou ldquoIdentity-based en-cryption with outsourced revocation in cloud computingrdquoIEEE Transactions on Computers vol 64 no 2 pp 425ndash4372015

[2] J Li Y K Li X Chen P P C Lee and W Lou ldquoA hybridcloud approach for secure authorized deduplicationrdquo IEEETransactions on Parallel and Distributed Systems vol 26no 5 pp 1206ndash1216 2015

[3] M Feki B Robert G Gelle and M Colas ldquoSecure digitalcommunication using discrete-time chaos synchronizationrdquoChaos Solitons amp Fractals vol 18 no 4 pp 881ndash890 2003

[4] V T Pham S Vaidyanathan C Volos S Jafari andT Kapitaniak ldquoA new multi-stable chaotic hyperjerk systemits special features circuit realization control and synchro-nizationrdquo Archives of Control Sciences vol 30 no 1pp 23ndash45 2020

[5] A Ouannas X Wang V T Pham G Grassi andV V Huynh ldquoSynchronization results for a class of frac-tional-order spatiotemporal partial differential systems basedon fractional lyapunov approachrdquo Boundary Value Problemsvol 2019 no 1 2019

[6] F Yu H Shen L Liu et al ldquoCCII and FPGA realization amultistable modified fourth-order autonomous Chuarsquos

Value of pixel (x y)

0

50

100

150

200

250Va

lue o

f pix

el (x

+ 1

y)

0 50 100 150 200 250

(a)

0 50 100 150 200 250Value of pixel (x y)

0

50

100

150

200

250

Valu

e of p

ixel

(x +

1 y

)(b)

Figure 8 Correlation analyses for the original and encrypted figures in Figure 6 (a) Correlation analysis for the original figure as shown inFigure 6(a) (b) Correlation analysis for the encrypted figure as shown in Figure 6(b)

Complexity 11

chaotic system with coexisting multiple attractorsrdquo Com-plexity vol 2020 Article ID 5212601 17 pages 2020

[7] Q Lai P D K Kuate F Liu and H H Lu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs vol 67 no 6 2019

[8] Q Lai A Akgul C B Li G H Xu and U Cavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and s-box designrdquo Entropy vol 20 no 12018

[9] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[10] R Wang M Li Z Gao and H Sun ldquoA new memristor-based5D chaotic system and circuit implementationrdquo Complexityvol 2018 Article ID 6069401 12 pages 2018

[11] Q Hong Y Li X Wang and Z Zeng ldquoA versatile pulsecontrol method to generate arbitrary multidirection multi-butterfly chaotic attractorsrdquo IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol 38 no 8pp 1480ndash1492 2019

[12] P S Shabestari Z Rostami V T Pham F E Alsaadi andT Hayat ldquoModeling of neurodegenerative diseases usingdiscrete chaotic systemsrdquo Communications in gteoreticalPhysics vol 71 no 10 pp 1241ndash1245 2019

[13] Q Jiang J Jiang and J Li ldquoAn image encryption algorithmbased on high-dimensional chaotic systemsrdquo in Proceedings ofthe IEEE International Conference on Signal ProcessingCommunications and Computing pp 1ndash4 Hong Kong ChinaAugust 2016

[14] S Singh M Ahmad and D Malik ldquoBreaking an imageencryption scheme based on chaotic synchronization phe-nomenonrdquo in Proceedings of the 2016 Ninth InternationalConference on Contemporary Computing (IC3) Noida IndiaAugust 2016

[15] P Chen S Yu B Chen L Xiao and J Lu ldquoDesign and SOPC-based realization of a video chaotic secure communicationschemerdquo International Journal of Bifurcation and Chaosvol 28 no 13 Article ID 1850160 24 pages 2018

[16] P R Sankpal and P A Vijaya ldquoImage encryption Usingchaotic maps a surveyrdquo in Proceedings of the 2014 Fifth In-ternational Conference on Signal and Image Processingpp 102ndash107 IEEE Bangalore India January 2014

[17] A Rani and B Raman ldquoAn image copyright protectionsystem using chaotic mapsrdquo Multimedia Tools and Applica-tions vol 76 no 2 pp 3121ndash3138 2016

[18] N Shyamala and K Anusudha ldquoReversible chaotic encryp-tion techniques for imagesrdquo in Proceedings of the 2017 FourthInternational Conference on Signal Processing Communica-tion and Networking (ICSCN) Chennai India March 2017

[19] S Chen S Yu J Lu G Chen and J He ldquoDesign and FPGA-based realization of a chaotic secure video communicationsystemrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 28 no 9 pp 2359ndash2371 2018

[20] Z Lin S Yu J Lu S Cai and G Chen ldquoDesign and ARM-embedded implementation of a chaotic map-based real-timesecure video communication systemrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 25 no 7pp 1203ndash1216 2015

[21] D Chang Z Li M Wang and Y Zeng ldquoA novel digital pro-grammable multi-scroll chaotic system and its application inFPGA-based audio secure communicationrdquo AEUmdashInternational

Journal of Electronics and Communications vol 88 pp 20ndash292018

[22] F J Farsana and K Gopakumar ldquoPrivate key encryption ofspeech signal based on three dimensional chaotic maprdquo inProceedings of the International Conference on Communica-tion amp Signal Processing pp 2197ndash2201 Chennai India April2017

[23] H Jia Z Guo S Wang and Z Chen ldquoMechanics analysis andhardware implementation of a new 3D chaotic systemrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 13Article ID 1850161 14 pages 2018

[24] E Tlelo-Cuautle L G De La Fraga V-T Pham C VolosS Jafari and A D J Quintas-Valles ldquoDynamics FPGA re-alization and application of a chaotic system with an infinitenumber of equilibrium pointsrdquo Nonlinear Dynamics vol 89no 2 pp 1129ndash1139 2017

[25] A Akgul H Calgan I Koyuncu I Pehlivan andA Istanbullu ldquoChaos-based engineering applications with a3D chaotic system without equilibrium pointsrdquo NonlinearDynamics vol 84 no 2 pp 481ndash495 2016

[26] Q Lai X Zhao K Rajagopal G Xu A Akgul andE Guleryuz ldquoDynamic analyses FPGA implementation andengineering applications of multi-butterfly chaotic attractorsgenerated from generalised sprott c systemrdquo Pramana vol 90no 1 p 12 2018

[27] W S Sayed M F Tolba A G Radwan and S K Abd-El-Hafiz ldquoFPGA realization of a speech encryption system basedon a generalized modified chaotic transition map and bitpermutationrdquo Multimedia Tools and Applications vol 78no 12 pp 16097ndash16127 2019

[28] Q Wang S Yu C Li et al ldquo+eoretical design and FPGA-based implementation of higher-dimensional digital chaoticsystemsrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 63 no 3 pp 401ndash412 2016

[29] E Dong Z Wang Z Chen and Z Wang ldquoTopologicalhorseshoe analysis and field-programmable gate arrayimplementation of a fractional-order four-wing chaoticattractorrdquo Chinese Physics B vol 27 no 1 pp 300ndash306 2018

[30] Z Hua B Zhou and Y Zhou ldquoSine chaotification model forenhancing chaos and its hardware implementationrdquo IEEETransactions on Industrial Electronics vol 66 no 2 pp 1273ndash1284 2019

[31] I Ozturk and R Kılıccedil ldquoA novel method for producing pseudorandom numbers from differential equation-based chaoticsystemsrdquo Nonlinear Dynamics vol 80 no 3 pp 1147ndash11572015

[32] R Wang Q Xie Y Huang H Sun and Y Sun ldquoDesign of aswitched hyperchaotic system and its applicationrdquo Interna-tional Journal of Computer Applications in Technology vol 57no 3 pp 207ndash218 2018

[33] R Wang H Sun J Wang L Wang and Y Wang ldquoAppli-cations of modularized circuit designs in a new hyper-chaoticsystem circuit implementationrdquo Chinese Physics B vol 24no 4 Article ID 020501 2015

[34] Z Lin S Yu and J Li ldquoChosen ciphertext attack on a chaoticstream cipherrdquo in Proceedings of the 30th Chinese Control andDecision Conference pp 1238ndash1242 Shenyang China June2018

[35] G Qi G Chen M A Van Wyk and B J Van Wyk ldquoOn anew hyperchaotic systemrdquo Physics Letters A vol 372 no 2pp 124ndash136 2008

[36] Z Hu and C-K Chan ldquoA real-valued chaotic orthogonalmatrix transform-based encryption for OFDM-PONrdquo IEEE

12 Complexity

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13

correlation dramatically decreases when comparing twofigures before and after encryption as shown in Figure 6

4 Conclusions

+is paper proposed a new chaotic system generated byusing semitensor product on two chaotic systems and therelated dynamic characteristics are analyzed +e new sys-tem is employed in video encryption as well and the pro-posed method can generate 8 or even 12 state variables whencompared to Qi system and Lorenz system which onlygenerate 7 state variables at most in one iteration period+eproposed method can improve the speed of random se-quence generation +e NIST test results demonstrate thatthe pseudorandomness of new system is better than that ofsingle Qi system and single Lorenz system

+e proposed encryption algorithm based on semitensorproduct can be used in other chaotic systems +e synchro-nization of the new system can be implemented by synchro-nizing two original chaotic systems separately In this paperFPGA is used to implement the generation of the new chaoticsystem and to encrypt video data +e corresponding statisticsand differential and correlation analyses were also conductedwhich demonstrates that the new system has obvious advan-tages such as better random features better resistance todifferential attacks and lower pixel correlation for encryptedimages +e future work will focus on the decryption of videoinformation by the proposed chaotic system generated by thesemitensor product method in hardware

Data Availability

+e figuresrsquo zip data used to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work was supported by the Fundamental ResearchFunds for the Central Universities-Civil Aviation Universityof China (3122019044)

References

[1] J Li J Li X Chen C Jia and W Lou ldquoIdentity-based en-cryption with outsourced revocation in cloud computingrdquoIEEE Transactions on Computers vol 64 no 2 pp 425ndash4372015

[2] J Li Y K Li X Chen P P C Lee and W Lou ldquoA hybridcloud approach for secure authorized deduplicationrdquo IEEETransactions on Parallel and Distributed Systems vol 26no 5 pp 1206ndash1216 2015

[3] M Feki B Robert G Gelle and M Colas ldquoSecure digitalcommunication using discrete-time chaos synchronizationrdquoChaos Solitons amp Fractals vol 18 no 4 pp 881ndash890 2003

[4] V T Pham S Vaidyanathan C Volos S Jafari andT Kapitaniak ldquoA new multi-stable chaotic hyperjerk systemits special features circuit realization control and synchro-nizationrdquo Archives of Control Sciences vol 30 no 1pp 23ndash45 2020

[5] A Ouannas X Wang V T Pham G Grassi andV V Huynh ldquoSynchronization results for a class of frac-tional-order spatiotemporal partial differential systems basedon fractional lyapunov approachrdquo Boundary Value Problemsvol 2019 no 1 2019

[6] F Yu H Shen L Liu et al ldquoCCII and FPGA realization amultistable modified fourth-order autonomous Chuarsquos

Value of pixel (x y)

0

50

100

150

200

250Va

lue o

f pix

el (x

+ 1

y)

0 50 100 150 200 250

(a)

0 50 100 150 200 250Value of pixel (x y)

0

50

100

150

200

250

Valu

e of p

ixel

(x +

1 y

)(b)

Figure 8 Correlation analyses for the original and encrypted figures in Figure 6 (a) Correlation analysis for the original figure as shown inFigure 6(a) (b) Correlation analysis for the encrypted figure as shown in Figure 6(b)

Complexity 11

chaotic system with coexisting multiple attractorsrdquo Com-plexity vol 2020 Article ID 5212601 17 pages 2020

[7] Q Lai P D K Kuate F Liu and H H Lu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs vol 67 no 6 2019

[8] Q Lai A Akgul C B Li G H Xu and U Cavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and s-box designrdquo Entropy vol 20 no 12018

[9] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[10] R Wang M Li Z Gao and H Sun ldquoA new memristor-based5D chaotic system and circuit implementationrdquo Complexityvol 2018 Article ID 6069401 12 pages 2018

[11] Q Hong Y Li X Wang and Z Zeng ldquoA versatile pulsecontrol method to generate arbitrary multidirection multi-butterfly chaotic attractorsrdquo IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol 38 no 8pp 1480ndash1492 2019

[12] P S Shabestari Z Rostami V T Pham F E Alsaadi andT Hayat ldquoModeling of neurodegenerative diseases usingdiscrete chaotic systemsrdquo Communications in gteoreticalPhysics vol 71 no 10 pp 1241ndash1245 2019

[13] Q Jiang J Jiang and J Li ldquoAn image encryption algorithmbased on high-dimensional chaotic systemsrdquo in Proceedings ofthe IEEE International Conference on Signal ProcessingCommunications and Computing pp 1ndash4 Hong Kong ChinaAugust 2016

[14] S Singh M Ahmad and D Malik ldquoBreaking an imageencryption scheme based on chaotic synchronization phe-nomenonrdquo in Proceedings of the 2016 Ninth InternationalConference on Contemporary Computing (IC3) Noida IndiaAugust 2016

[15] P Chen S Yu B Chen L Xiao and J Lu ldquoDesign and SOPC-based realization of a video chaotic secure communicationschemerdquo International Journal of Bifurcation and Chaosvol 28 no 13 Article ID 1850160 24 pages 2018

[16] P R Sankpal and P A Vijaya ldquoImage encryption Usingchaotic maps a surveyrdquo in Proceedings of the 2014 Fifth In-ternational Conference on Signal and Image Processingpp 102ndash107 IEEE Bangalore India January 2014

[17] A Rani and B Raman ldquoAn image copyright protectionsystem using chaotic mapsrdquo Multimedia Tools and Applica-tions vol 76 no 2 pp 3121ndash3138 2016

[18] N Shyamala and K Anusudha ldquoReversible chaotic encryp-tion techniques for imagesrdquo in Proceedings of the 2017 FourthInternational Conference on Signal Processing Communica-tion and Networking (ICSCN) Chennai India March 2017

[19] S Chen S Yu J Lu G Chen and J He ldquoDesign and FPGA-based realization of a chaotic secure video communicationsystemrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 28 no 9 pp 2359ndash2371 2018

[20] Z Lin S Yu J Lu S Cai and G Chen ldquoDesign and ARM-embedded implementation of a chaotic map-based real-timesecure video communication systemrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 25 no 7pp 1203ndash1216 2015

[21] D Chang Z Li M Wang and Y Zeng ldquoA novel digital pro-grammable multi-scroll chaotic system and its application inFPGA-based audio secure communicationrdquo AEUmdashInternational

Journal of Electronics and Communications vol 88 pp 20ndash292018

[22] F J Farsana and K Gopakumar ldquoPrivate key encryption ofspeech signal based on three dimensional chaotic maprdquo inProceedings of the International Conference on Communica-tion amp Signal Processing pp 2197ndash2201 Chennai India April2017

[23] H Jia Z Guo S Wang and Z Chen ldquoMechanics analysis andhardware implementation of a new 3D chaotic systemrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 13Article ID 1850161 14 pages 2018

[24] E Tlelo-Cuautle L G De La Fraga V-T Pham C VolosS Jafari and A D J Quintas-Valles ldquoDynamics FPGA re-alization and application of a chaotic system with an infinitenumber of equilibrium pointsrdquo Nonlinear Dynamics vol 89no 2 pp 1129ndash1139 2017

[25] A Akgul H Calgan I Koyuncu I Pehlivan andA Istanbullu ldquoChaos-based engineering applications with a3D chaotic system without equilibrium pointsrdquo NonlinearDynamics vol 84 no 2 pp 481ndash495 2016

[26] Q Lai X Zhao K Rajagopal G Xu A Akgul andE Guleryuz ldquoDynamic analyses FPGA implementation andengineering applications of multi-butterfly chaotic attractorsgenerated from generalised sprott c systemrdquo Pramana vol 90no 1 p 12 2018

[27] W S Sayed M F Tolba A G Radwan and S K Abd-El-Hafiz ldquoFPGA realization of a speech encryption system basedon a generalized modified chaotic transition map and bitpermutationrdquo Multimedia Tools and Applications vol 78no 12 pp 16097ndash16127 2019

[28] Q Wang S Yu C Li et al ldquo+eoretical design and FPGA-based implementation of higher-dimensional digital chaoticsystemsrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 63 no 3 pp 401ndash412 2016

[29] E Dong Z Wang Z Chen and Z Wang ldquoTopologicalhorseshoe analysis and field-programmable gate arrayimplementation of a fractional-order four-wing chaoticattractorrdquo Chinese Physics B vol 27 no 1 pp 300ndash306 2018

[30] Z Hua B Zhou and Y Zhou ldquoSine chaotification model forenhancing chaos and its hardware implementationrdquo IEEETransactions on Industrial Electronics vol 66 no 2 pp 1273ndash1284 2019

[31] I Ozturk and R Kılıccedil ldquoA novel method for producing pseudorandom numbers from differential equation-based chaoticsystemsrdquo Nonlinear Dynamics vol 80 no 3 pp 1147ndash11572015

[32] R Wang Q Xie Y Huang H Sun and Y Sun ldquoDesign of aswitched hyperchaotic system and its applicationrdquo Interna-tional Journal of Computer Applications in Technology vol 57no 3 pp 207ndash218 2018

[33] R Wang H Sun J Wang L Wang and Y Wang ldquoAppli-cations of modularized circuit designs in a new hyper-chaoticsystem circuit implementationrdquo Chinese Physics B vol 24no 4 Article ID 020501 2015

[34] Z Lin S Yu and J Li ldquoChosen ciphertext attack on a chaoticstream cipherrdquo in Proceedings of the 30th Chinese Control andDecision Conference pp 1238ndash1242 Shenyang China June2018

[35] G Qi G Chen M A Van Wyk and B J Van Wyk ldquoOn anew hyperchaotic systemrdquo Physics Letters A vol 372 no 2pp 124ndash136 2008

[36] Z Hu and C-K Chan ldquoA real-valued chaotic orthogonalmatrix transform-based encryption for OFDM-PONrdquo IEEE

12 Complexity

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13

chaotic system with coexisting multiple attractorsrdquo Com-plexity vol 2020 Article ID 5212601 17 pages 2020

[7] Q Lai P D K Kuate F Liu and H H Lu ldquoAn extremelysimple chaotic system with infinitely many coexistingattractorsrdquo IEEE Transactions on Circuits and Systems IIExpress Briefs vol 67 no 6 2019

[8] Q Lai A Akgul C B Li G H Xu and U Cavusoglu ldquoA newchaotic system with multiple attractors dynamic analysiscircuit realization and s-box designrdquo Entropy vol 20 no 12018

[9] Q Lai B Norouzi and F Liu ldquoDynamic analysis circuitrealization control design and image encryption applicationof an extended lu system with coexisting attractorsrdquo ChaosSolitons amp Fractals vol 114 pp 230ndash245 2018

[10] R Wang M Li Z Gao and H Sun ldquoA new memristor-based5D chaotic system and circuit implementationrdquo Complexityvol 2018 Article ID 6069401 12 pages 2018

[11] Q Hong Y Li X Wang and Z Zeng ldquoA versatile pulsecontrol method to generate arbitrary multidirection multi-butterfly chaotic attractorsrdquo IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol 38 no 8pp 1480ndash1492 2019

[12] P S Shabestari Z Rostami V T Pham F E Alsaadi andT Hayat ldquoModeling of neurodegenerative diseases usingdiscrete chaotic systemsrdquo Communications in gteoreticalPhysics vol 71 no 10 pp 1241ndash1245 2019

[13] Q Jiang J Jiang and J Li ldquoAn image encryption algorithmbased on high-dimensional chaotic systemsrdquo in Proceedings ofthe IEEE International Conference on Signal ProcessingCommunications and Computing pp 1ndash4 Hong Kong ChinaAugust 2016

[14] S Singh M Ahmad and D Malik ldquoBreaking an imageencryption scheme based on chaotic synchronization phe-nomenonrdquo in Proceedings of the 2016 Ninth InternationalConference on Contemporary Computing (IC3) Noida IndiaAugust 2016

[15] P Chen S Yu B Chen L Xiao and J Lu ldquoDesign and SOPC-based realization of a video chaotic secure communicationschemerdquo International Journal of Bifurcation and Chaosvol 28 no 13 Article ID 1850160 24 pages 2018

[16] P R Sankpal and P A Vijaya ldquoImage encryption Usingchaotic maps a surveyrdquo in Proceedings of the 2014 Fifth In-ternational Conference on Signal and Image Processingpp 102ndash107 IEEE Bangalore India January 2014

[17] A Rani and B Raman ldquoAn image copyright protectionsystem using chaotic mapsrdquo Multimedia Tools and Applica-tions vol 76 no 2 pp 3121ndash3138 2016

[18] N Shyamala and K Anusudha ldquoReversible chaotic encryp-tion techniques for imagesrdquo in Proceedings of the 2017 FourthInternational Conference on Signal Processing Communica-tion and Networking (ICSCN) Chennai India March 2017

[19] S Chen S Yu J Lu G Chen and J He ldquoDesign and FPGA-based realization of a chaotic secure video communicationsystemrdquo IEEE Transactions on Circuits and Systems for VideoTechnology vol 28 no 9 pp 2359ndash2371 2018

[20] Z Lin S Yu J Lu S Cai and G Chen ldquoDesign and ARM-embedded implementation of a chaotic map-based real-timesecure video communication systemrdquo IEEE Transactions onCircuits and Systems for Video Technology vol 25 no 7pp 1203ndash1216 2015

[21] D Chang Z Li M Wang and Y Zeng ldquoA novel digital pro-grammable multi-scroll chaotic system and its application inFPGA-based audio secure communicationrdquo AEUmdashInternational

Journal of Electronics and Communications vol 88 pp 20ndash292018

[22] F J Farsana and K Gopakumar ldquoPrivate key encryption ofspeech signal based on three dimensional chaotic maprdquo inProceedings of the International Conference on Communica-tion amp Signal Processing pp 2197ndash2201 Chennai India April2017

[23] H Jia Z Guo S Wang and Z Chen ldquoMechanics analysis andhardware implementation of a new 3D chaotic systemrdquo In-ternational Journal of Bifurcation and Chaos vol 28 no 13Article ID 1850161 14 pages 2018

[24] E Tlelo-Cuautle L G De La Fraga V-T Pham C VolosS Jafari and A D J Quintas-Valles ldquoDynamics FPGA re-alization and application of a chaotic system with an infinitenumber of equilibrium pointsrdquo Nonlinear Dynamics vol 89no 2 pp 1129ndash1139 2017

[25] A Akgul H Calgan I Koyuncu I Pehlivan andA Istanbullu ldquoChaos-based engineering applications with a3D chaotic system without equilibrium pointsrdquo NonlinearDynamics vol 84 no 2 pp 481ndash495 2016

[26] Q Lai X Zhao K Rajagopal G Xu A Akgul andE Guleryuz ldquoDynamic analyses FPGA implementation andengineering applications of multi-butterfly chaotic attractorsgenerated from generalised sprott c systemrdquo Pramana vol 90no 1 p 12 2018

[27] W S Sayed M F Tolba A G Radwan and S K Abd-El-Hafiz ldquoFPGA realization of a speech encryption system basedon a generalized modified chaotic transition map and bitpermutationrdquo Multimedia Tools and Applications vol 78no 12 pp 16097ndash16127 2019

[28] Q Wang S Yu C Li et al ldquo+eoretical design and FPGA-based implementation of higher-dimensional digital chaoticsystemsrdquo IEEE Transactions on Circuits and Systems I RegularPapers vol 63 no 3 pp 401ndash412 2016

[29] E Dong Z Wang Z Chen and Z Wang ldquoTopologicalhorseshoe analysis and field-programmable gate arrayimplementation of a fractional-order four-wing chaoticattractorrdquo Chinese Physics B vol 27 no 1 pp 300ndash306 2018

[30] Z Hua B Zhou and Y Zhou ldquoSine chaotification model forenhancing chaos and its hardware implementationrdquo IEEETransactions on Industrial Electronics vol 66 no 2 pp 1273ndash1284 2019

[31] I Ozturk and R Kılıccedil ldquoA novel method for producing pseudorandom numbers from differential equation-based chaoticsystemsrdquo Nonlinear Dynamics vol 80 no 3 pp 1147ndash11572015

[32] R Wang Q Xie Y Huang H Sun and Y Sun ldquoDesign of aswitched hyperchaotic system and its applicationrdquo Interna-tional Journal of Computer Applications in Technology vol 57no 3 pp 207ndash218 2018

[33] R Wang H Sun J Wang L Wang and Y Wang ldquoAppli-cations of modularized circuit designs in a new hyper-chaoticsystem circuit implementationrdquo Chinese Physics B vol 24no 4 Article ID 020501 2015

[34] Z Lin S Yu and J Li ldquoChosen ciphertext attack on a chaoticstream cipherrdquo in Proceedings of the 30th Chinese Control andDecision Conference pp 1238ndash1242 Shenyang China June2018

[35] G Qi G Chen M A Van Wyk and B J Van Wyk ldquoOn anew hyperchaotic systemrdquo Physics Letters A vol 372 no 2pp 124ndash136 2008

[36] Z Hu and C-K Chan ldquoA real-valued chaotic orthogonalmatrix transform-based encryption for OFDM-PONrdquo IEEE

12 Complexity

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13

Photonics Technology Letters vol 30 no 16 pp 1455ndash14582018

[37] D Cheng H Qi and Y Zhao An Introduction to Semi-TensorProduct of Matrices and Its Applications World ScientificSingapore 2012

[38] T Akutsu M Hayashida W-K Ching and M K NgldquoControl of boolean networks hardness results and algo-rithms for tree structured networksrdquo Journal of gteoreticalBiology vol 244 no 4 pp 670ndash679 2007

[39] D Cheng ldquoOn finite potential gamesrdquo Automatica vol 50no 7 pp 1793ndash1801 2014

[40] X Liu and J Zhu ldquoOn potential equations of finite gamesrdquoAutomatica vol 68 pp 245ndash253 2016

[41] H Li G Zhao M Meng and J Feng ldquoA survey on appli-cations of semi-tensor product method in engineeringrdquoScience China Information Sciences vol 61 no 1 pp 24ndash402018

[42] D Cheng and H Qi Semi-Tensor Product of Matrices-gteoryand Applications Science Publication Washington DC USA2nd edition 2011

Complexity 13