100

CHAPTER 6

EC BASED KEY GENERATION FOR SYMMETRIC ENCRYPTION

6.1 INTRODUCTION

Stream ciphers are cryptographic primitives used to ensure privacy

in digital communication. Security of stream cipher depends on the generation

of unpredictable sequence called key stream that must be of sufficient size and

randomness (Goldreich et al 1986). Hence, key stream generator is incredibly

a vital building block for stream cipher applications. This chapter presents the

implementation of stream cipher, where the key stream is generated based on

the properties of LFSR and cyclic EC over a finite prime field.

The process of encryption / decryption of an image in spatial

domain are illustrated and the key file parameters needed for generating the

key stream to other parties are also encrypted using ECC. Therefore, the

encrypted key file parameters are only transmitted and not the entire full

length key. Whereas ECC is replacing RSA for key exchange, EC based

stream cipher offers a good choice for encryption in real time application. The

strength of the proposed cipher lies in the generation of random sequence

using LFSR over GF(p), the difficulty of ECDLP for determining the key file

parameters and no need to transmit the entire key stream in the encryption

process. This chapter also discusses the security aspects of the proposed cipher

which is secure against all kinds of attacks.

101

The rest of the chapter is organized as follows. Section 6.2

describes the concepts of pseudo random sequence generation.

Section 6.3 discusses security issues related to image encryption techniques.

Section 6.4 proposes an efficient EC based stream cipher for confidential

communication. The algorithm is also defined. Section 6.5 gives experiment

results. Section 6.6 evaluates the security analysis, and finally section 6.7

concludes this chapter.

6.2 PSEUDO RANDOM SEQUENCE GENERATION

Random numbers are of crucial importance in almost every aspect

of modern digital cryptography, responsible for the strength of cryptographic

primitives in securing precious information by rendering it unknown,

unguessable, unpredictable and unrepeatable for an adversary. One of the most

important roles randomness plays in cryptography is represented by

cryptographic keys which determine the transformation of the plaintext into

cipher text and vice versa. Considering that both the encryption and the

decryption algorithms are publicly known together with all the cipher texts

transmitted between the sender and receiver, the security of the whole

cryptosystem is dependent on how the key information is managed, generated,

agreed on, applied, stored and destroyed. The knowledge of the key entails the

access to the secret message, thus the choice of the key space and the key

derivation method is critical.

Cryptographic keys must be unpredictable for the adversary

meaning a high information content and high uncertainty, and the measure of

these properties is entropy. Thus sequences chosen for cryptographic keys

must also exhibit independency of values, uniform distribution and

irreproducibility. As a result, what cryptography needs for its keys is

102

randomness. But randomness comes in many flavours and deciding on a

certain source of randomness for a given application is a difficult task,

considering the quality and quantity of randomness necessary for a key to

withstand possible attacks. Furthermore, aspects such as the performance

implications of working with a long key sequence and the effort of managing

such a key also have to be taken into account.

One of the properties that a key sequence provides is to classify the

cipher using these keys in different security categories. The most demanding

requirements on the quality of the key sequence are stated by unconditional

security. The most notable work in this area is Shannon’s demonstration that a

vernam cipher which uses a perfectly random key and applies it only once,

known as OTP, is unbreakable.

The need for design of efficient and secure pseudo random

sequence generators remains an ongoing challenge and an important field in

cryptographic research up to the present day. The cryptosystems that exploit

the idea of the pseudo random sequence generators are the stream ciphers. The

cryptographically random generators follow this property: from a piece of an

arbitrary long sequence, it is computationally impossible to predict the

following bit of the sequence. The requirement is that the complete sequence

cannot be computed from a piece of it, and at the same time, it can be

completely regenerated from the seed. Pseudo random generators allow

ciphering messages of arbitrary length combining the message with the

sequence using the exclusive OR operation byte to byte.

Considerable research has been made in the design and analysis of

pseudo random generators over the last decade (Rueppel 1986). Kaliski(1986)

discussed how to generate a pseudo random sequence from elliptic curves,

103

wherein randomness criteria based on the computational difficulty of the DL

over the elliptic curves is used. Guang et al (1999) use the unconditional

randomness criteria to measure the EC sequences and the generation of binary

sequences by applying trace functions to EC over GF(2m). Deepthi et al (2009)

presented stream cipher based on elliptic curve point multiplication over

GF(2m). Pareek et al (2010) discussed the random bit generators based on

properties of chaotic maps.

Thus there are ongoing research efforts that aim to reveal secure

ways of using PRNG in cryptography and this search has brought about

several PRNG designs that are considered cryptographically secure PRNG

built on cryptographic primitives such as hash functions or block ciphers,

mathematical problems considered to be extremely difficult such as EC

generators, or integer factorization. In this work, the focus is on stream cipher

built using LFSR and cyclic EC over a finite prime field.

6.2.1 Linear Feedback Shift Register

Linear feedback shift register is the basis for generating key

sequences of stream ciphers because they generate sequences having long

periods and good randomness and statistical properties. The LFSR was first

presented by Golomb (1982) as pseudo random number generator. The secret

key in these ciphers is the LFSR’s initial state. The pseudo random number

generators based on LFSRs are very quick, easy and secure in the

implementation of hardware and software (Schneier 1996). This work aims to

extend the period of a LFSR in order to make them more secure.

Any LFSR can be characterized as a polynomial of variable x,

referred to as the generator polynomial:

104

G(x) = gmxm + gm-1xm-1 + ... + g2x2 + g1x + g0 (6.1)

The coefficients gi denotes the tap weights. The order of the polynomial, m,

specifies the number of LFSR stages. Finite field is used to attain m-sequence

feedback taps. As an example of polynomial representation, the generator

polynomial G(x) = g4x4 + g3x3 + g2x2 + g1x + g0 represents an LFSR with

feedback taps 2 and 1, denoted in Figure 6.1. These taps are elected based on

the primitive polynomial.

Figure 6.1 Linear feedback shift register

To determine maximum length tap sequences, the following points

are to be considered.

The polynomial is primitive.

Maximal length tap sequences constantly have an even number of taps.

The tap values in a maximal length tap sequence are all relatively

prime.

Recent works point out that the key stream generation plays a major

role in the performance of a secure stream cipher. The key stream generation

presented in this work is a variation of the above methods, where the key

streams are generated based on the combination of LFSR and cyclic EC over a

finite prime field.

g4

Seed

mod p

Random sequence (ri)g3 g2 g1

105

6.3 SECURITY ISSUES IN IMAGE ENCRYPTION

Images are widely used in various areas and hence the protection of

image data from unauthorized access has become a major concern. Image

encryption plays an important role in real time multimedia applications

because of large data sizes. Therefore, communication security of digital

images and textual digital media can be accomplished by means of standard

symmetric key cryptography. Such media can be treated as binary sequence

and the whole data can be encrypted using a cryptosystem such as Advanced

Encryption Standard (AES) or Data Encryption Standard (DES) (Stinson

2002).

In general, when the multimedia data is static (not a real time

streaming) it can be treated as a regular binary data and the conventional

encryption techniques can be used. Deciding upon what level of security is

needed is harder than it looks. To identify an optimal security level, the cost of

the multimedia information to be protected and the cost of the protection itself

are to be compared carefully.

At present, there are many available image encryption algorithms

such as magic cube transformation (Jun et al 2002), baker’s transformation

(Feng 2003), affine transformation (Guibin et al 2003) and tangram algorithm

(Ding et al 2005). In some algorithms, the secret key and algorithm cannot be

separated effectively. This does not satisfy the requirements of the modern

cryptographic mechanism and are prone to various attacks. In recent years,

image encryption has been developed to overcome the above disadvantages as

discussed in (Stinson 2002, Gang et al 2002).

106

Various image encryption schemes have been proposed by several

researchers to overcome image encryption problems. Sinha et al (2003) have

proposed a technique to encrypt an image for secure image transmission.

Image encoding is done by using an appropriate error control code like a Bose

Chaudhuri Hochquenghem (BCH) code. Shin et al (2003) proposed the

multilevel image encryption by using binary phase exclusive OR operation

and image dividing technique. Salleh et al (2003) discussed a chaos based

symmetric key encryption algorithm for securing images.

Mitra et al (2006) proposed an image encryption using permutation

method where the image can be sighted as an arrangement of bits, pixels and

blocks. El-din et al (2007) presented a feedback stream cipher based on chaos

for image encryption. Zeghid et al (2007) illustrated a modified version of

AES algorithm for image encryption by introducing a key stream generator.

Ismail et al (2010) established a chaos based stream cipher, where the key is

tailored after encryption of each pixel of the plain image. Based on the study,

EC based key stream is generated and is used to enhance the security of the

stream cipher system further.

6.4 EC BASED STREAM CIPHER

Generally, elliptic curve is used in public key cryptosystems. The

security of ECC is based on discrete logarithmic problem and has advantages

over RSA scheme (Koblitz 1987). In this section, the use of ECs over finite

prime field in stream cipher cryptosystems is discussed.

107

6.4.1 Methodology

Stream ciphers are a symmetric key cryptosystem used to encrypt

large amounts of data very fast. Figure 6.2 presents the block diagram of the

proposed stream cipher. Let the source be A and destination be B. Here,

plaintext message is represented as a stream of characters, Mi. The key stream

Ki is chosen from the EC points. The same key stream is exploited for

encryption and decryption procedure. The cipher stream is indicated by Ci.

The length of the message is taken as n bytes.

Figure 6.2 Elliptic curve based stream cipher system

Assume that user A wants to encrypt and transmit the message to

user B, it does the following step by step procedure.

1. Generate elliptic curve y2 = x3 + ax + b by choosing the appropriate

parameters a and b over GF(p). The algorithm ‘genPoints’ described in

Section 4.3.1 is used to generate EC points Pi.

EC based Key Stream Generator

over GF (p)

EC based Key Stream Generator

over GF (p)

Key file parameters (a, b, p, x, seed)

E

||

EPB

(keyfile)

Ci

Message (Mi)

Ki Ciphertext (Ci)

Key file parameters (a, b, p, x, seed)

D

PB nB

Message (Mi)Ki

EPB (keyfile)

User A User B

108

2. Generate random numbers ri using LFSR. For LFSR, choose a primitive

or irreducible polynomial of order m over GF(p) so that maximal length

sequence can be obtained of period pm-1.

3. Map the generated random numbers ri to the EC points Pi using scalar

multiplication described in Section 3.3.

4. Get a symmetric key Ki which is either x or y coordinate or Least

Significant Bit (LSB) or Most Significant Bit (MSB) of eight

consecutive bits of x or y coordinates from the computed stream of EC

points.

5. Encrypt the message Mi using the symmetric key Ki.

6. Communicate parameters needed for generating the key stream to other

parties using ECC based public key system along with the cipher

message.

7. Decrypt the cipher message Ci using the same symmetric key Ki

generated by the other parties.

6.4.2 Algorithm

The proposed algorithm ‘ecStream’ is used for enciphering and

deciphering the message using key stream generated from EC points. At the

encryption end, symmetric key stream is exclusive OR with the plaintext

stream to produce the cipher stream. At the decryption end, the cipher stream

is exclusive OR with the same key stream to recover the plaintext stream.

Hence, stream ciphers can be sighted as computational analogy of an OTP

cipher, replacing a long secret key by short secret key file parameters. The key

file parameters needed for generating the key stream is ciphered using ECC.

This algorithm is pertinent for messages like text, image, video and speech

samples.

109

Algorithm ecStream( )

// Input : EC parameters (a, b, p, G), Coefficient x, seed, Input message Mi

// Output : Key stream Ki, Mi – Decrypted Cipher message Ci

{

// Key Distribution

// Let UA and UB be legitimate users

UA = {PA, nA} // Key pair for UA

UB = {PB, nB} // Key pair for UB

// Send the Public key of UB to UA

Send (PB, UA);

// Send the Public key of UA to UB

Send (PA, UB);

// Encryption at UA

Pi = genPoints (a, b, p)

Construct an irreducible polynomial using coefficient as x and a seed.

Generate ri from this irreducible polynomial

Qi = ri Pi // Scalar Multiplication

Ki = Qi {x, y, LSB(x), LSB(y), MSB(x), MSB(y)}

Ci = Mi Ki // Encrypt the message

Key file parameters are converted into EC points as Pkp

k = random ( )

EPB (keyfile) = {kG, Pkp+ kPB} // Encrypt the key file parameters

Send (Ci, EPB (keyfile), UB)

// Decryption at UB

Pkp= Pkp+ kPB – nBkG // Decrypt the encrypted key file

Compute key file parameters from EC points (Pkp) using discrete

logarithm

110

Pi = genPoints (a, b, p)

Construct an irreducible polynomial using coefficient as x and a seed.

Generate ri from this irreducible polynomial

Qi = riPi // Scalar Multiplication

Ki = Qi {x, y, LSB(x), LSB(y), MSB(x), MSB(y)}

Mi = Ci Ki // Decrypt the cipher message

}

The challenge of this proposal lies in the generation of key stream

and the secure distribution of the parameters needed for generating the key

stream through unsecured channels. In several of the existing algorithms, the

key will be of equal length as the message. In the proposed scheme presented

in this chapter, the key stream is never transmitted. Instead the key file needed

for generating the key stream is encrypted using ECC technique and

transmitted along with the cipher stream. The key file contains the parameters

a, b, p, x, and seed values needed for generating the key stream. For the EC

based stream cipher, the secret key is tailored after encryption of each pixel of

the plain image, and for each cipher image the key file parameters are also

changed. These factors are used to enhance the security of the proposed EC

based stream cipher.

In the proposed work, the security using ECC is realized on byte basis

only. Hence the work carried out for the image could be extended for the

video applications also. However for any embedded system work

environment, fast computing techniques with the help of hardware processing

system could be deployed.

111

6.5 RESULTS AND DISCUSSION

For demonstration purposes typical EC is represented by

y2 mod 841 = x3 - 5x + 25 mod 841 where a = -5, b = 25 and p = 841. The

generated points on the EC can be represented as (0, 5), (0, 836), (2, 368),

(2, 473), (5, 258), (5, 583), (8, 85), (8, 756), (9, 338), (9, 503), (11, 372),

(11, 469), and so on. The base point G is selected as (0, 5). Pi is affine point,

which is picked out of a series of affine points evaluated for the given EC.

However, for the purpose of individual identity, Pi is chosen differently for

every random number. Varying values of Pi can be chosen as part of an

exercise to work with ECC process on the given EC.

In the proposed EC based key stream generation, a sequence of

random numbers ri using LFSR is generated, which needs to be kept secret.

For LFSR, first construct an irreducible polynomial x4 + 14x3 - 940x2 + 1082x

+ 1076 of order 4 over GF(587) using randomly selected coefficient x as 1076

and a seed value as 1456. Then generate random sequences from this

irreducible polynomial that are represented as 495, 234, 14, 31, 261, 228, 17,

282, 481, 245, 267, 251, 276, 510, and so on.

Next the generated random number ri and the EC point Pi are scalar

multiplied, which is carried out with a series of doubling and additions,

depending on the value of ri. Efficient procedure can be adapted for optimal

number of doublings and additions. For example, the first EC point (0, 5) is

multiplied with the random number 495 to get an affine point (0, 836) which is

also an EC point. Similarly, other EC points are multiplied with consecutive

random numbers to obtain a point on EC. Some of them are tabulated in the

following Table 6.1.

112

The computed EC point (Qi) should fit into the EC. This conversion

is done for two reasons. First, the random sequence is mapped into an affine

point on an EC. Second, it will be entirely concealed from the hacker. These

steps are introduced to add some level of complexity for the key stream

generation process.

Table 6.1 Computed EC point

Generated Random Sequence

(ri)

Generated EC Point (Pi)

Computed EC Point (Qi)

495 (0, 5) (0, 836) 234 (0, 836) (603, 358) 14 (2, 368) (817, 525) 31 (2, 473) (171, 155) 261 (5, 258) (58, 24) 228 (5, 583) (544, 329) 17 (8, 85) (574, 338) 282 (8, 756) (116, 295) 481 (9, 338) (727, 600) 245 (9, 503) (669, 716) 267 (11, 372) (564, 633) 251 (11, 469) (292, 212) 276 (18, 411) (150, 32) 510 (18, 430) (646, 727)

From the computed EC points Qi, choose a symmetric key Ki from

any one of the methods specified in Table 6.2. For example, in method 2,

y coordinate of computed EC points is taken as a key stream Ki means the key

stream are 836, 358, 525, 155, 24, 329, 338, 295, 600, 716, 633, 212, 32, 727,

and so on. The same key stream Ki is exploited for encryption and decryption

procedure. The key stream generator plays a major role in a stream cipher for

the overall security.

113

Using the generated key stream Ki, any sort of messages like

document, text, or image can be encrypted as well as decrypted byte by byte.

In this work, (225×225) bmp standard test image of lena that has the size of

148 KB is considered as an example. The selection of key dictates the

complexity of encryption algorithm for breaking. A novel key stream

generation method based on LFSR and EC over finite prime field is

introduced, which is not part of any of the existing work on stream cipher key

generation.

Table 6.2 Key selection methods

Methods Method1

Method2

Method3

Method4

Method5

Method6

Ki Qi(x) Qi(y) LSB {Qi(x)}

LSB {Qi(y)}

MSB {Qi(x)}

MSB {Qi(y)}

where

Qi(x) - x coordinates of random EC point.

Qi(y) - y coordinates of random EC point.

LSB{Qi(x)} - LSB of x coordinates of eight consecutive random EC point.

LSB{Qi(y)} - LSB of y coordinates of eight consecutive random EC point.

MSB{Qi(x)} - MSB of x coordinates of eight consecutive random EC point.

MSB{Qi(y)} - MSB of y coordinates of eight consecutive random EC point.

Here, the secret key stream is tailored after encryption of each pixel

of the plain image and for each cipher image, the key file parameters are also

changed. The key file parameters are also encrypted using ECC based

114

technique and sent along with the cipher image. The discrete logarithm

concept is applied to recover the value of key file parameters. This also

increases the security of the proposed EC based stream cipher.

The software implementation of the EC based key generation for

stream cipher is done using Java. The input image and respective cipher

images are shown in Figure 6.3. The corresponding histograms are shown in

Figure 6.4.

Figure 6.3 Input and cipher images of bmp lena image

Figure 6.4 Histograms of input and cipher images of lena image

115

Table 6.3 lists the values of entropy and correlation between two

adjacent pixels. Computations for the plain and cipher images are carried out

using the procedure given in equation (6.2 - 6.6). The encryption time taken by

method 2 of the proposed stream cipher is 2620 ms. The selection of the

methods in Table 6.2 is of arbitrary choice and only the key size decides the

complexity of the encryption algorithm. From this, it is observed that the

proposed stream cipher can be efficiently used in real time multimedia and

wireless applications because it has simple structure and generates a key

stream faster than other generators.

6.6 SECURITY ANALYSIS

Application of computing power to encryption schemes is a

potential area of research. A good encryption scheme is the need of the hour

against the background of multi-nationalism. Globalization has opened up

frontiers, but cryptography has become more essential in the modern times. A

good encryption scheme should be insulated against possible attacks. An

analysis of encryption schemes such as key space analysis, statistical analysis,

correlation analysis and key sensitivity analysis ensures right development of

the security system.

6.6.1 Key Space Analysis

The key space that is being used for encryption must be large

enough to prevent the brute force attackers to intrude. For, the proposed EC

based stream cipher has a flexible, moderately large key space, which

comprises number of stages of LFSR over GF(p), initial values of LFSR,

feedback coefficients, possible elliptic curves and the base point. Hence for

116

this image encryption, this large key space is sufficient which is immune to all

kinds of brute force attacks.

6.6.2 Statistical Analysis

Statistical analysis generally depends on the measure of the

randomness of the cipher image. Also, it works on the relative frequency of

the occurred cipher image. It is eminent that a lot of ciphers have been

successfully analyzed with the help of statistical analysis and numerous

statistical attacks have been formulated on them. Hence, a perfect cipher

should be vigorous against any statistical attack. The following aspects related

to statistical attack are considered in this work.

6.6.2.1 Histograms

To prevent the leakage of information to an adversary, it is

important to ensure that cipher image does not have any statistical

resemblance to the input image. An image histogram shows how pixels in an

image are distributed by plotting the number of pixels at each intensity level.

In this work, the histograms are plotted for input and cipher images

as shown in Figure 6.4. The histogram of the input image has large spikes.

But, the histogram of the cipher image is nearly smooth and uniform,

representing almost equivalent probability of occurrence of each intensity

level. They are considerably different and tolerate no statistical similarity to

the input image. Hence, this does not give any hint to use any statistical attack

on the proposed stream cipher.

117

6.6.2.2 Entropy

The recital of the encryption algorithms is measured by computing

entropy of the input and the cipher images and then comparing them. Entropy

is defined to express the measure of uncertainty. The entropy Em of the image

is calculated as: 255

02 )(

1log)(i

m iPiPE (6.2)

whereimagetheinpixelofnumberTotalpixelaofoccurrenceofNumberiP )(

Table 6.3 Entropy and correlation of plain and cipher image

Table 6.3 gives the values of entropy calculated for the input and

the cipher images as stated by the formula specified in equation (6.2). The

entropy of the input image is 7.4569. It can be noticed from the table that the

entropy of the cipher image is extremely close to the theoretical value of 8.

Therefore, the information leakage in the proposed cipher is negligible and it

is secure upon the entropy attack.

Image Entropy Adjacent Pixels Correlation Coefficient

Vertical Horizontal Diagonal

Plain image 7.4569 0.9161 0.9561 0.8824

Cipher image 7.9952 1.586e- 4 0.0072 -0.0428

118

6.6.2.3 Randomness tests

The proposed key stream generator is based on the arithmetic

operation of EC and the properties of LFSR. The random sequences are

unpredictable and the period of the sequences is analysed theoretically. In

addition, sequences produced by the proposed key stream generator have

passed the Federal Information Processing Standards (FIPS) 140-2 statistical

tests of the Cryptographic Standards and Validation Programs (CSVP) at

NIST. As a result, statistical attacks are difficult to perform in the proposed

key stream generator. Based on these aspects, it is observed that the proposed

stream cipher is resistant against statistical attack.

6.6.3 Correlation Analysis

The correlation between two neighbouring pixels in horizontal,

vertical and diagonal orientations of input and cipher image is analysed. The

process is as follows: First, randomly pick M pairs of neighbouring pixels

from an image. Afterwards, calculate their correlation coefficient using the

following equation (6.3 - 6.6). Here, x and y are intensity values of two

neighbouring pixels in the image.

M

iix

MxE

1

1)( (6.3)

2

1)(

1)(

M

ii xEx

MxD (6.4)

)()(1),cov(1

yEyxExM

yx i

M

ii (6.5)

119

)()(),cov(yDxD

yxrxy (6.6)

To test correlation, 500 pairs of two neighbouring pixels are

selected randomly from the image. Table 6.3 presents the correlation between

two adjacent pixels for the plain and cipher images. It is observed that the two

neighbouring pixels in the input image are highly correlated, while there is a

negligibly less correlation between the two neighbouring pixels in the cipher

image.

6.6.4 Key Sensitivity Analysis

Even a change in a single bit of key will make a completely

different cipher image for the intruders to guess the key. This makes the

encryption procedure sensitive enough to the secret key. To prove the

heftiness of the proposed cipher, key sensitivity analysis is performed with the

following procedure.

Figure 6.5 Key sensitivity test

First, the input image in Figure 6.5(a) is encrypted by using the

secret key that is the initial values of LFSR as (14, -940, 1082, 1076) and the

cipher image 1 as shown in Figure 6.5(b). Then the initial values of LFSR is

changed to (14, -940, 1082, 1077) and the cipher image 2 as shown in

(c) Cipher image 2 (a) Input image (b) Cipher image 1

120

Figure 6.5(c). Finally, the two cipher images are compared. It is not easy to

compare the cipher images by simply observing these images. Thus for

comparison, correlation between the matching pixels of the two cipher images

is calculated.

Table 6.4 Entropy and correlation between two cipher images

The results of the correlation coefficients between the matching

pixels of the two cipher images using method 2 are given in Table 6.4. It is

clear that no correlation exists among cipher images corresponding to small

change in the key.

6.6.5 Algebraic Attack

Algebraic attack is a technique of cryptanalysis against a cipher. If

the key stream is linearly narrated even if the period is large, by knowing a

small section of key stream, it is likely to make a set of linear simultaneous

equations and solve for the entire key stream. In the case of random sequence

of EC points, the key stream is nonlinearly related. Therefore, it may not be

probable to make finite number of proper equations whose solution finally

escorts to knowledge of entire sequence. Therefore, the nonlinear key stream

Image Entropy Adjacent Pixels Correlation Coefficient

Vertical Horizontal Diagonal

Cipher image 1 7.9952 1.586e- 4 0.0072 -0.0428

Cipher image 2 7.9951 0.0134 0.0107 0.0372

121

makes the relation between input and cipher image nonlinear and provides

resistance against algebraic type attack.

From the above analysis, it is concluded that the proposed EC

based stream cipher is secure against brute force, statistical, correlation, key

sensitivity and algebraic attacks.

6.7 SUMMARY

In this chapter, EC based key generation for stream cipher is

proposed. The key streams are generated based on the combination of LFSR

and cyclic EC over a finite prime field. In this work, (225×225) bmp standard

test image of lena that has the size of 148 KB is considered as an example.

The input image and the respective cipher image histograms are conversed. It

is seen that cipher image does not have residual information and the histogram

is nearly smooth and uniform, offering good security for images. The entropy

and the correlation between two neighbouring pixels for the input and cipher

images are computed and analysed. The proposed scheme key space is

sufficient to resist all sorts of brute force attacks. Hence, the proposed EC

based image encryption algorithm is protected against brute force, statistical,

correlation, key sensitivity and algebraic attacks.

For the proposed cipher, the secret key stream is altered after

encryption of each pixel of the plain image and for each cipher image, the key

file parameters are also changed. The key file parameters are also encrypted

using ECC based technique and sent along with the cipher image. It is difficult

for an adversary to determine the key file parameters since the ECDLP is

considered difficult. These factors are used to enhance the security of the

proposed EC based stream cipher. The encryption time required for the

122

proposed scheme is estimated. It can be scrutinized that the proposed EC

based stream cipher can be a potential candidate for real time multimedia

applications.