manoj dr brau thesis

8/3/2019 Manoj DR BRAU Thesis

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RSA Cryptosystem

ElGamal Cryptosystem

Messey - Omura Cryptosystem

Knapsack Cryptosystem

Construction of Knapsack Cryptosystem

Quadratic Residue Cryptosystem

Hybrid Cryptosystem: Diffie - Hellmans key Exchange

Digital Signatures

A Classification of Digital Signature Schemes

Digital Signature Schemes with Appendix

Digital Signature Schemes with Message Recovery

RSA Signature Scheme

Feige Fiat Shamir Signature scheme

ElGamal Digital Signature Scheme

The Digital Signature Algorithm

The Schnorr Signature Scheme

The ElGamal Scheme with Message Recovery

Nyberg Rueppel Digital Signature Scheme

Digital Signature with Additional Functionality

Multi-signature scheme

Group Signature

Threshold Signature Scheme

Undeniable Signature Scheme

Blind Signature Scheme


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Proxy Signature Scheme

Directed Signature Scheme

Abstract of the Thesis

Introduction

A Directed Signature Scheme

Security of the Proposed Scheme

A Directed Signature with Threshold Verification

An Application to Threshold Cryptosystem

Introduction

A Directed Delegated Signature Scheme


Remarks

IntroductionDirected Threshold Signature Scheme


Remarks

Introduction

Threshold Signature Scheme with

Threshold Verification

Security of the Proposed scheme

Remarks


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Introduction

Directed - Threshold Multi Signature Scheme

Security of the Proposed SchemeRemarks

Introduction

Directed - Threshold Multi Signature Scheme Without SDC


Remarks

Introduction.

Generalized Directed - Threshold Multi - Signature Scheme


Remarks.

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A thesis submitted for the partial

fulfillment of the degree of

in

MATHEMATICS

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**Declaration**

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DEPARTMENT OF MATHEMATICS

Institute of Basic Science

Dr. B.R. Ambedkar University

Khandari, Agra-282002

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Certificate

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A CRYPTOGRAPHIC

STUDY OF SOME DIGITAL SIGNATURE SCHEMES

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Sunder Lal!"!"!"!"


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**Acknowledgment**

I am grateful to my Supervisor Dr. Sunder Lal, Prof. & Head, Department of

Mathematics, Institute of Basic Science, Dr. B.R. Ambedkar University, Agra, who

spares his valuable time in guiding me for my research work. He encourages me always.

I am short in word to express his contribution to this thesis through criticism, suggestions

and discussions. My sincere thanks are to Dr. Sanjay Choudhary and Dr. Sanjeev

Sharma, both senior lectures in the Department of Mathematics, Institute of Basic

Science, Dr. B.R. Ambedkar University, Agra, for their kind suggestions.

I am deeply indebted to Dr. J.P. Arya, HOD, Department of Mathematics, D.A.V.

College Muzaffarnagar, who had laid the foundation of my M.Phil. degree and then

encouraged me for Ph.D. degree.

There is no word to express my feeling for my family members and relatives,

especially to my parent for their hidden cooperation and to my wife Chhaya for her

enthusiastic inspirations, round the clock cooperation and help me in many ways. Really,

it is not possible to express the love and affection to sweet and little daughter Aayushi

Raghuvanshi, who is the driver of my success, to make the path for my Ph.D. work.

I am thankful to all the faculty members and staff of Hindustan College of Science &

Technology, Farah Mathura, Ravindra Kumar (T&P Officer), Jagadeesh. G. (Lecturer,

Computer Science), Sri Gopal Sharma, Sri Sarvesh B. Singh for their kind cooperation in

this electronic age, through computer, printer etc. I am also thankful to my research

colleagues Ms. Meeta Gurmukh, Mr. Atul Churvedi, Mr. Anil Agarwal and Mr. Amit K

Awasthi for their result oriented discussion.

At last but not the least, my sincere thanks to the writers of the books and the

research papers, which, I have consulted during the course of my research work.

&!'()!**+ ,./


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Chapter 0Introduction


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%$&%-%&&&&&&&%&-$%-@$&&

%%%%####,&&-%%%&

- $ & -$% $- $ B& & B& & B& & B& &

-%&&$%,&&-%&&$%,&&-%&&$%,&&-%&&$%,##'&%&

1%& && , 3#1%& && , 3#1%& && , 3#1%& && , 3# & %

$&%-%&&&&

+& % % % % #### ,& & -

%% %& - $ & -$% $-

$$%%,&&$%%,&&$%%,&&$%%,##

%$%-&

& $ ' % - % &


42/122

41

Chapter 7

@&&&$%)$&--

-% $ ' & , % & %

$ %&,& && $A$ %&-- && $ %&$ &7C%7C%7C%7C%AAAA+&&G$+&&G$+&&G$+&&G$AAAA

3$ 3%&3$ 3%&3$ 3%&3$ 3%& +& $ & C %A && $ A

$ %& & & % &&A $ P

$%&-$&%&-(%-&&$-$

% generated only by some specified subsets of members according to the signature

policy.

' & &$% & $ % &$% & $ %

C3-&)$&---%%

$& && - &

&-%$&%%

&-%&$,--7C%7C%7C%7C%AAAA&&&&&&&&

$$$$AAAA$%&$%&$%&$%&

573&&C3"3&

-% $ +& $- $ % &

%-&"3@&73&

-%$

$%$%,$&&$&$%$&$%$&$%$&$%

3#3#3#3#&&&&&$-%&&I%&

% % % % #### ,& & - %% %&

-$&-$%$-$'&%&

+&3#3#3#3#$-%%-%&

&&

+&%%%%####,&&-%%

%& - $ & -$% $- $ $$ $ $

%%,&&%%,&&%%,&&%%,##


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42

%$%-&

&$'%-%&

&&%%$%%-%$%

&$-%-&%&

The thesis ends with a list of references (papers, books and websites) that have been

consulted from time to time while completing the work.


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43

Chapter 1



45/122

44


1.1. Introduction

,& %$ $ & % & % $ %&

%$%&$&$,&&&

$ % ' & %&-, -- % $ %& % $ %& % $ %& % $ %&

%$ --% ' & %& & - % %&% &

$,&&&-$%

1.2. A Directed Signature Scheme

3$--&$,$&&

%%%&$%-&$

&-#,&%+&%-%

,

1.2.1. Signature Generation by A

-%1a

K 2a

K Q)%-$

@J 2aK ->J 1aK 2aK -

"&-$%&$%

A,&&$%&%-$%$J& 1aK

%%-$ AS J 1aK U Ax Ar )" Ax &-&

K AS @>L&$&

1.2.2. Signature Verification by B

&- Bx %-$J>@ Bx -%

Ar J&


46/122

45

%&%&%$% AS ?

Ar -$'&

&K AS @>L$

1.2.3. Proof of Validity by B to C

-% KQ)%-$@#J K ->#J#K-

#

#$@#>#-%@>%&%&$$

& % +& $ %-%, $A

%;

1.3. Security Discussions

'&$A%,&%$&%$--%3$%3$%3$%3$3%&3%&3%&3%&

#&% Ax 1aK % $ Ar &

)$AS J 1aK U Ax Ar )O

"&$$,-&+&$)$

%-$%%&% Ax

1aK %$ Ar &)$

#-&O

-&%,

1K 2K Q)%%$

@J 2K ->J 1K 2K -J& 1K

$,&$,&% Ax %$ AS

&%)$ AS ?

E Ar F-

%%%%#$K AS @>L&)$

AS ?

E Ar F-O


47/122

46

%&%-$&&&$ Ar

$%& & Ar J & ) .$ %-$ & AS

)$&%&-+&$&%,

$%%$

Illustration

+$&%&-J)J;;J+&%-$%

$,

3%-$%3%-$%3%-$%3%-$%

0 ;

?

#


48/122

47

E:?F@$&$&$-7--

iRx -$%

iRy

+&%&%&,-

1.4.1. Signature Generation by A

-%1a

K 2a

K Q)%-$

@J 2aK ->J 1aK -

A,&&$%&%-$%$

J&> AS J 1aK U Ax Ar )

" Ax &-&

%%-(J1a

K U;(UNA;(A;),&1a

K J

%-$-$%$iR

v %&&$-7

iRv J

iRu 2a

i

K

Ry -

"iR

y -$%iR

u &-$%$%,&%&$

&$-7

K AS @n

iRiv 1}{ = L&$-7&$

1.4.2. Signature Verification by the Organization R

$" $ $-7 $ % &

$@$ & & % ,&%&%%-

%-$ %& $ " & $ +&

-%,

1%&$"%&R&%&iR

u JiR

v iRx

RW -

1%&$"&R&&,


49/122

48

iRMS J

iRu q

uu

u

ji

i

RR

Rt

ijj

mod,1

=

%1%&$"$&R&&,iR

MS %%$&-

$iR

R J iRMS )&%

+&%%-$J pRk

i

Rimod

1

=

%J&)

+&%%&%&%$% AS ?

ArAy -$

'&&KK AS @n

iRiv 1}{ = L$&

+&--%&&&,%&%%*

+&$3#,&&&$

%& & $ 3# & & %$

-$&%&%&$&

+&%&%$-%&

%&$

+&%&%$-%&&

+&%$&%$-%&A

%+&(&

+& % %& & && $ &

%$

+&)$&$&$-7---$%-

iRx

iRy

1.5. Application to Threshold Cryptosystem

+&&%- %P%-+& $$,&

-%)$&%&-+&C&


50/122

49

-$%+&%$&$C$%&

,&$%&-%$,& P;

$,&&%%$&%

44 E;F - &%%- &&

%-+&I,&%-,*

+&$-%(/$$- &$%

%&$-%%&$&

+&%$-(+&%&

$ & $- %& A$- %$

-%)$D$$-I%&%%

.$%$%&%$&&%-$&,3$--$,%-&

&$$-7$&$%-%

& +& %- %- H & %

$-7 % % > -% +& %

KK AS @ %n

iRiv 1}{ = L&$-7,&%J1HHJ&>

% % & %- H J & +& & %

%-&H%J

+& -- && %- & & , &

%-&%%-

+& $- % ( % %& %& $&

%$%

+&($-+&&

&%-

$ % %-& ( %& $- ,& %$-%


51/122

50

Chapter 2

A Directed Delegated Signature Scheme


52/122

51

A Directed-Delegated Signature Scheme

2.1 Introduction

&%-%%&,'&%%,

% & )$ -( $ %&& &$&

,&%& %- % -

$%&%$E0F,&%&&%-

,'&%&-%%

,& & - &% & $- & $

+&&&%-,&$

&%$%&&&%-%&$&

,&&&-&-(

. & & & & $ ,& & &

$%%$$#$,&

, , & %- & &

$ % & $ , % & -( $

%&,&%$%&%%%%AAAA$$$$

%&%&%&%&

'&%&-,--%P$%&,&%&&

,-&+&-&A+&-&

-($%&%-&

, +& % -& A ' &

%,$C&-%-$&-&

' & %&- , $ --% & -- %%%%AAAA

$ %& $ %& $ %& $ %& % % $& ,&

$$&$%%&&+

%%&$#,%%&%%&%&


53/122

52

$,&$&&-$#%-&&$

&-4,&%

2.2. A Directed Delegated Signature Scheme

# $ ,& & B7.I &- ,&%& % '3%&%$- +& &- & #G. ,& & -

-" $ %$ $ %% & &

-#,%&$&R&%&%A$-&R&,

%% & % ,&$ & &- $ # "'>

- & & %$ & & -

&-&B7.%&4@&-G%

$%'3-#&-&

%%-

+&$$%&-%A$

%&"&%#%%&$%-

&-4,&%3%%$%

,& & &- % # , % & & %

%%&,&$&$%+&,

(-&,-(%$%&%-$

%$%

C&%&%$---"

$$&,-%%&-

$-)-%-P;Q-,&)

,&&$%&

2.2.1.Signature Key Delegation by A

;%Ak Q)%-$J Ak -

%aQ)%-$Ja-

'Q)X&&,-

%-$J(U Ak ),

0%-$3JUa)%&%3J-


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53

')$&%%-WI$

B, &&- - &-$% & %&%$- -#

---%-#+&$

%&,-%

2.2.2. Signature Generation by B for C

-%1b

K 2b

K Q)%-$

@J 21 bb KK -Q%J# 1bK -

%-$ Br J&Q%@ BS J 2bK 3)

%KB

S @B

r L##G.I$

2.2..3. Signature Verification by C

#%-$bJ BS Ay Br @-Q%Jb Cx -

#%&%& Br J&Q%@&$

2.2.4. Proof of Validity by C to Y

#KAS

@

A

r

bL4

4%&% Ar J&Q@)

'&&4-&-%/&,&(-

%#C,&-4&bQ#J#,

4%&$Q-%-$,Jbu v -,#

#%&aQ-%-$cJ, -Jc Bx -

4

4$,&%%&,Jbu v -

#a4,&%&&%&

cJbu v U-JQ#u # v U -


55/122

54


'&%,%$-%

'&-(&&&%%&&

&R&$&R&%&%

+& $ & - & (% $ & - +&

&,,-

$&-%$&%+&&--&

% I %&% %

I

# 2bK 3%$-(&)$

BS J 2bK 3)O

"& $ $,- ,+&$ )$

%-$%%&%

2bK 3

%%%%#-&O

-&%,

1iK

2iK Q)$,&$,&%-a

%$ -( $ 3 BS &

%)$

Q%J BS Ay Br @ Cx - Br J&Q%@

#$K BS @ Br L$&)$

bJ BS Ay Br @-O

+ %-$ & BS & )$ )$ &

%&-'% *S K *S

@ Br L&%%-$b XJE*S Ay Br @F

-QXJbX Bx -%%&%


56/122

55

Br J&QX@%&

Illustration

@$&%&$-+-J)J;;J


57/122

56

2.4. Remarks

'&%&-,&%$%A$%&,&%&

$$ &%,&& $% '

&%&&$%#&$%&$%

-%B % %&% &$ ,&$& %A- +&

%-&&$,&%

@&-%$%$%&%$%&,&%&

% & - "% & %$ & %&

& & %& % & 3% &

,&I % ,&%&

%$&&%&&%&


58/122

57

Chapter 3

Directed Threshold Signature Scheme


59/122

58

Directed Threshold Signature Scheme

3.1. Introduction

'$&&C

)$&--%--'&%&$

& % & & - %

(-&-%%,&%&)$&$

& - 3$%& -% %$ - & -

$)$$&$%&

& ,& & $ +&&+&&+&&+&& $$$$

, & -+& &&$ %& $

&-+&&$%&%%-&&

%--&$%E;F';::;

EF--&&&$%&&3

$-

'&%&-,--%%%%AAAA&&$%&&&$%&&&$%&&&$%&

3&I&&$%&E:=F3%&I$%&E:0F

3.2. Directed -Threshold Signature Scheme

57$-$$,&%& %

& $ $ +&$ % &

$ & % - & - # ,&

%B%%&%&&$,&$&&-

$ %$%,$ & (% $ &$

%3#,&%&&$-%-&%&

7 5" $ 7 % @ $ &(% %# ,& %%- $ %&

$&$$-"1&&&$-&)$$&,&

&$-%+& &%$)$$

&&%,


60/122

59

+&%&%&,-

3.2.1. Group Secret Key and Secret Shares Generation

3#%&$--$%--)%,&&

$%&3#%-

(JU;(UNA;(A;),&J Gx J

"Gx &%&$-7

3#%-$&$--$% Gy Gy J-

%3#%-$%&%&&$-7

J$)"$&-$%$%,&$&$-7

3#%&$%

3.2.2. Signature Generation by any tUsers

'$&$-$&

&$$,-*

1%&% 1iK 2iK Q)%-$

,J 2iK 1iK -CJ By 2iK -

1%& , - C % %&

".%,C%-$

Q@

@J qwHi

i mod

QJ qz

Hi

i mod

J&Q@)

%1%&&R&&G3J quu

u

ji

jt

ijj

mod,1

=


61/122

60

1%& $ &R&&G3 1i

K

%%$&-$J1i

K PG3)

1%& &R&-$&%#

,&%%&-$-$%&$-$

3J=

t

i ,1

)

#K3@L$&$-7&

%-$bJES G

y R @F-QJb Bx -

%&%&$J&Q@)

3.2.4. Proof of Validity By B to Any Third Party C

+&-&-%$%A0


,, %$ %$-% & --%%%% AAAA +&& 3$+&& 3$+&& 3$+&& 3$

3%&3%&3%&3%&

'-&$-%&$--$%

Gy OB%$&%$%&-

# &% & 7&-$%$ $ OB

%$%%-

(c). Can one retrieve the secret shares vi ,integer1i

K and partial signature si , from the

equation si =

1iK P

MS

i.R mod q. ?

"&$$,-,+&$)$

%-$%%&%&

1i

K -$7


62/122

61

(d). Can the designated combiner DCretrieve the group secret key f(0) or any partial

information from the equation, S = =

t

i ,1

si mod q ?

+&%-$

#-"O

-&& %,

1i

K 2i

K Q)%,C$,&$,&

%&%$-$&

%)$

QJES G

y R @F Bx -,&3J=

t

i ,1

)

#$K3@L&,)$

bJES Gy R @F-O

+%-$3 &)$&%&

-'% *S K *S @QL&

%,$%-$

bXJE*S

Gy R @F-QXJbX Bx -%&%

Br =?

&QX@

+&%%&&&$

#&&%%$%$%&-

(O

%%&,)$

(J quu

uxuf

ji

jt

ijj

t

i

i mod)(,11

==


63/122

62

&%- (%%$%,&&,

%&$7

3 C & && , %& & &

- &%$%$$&,$

&%&%-,&%&,%$$

+& % , & %$ $ %& & & &

$ $ & & %$ & $

& & && $", ,&- $ &

$%%-%-$&%

Illustration

$,$&$$&.$$$

# 1 ,$ % & %

$&$,&%-$%- Bx JJJJ


64/122

63

+&$%1a

K J2a

K J?%-$ 1w J 1z J;

+&$%1f

K J82f

K J:%-$ 4w J0 4z J:

% & & $ 1w 4w 1z 4z -$%

&$&% %&.% 1w 4w 1z 4z

%&$"%-$&-$%@J;QJ;


65/122

64

$%&$-%&,$

&$-%$$%%$

'&%##,&%%&-$&@

&$&&%%,&$

% %-$ & & $ ) ' ) - & &

%%$ & (- -f ),&%& - +&

- &5 - %%$ &&,,

, (%- ,& ) J ,&%& % , # &

$,& =

t

ijj

ji uu,1

)( )%-'&%&,

$ &$-% =

t

ijjji uu,1 )( )+& -

% & -) $ %& - $ &

%$

%&&$&-%$&

,&-+&, Q$%

- & $ 3$ % & - &

%-%C,&-&$


66/122

65

Chapter 4

Threshold Signature Scheme with Threshold

Verification


67/122

66

Threshold Signature Scheme with

Threshold Verification

4.1. Introduction

'$&&-",

,& & C & C

)$ & -- % -- '& % &

$ % & % &&-+&%&---+&&$%&+&&$%&+&&$%&+&&$%&,&,&,&,&

&&%&&%&&%&&%

4.2. Threshold Signature Scheme with Threshold

Verification

#&C3&C

)$&--$"3

$-73$&C3

. & & & & $- , & $

$%&,&&$%$"

$$-7$&C&

&&%%&$-$-

@$&&%&C3 Sx &-$%

Sy,&

SyJ

Sxg -

,&Sx

Q)

3&C

-

-Rx Ry - Rx -$% Ry J

Rxg -'

$&&C-- Ax Ay ,& Ax % Ay J

xg --$%


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67

@$&$&&&C3&%$

%#+#&$-%-&,$-&

%&+&%&%&,-

4.2.1.Group Secret Key and Secret Shares Generation for theOrganization S

#+#%&$--$%--)%,

&&$%+#%-3(&$-73

3(JU;(UNA;(A;),&J Sx J3

#+#%-$&$--$% Sy J3-

(c). #+#%KQ)%-$-$%$@J K -

#+#%-$-$%$iS

v %&&$-73

iSv J3

iSu

K

Siy -

"iS

y -$%iS

u &-$%$%,&%&$&

$-73

#+#K iSv @L%&$&$-73&$&-$%%&

4.2.2. Group Secret Key and Secret Shares Generation for the

Organization R

#+#%-(&$-7

(JU;(UNA;(A;),&J Rx J

#+#%-$&$--$% Ry J-

%#+#%-$-$%$iR

v %&&$-7

iR

v JiR

u K

Riy -


69/122

68

"iR

y -$%iR

u &-$%$%,&%&$

&$-7

#+#KiR

v @L%&$&$-7&$&-$%%&

4.2.3.Signature Generation by any tUsers

'"3$&C 3$,&

&C&&$&

&,-*

1%&$"3%1i

K 2i

K Q)%-$

$J 2iK -J 1iK -,J 1iK Ry 2iK -

1%&$%$,-$%%&$"3

.%$,%&"3%-$&-$%

3>3@3&&$3

3J quSHi

i mod

>3J qvSHi

i mod

@3J qwSHi

i mod 3J&>3)

%1%&$"3%&R&%&3iS

u JiS

v iSx

W -

1%&$"3&R&&,iS

MS J3iS

u quu

u

ji

j

SS

St

ijj

mod,1

=

$ &R& &,iS

MS %& $ "3 %-$ &R&

-$J 1iK U iSMS 3)

1%&$ "3 &R&- $ +#,& -$%

$-$33J=

t

i ,1

)


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69

#+#K333@3L&%%%%####CCCC

$&$-3&

4.2.4. Signature Verification by the Organization R

$"$$-7%&$@

$ & & % % % % #### % &% &% &% &

&$-&$-&$-&$-7777&&&C&&&C&&&C&&&C,&%%-

%-$ %&$ " & $ +&

-%,

1%&$"%&R&%&iR

u JiR

v iRx

W -

1%&$"&R&&,iR

MS JiR

u quu

u

ji

i

RR

R

k

ijj

mod,1

=

%1%&$"&&,iR

MS

#%-$J=

k

i

iRMS

SSUW

1

. )%

3J&)

#%&%&%$% SS ?

SR

Sy -$'&

&K333@3L$&


' & $A% , & %$ & %$ -% -- +&&+&&+&&+&&

3$3%&,&&&%3$3%&,&&&%3$3%&,&&&%3$3%&,&&&%",&%$-

%$&%$%%$$

#&CI% Sx Rx &$--$%

Sy Ry -%O

+& %$ % & -B % &

% Sx Rx % 3 &%%

-+#.&&&$&-$% Sy Ry


71/122

70

&%Sx Rx %$&%$

%&-

# &% & 3 iS

u 73 &

)$iS

v J3iS

u KSiy -O

B %$ 3 %%- H

%%+#3%&

%&iR

u 7&)$

iRv J

iRu

K

Riy -

(c). # & % & 3 iS

u 73 &

)$3iS

u JiS

v iSx

W -O

.&$%%&%&3iS

u %$3

%%-iS

x %&$733

%&%&iR

u 7&)$

iR

u JiR

v iRx

W -

#&&,iS

MS 1i

K &&&$3

-$73&)$

J1i

K UiS

MS 3)O

+&%-%-$

%%&iS

MS 1i

K &&&$3-$73

#+#-&)$

33J=

t

i ,1

)O

.$ ,$ %-$ #+#


72/122

71

#-"3O

-&& "3%

, 1i

K 2i

K Q) %$ , $,&$

, & % & 3 iSu %$ -

$&%)$

33J=

t

i ,1

) SS ?

SR

Sy -

# $ K33 3 @3 L & , )$

SS SRSy -O

%&%-$&&&$

3$%&&3J&)

.$ %-$ & SS )$ & %

&-.&&&&%% SS

&$3&&)$

SS SRSy -

",%%&A,--&&&$% &)$

-+&$&%,$%%$

&&&&&%%$%$%&-3(O

%%&)$3(J quu

uxuf

jSiS

jSt

ijj

t

iiSS

mod)(,11

==

&%

-3(%%$%,&&, %&

3 iSu 73 3C&&&,%&

& &%$&%-3+&%

&, , & %$ $ %& & & &

$ $ & & %$ & $

&&&&$


73/122

72

Illustration

$ 3$-- =SG ? =SH 0 =RG 51

31

iS

y iS

x

iSu

3iS

u iS

v

P3; ? ; = 0

P3 ? ; : 0

P3 = ;0 = =

P30

;= ;


74/122

73

>51

31iR

y iR

x iR

u iR

u iR

v

P; : ;8 ;; ; ;0

P 0 : 8 : 8

P ? ;; = ; ;


75/122

74

%3R

u J;%-$3R

MS J0

%0%4R

u J;8%-$4R

MS J;;

8%5R

u J;%-$5R

MS J:


76/122

75

Chapter 5

Directed-Threshold Multi-Signature Scheme


77/122

76

Directed-Threshold Multi-Signature Scheme

5.1. Introduction

'&&%&'&&%&'&&%&'&&%&,& &&%%$&%-&&&&$'&%&

%$&-&$

- % & $ ,& & &&

%&--&-%

'$$%&'$$%&'$$%&'$$%& &$$&

&&$$&,&&&-

& $$ $% -$

&&$ %&$# &$,&

$- $ ,$ & $$ +&

&$-$-$&&-$%%

@&%%&&

&%&$-$&&,,&&

$ % & $- %$ & &

&-'&%&$$%&%&-

&,&&&$%&

.&&&&$,&&

&$%)$&--%

--#&,--$%&

%A+&&G$A3$3%&

5.2. Directed - Threshold Multi - Signature Scheme

' & %&- , % & && $&& $&& $&& $

%&$$%&$$%&$$%&$$%&%&%&%&,&%$%&,&%$%&,&%$%&,&%$%&--,-$%&%& %%%%AAAA+&&G$+&&G$+&&G$+&&G$AAAA3$3$3$3$

3%&3%&3%&3%&

$%$%,$&&$&$%$&$%$&$%$&$%

3#3#3#3#,&%&&$-%%&&&


78/122

77

% %% %#### ,& & - %% %&

-$&-$%$-$

+&%&%&,-

5.2.1.Group Secret Key and Secret Shares Generation for the

Organization S

3#3#3#3#%&$--$%--)%,

&&$%&3#%-

3(JU;(UNA;(A;),&J Sx J3

3#3#3#3#%-$&$--$% Sy J3-

(c). SDC randomly selects K Zq and computes a public value W = g K mod p.

3#3#3#3#%HQ)%-$JEHU3iS

u F)

""""iS

u &-$%$%,&%&$&-$%$%,&%&$&-$%$%,&%&$&-$%$%,&%&$&$-&$-&$-&$-77773333

3#%-$-$%$iS

v %&&$-73

iSv J

K

Siy -

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108/122

107

Appendix

3$&---$&R$-$%*3$&---$&R$-$%*3$&---$&R$-$%*3$&---$&R$-$%*

--$&*--$&*--$&*--$&*

%& ' In South East

Asian Journal of Mathematics and Mathematical

Science2 (1), !"#$%

(& In the

proceeding of National conference on Information

Security,Sponsored by DRDO,!!$&'!"$

-$-$-$-$-$%*-$%*-$%*-$%*

%& Communicated toAligarh

Math Bulletin.

(&

Communicated to J. of Applied and Pure

Mathematics, New Delhi.

)& ' Communicated to J.

of Natural Science Grukhul Khagri University

Haridwar.

*& 'Communicated to GANIT SANDESH, J. of Rajasthan Ganit

Parishad.

+&

Manuscript.


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References


110/122

109

'

1. Adleman L.M., Pomerance C.and Rumely R.S. (1983). On distinguishing prime from the

composite numbers, Annals of Mathematics - 117, p.p.173-206.

2. Bellare M. and Michali S. (1988). How to sign given any trapdoor function, Proceeding of20th

STOC-ACM, p.p. 32-42.

3. Biehl I., Buchmann, J. A., Meyer, B., Thiel, C. and Thiel, C. (1994). Tools for proving zero

knowledge, Advances in Cryptology -EuroCrypt - 94, Springer Verlag, p.p. 356-365.

4. Blakley G.R. (1979). Safeguarding cryptographic keys, Proceeding, AFIPS 1979 Nat. Computer

conference - 48, p.p. 313-317.

5. Blake I.F., Van Oorschot P.C. and S.Vanstone. (1986). Complexity issues for public key

cryptography, Performance limits in communication, Theory and Practice, NATO ASI Series

E: Applied Science - 142, p.p. 75 97.

6. Blum L., Blum M. and Shub M. (1986). A Simple unpredictable pseudorandom number

generator, SIAM Journal on Computing - 15 (2), p.p. 364-383.

7. Boyar J., Chaum D., Damgard I. and Pederson T. (1990). Convertible undeniable signatures,

Advances in Cryptology Crypto - 90, Springer Verlag, LNCS # 537, p.p.189-205.

8. Boyd C. (1986). Digital multi-signature, In Cryptography and Coding. Editors, Beker H.J. and

Piper F. C., Clarendon Press, London, p.p. 241 246.

9. Burmester M. V. D., Desmedt Y., Piper F. and Walker M. (1989). A general zero knowledge

scheme, Advance in Cryptology- Eurocrypt - 89, Springer Verlag, p.p. 122-133.

10. Camenish J.L., Piveteare J.M. and Stadler M.A. (1994). Blind signature based on discrete

logarithm problem, Advance in Cryptology-Eurocrypt - 94, Springer Verlag, p.p. 428-432.

11. Chang C.C., Jan J.K. and Kowng H.C. (1997). A digital signature scheme based upon the

theory of Quadric Residues, Cryptologia - 21 (1), p.p. 55- 69.

12. Chaum D. (1982). Blind signature for untraceable payments, Advances in Cryptology Crypto

- 82, Springer Verlag, p.p. 199-203.


111/122

110

13. Chaum D. and Van Autwerpan H. (1989). Undeniable signatures, Advance in Cryptology-

Eurocrypt - 89, Springer Verlag, p.p. 212-216.

14. Chaum D. (1990). Zero knowledge undeniable signatures, Advance in Cryptology-Eurocrypt -

90, Springer Verlag, LNCS # 473, p.p. 458-464.

15. Chaum D. (1991). Group signatures, Advance in Cryptology-Eurocrypt - 91, Springer Verlag,

p.p. 257-265.

16. Chaum D. (1995). Designed confirmer signatures, Advance in Cryptology-Eurocrypt - 94

Springer Verlag, LNCS # 950, p.p. 86-91.

17. Chaum-m. Li, Hwang T., Lee N. and Jiun-Jang Tsai (2000). (t, n) threshold multi-signature

scheme and generalized multi-signature scheme, where suspected forgery implies

traceability of the adversarial shareholders, Cryptologia 24(3), p.p. 250-268.

18. Chen L. and Pederson T.P. (1994). New group signature signatures, Advance in Cryptology -

Eurocrypt - 94, Springer Verlag, p.p.171-181.

19. Damgard I.B. (1987). Collision free hash function and public key signature scheme Advance

in Cryptology - Eurocrypt - 87, Springer Verlag, p.p. 203-216.

20. Desmedt Y. (1988). Society and group oriented cryptography, Advances in Cryptology

Crypto - 87, Springer Verlag, p.p. 120 - 127.

21. Desmedt, Y. and Frankel Y. (1990). Threshold cryptosystems, Advances in Cryptology Crypto

- 89, Springer Verlag, LNCS # 293, p.p. 307-315.

22. Desmedt, Y. and Frankel Y. (1991). Shared generation of authenticators and signatures,

Advances in Cryptology Crypto - 91, Springer Verlag, p.p. 457-469.

23. Desmedt Y. (1994). Threshold cryptography, European Transactions on Telecommunications

and Related Technologies - 5(4), p.p.35 43.

24. Diffie W. and Hellman M. (1976). New directions in Cryptography, IEEE Trans. InformationTheory - 31, p.p. 644 - 654.

25. Diffie W. (1988). The first ten years of Public Key Cryptography, In Contemporary

Cryptology: The Science of Information Integrity, Editor, Simmons G.J. IEEE Press, New York.

p.p 135-175.


112/122

111

26. Dowland P. S., Furenell S. M., Illingworth H. M. and Reynolds P. L. (1999). Computer crime

and abuse: A survey of public attitudes and awareness, Computer and Security - 18(8), p.p.

715-726.

27. Due Liem V. (2003). A new threshold blind signature scheme from Pairings.

www.caislab.icu.ac.kr

28. ElGamel T. (1985). A PKC and a signature scheme based on discrete logarithm, IEEE trans

information theory - 31, p.p. 469 - 472.

29. Fiat A. and Shamir A. (1986). How to prove yourself, Practical solution to identification and

signature problem, Advances in Cryptology - Crypto - 86 Springer and Verlag, LNCS # 263,

p.p. 186-194.

30. Frankal Y. and Desmedt Y. (1992). Parallel reliable threshold multi-signature, Tech report,

Dept of EE and CS, University of Wisconsin.

31. Gennaro R., Jarecki Hkrawczyk S. and Rabin T. (1996). Robust threshold DSS signature,

Advances in CryptologyEuroCrypt - 96, Springer Verlag, p.p. 354 - 371.

32. Gnanaguruparan G. and Kak S. (2002). Recursive Hiding of the secrets in the visual

Cryptography, Cryptologia - 25 (1), p.p. 68 75.

33. Goldrich O. (1986). Two remark concerning the GMR signature scheme, Advance in

Cryptology - Crypto 86, Springer and Verlag, p.p. 104 - 110.

34. Goldwasser, Michali S. and Yao A. (1983). Strong signature scheme, Proceeding of the 15th

STOC - ACM, p.p. 431 - 439.

35. Goldwasser S., Michali S. and Rivest R. (1985). A paradoxical signature scheme, 25th IEEE

symposium of foundation on the computing, p.p. 441 344.

36. Goldwasser S., Michali S. and Rivest R. (1998). A digital signature secure against adaptive

chosen message attacks, SIAM Journal on Computing - 17, p.p. 281-308.

37. Gordon J. A. (1985). Strong primes are easy to find, Advances in Cryptology -Eurocrypt 84,

Springer-Verlag, p.p. 216-223.

38. Guillou L.C. and Quisquater J.J. (1988). A paradoxical identity based on signature scheme

resulting from zero knowledge, Advances in Cryptology Crypto 88, Springer-Verlag, p.p. 216-

231.


113/122

112

39. Guillou L.C. and Quisquater J.J. (1988). A practical zero-knowledge protocol fitted to security

microprocessors minimizing both transmission and memory, Advances in Cryptology

Eurocrypt - 88, Springer-Verlag LNCS # 330, p.p.123 - 128.

40. Hard jono T. and Zheng Y. (1992). A practical digital multi-signature scheme based on

discrete logarithm, Advance in Cryptology Auscrypto 92 Springer and Verlag, p.p.16-21.

41. Harn L. and Kiesler T. (1989). New Scheme for Digital Multi-signature, Electronic Letters - 25

(15), p.p. 1002-1003.

42. Harn L. and Yang S. (1992). Group oriented undeniable signature scheme without the

assistance of a mutually trusted party, Advance in Cryptology- Auscrypt - 92, Springer and

Verlag, p.p. 133-142.

43. Harn L. (1993). (t,n) threshold signature and digital multi-signature. Proceeding of Workshop

on Cryptography & data security, Chung Cheng Institute of technology, ROC, p.p. 61 73.

44. Harn L. (1994). Group oriented (t, n) threshold signature scheme and digital multi-signature,

IEEE, Proc computer digit tech - 141(5), p.p. 307-313.

45. Hellman M. E. (1979). The mathematics of public key cryptography, Scientific American -

241, p.p. 130-139.

46. Hill L. (1929). Cryptography in an algebraic alphabet, American Mathematical Monthly 36,

p.p.15-30.

47. Hwang M., Lin I. and Lie E.J. (2000). A secure nonrepudiable threshold signature Scheme

with Known Signers, International Journal of Informatica - 11(2), p.p.1- 8.

48. Hwang T. and Chen C.C. (2001). A new proxy multi-signature signature scheme, International

Workshop on Cryptography and Network Security, Taipei, p.p. 26 28.

49. Hwang T., Li C. and lee N. (1993). Remark on the threshold RSA signature scheme, Advance

in Cryptology - Crypto 93, Springer and Verlag, p.p. 413-419.

50. Hwang T., Li C. and lee N. (1995). (t, n) Threshold signature scheme based on discrete

logarithm, Advance in Cryptology Eurocrypt - 94, Springer and Verlag, p.p. 191-200.

51. Itakura K. and Nakamura K. (1983). A public key cryptosystem, suitable for digital multi-

signatures, NEC Research and Develop, p.p.1- 8.


114/122

113

52. Jac C.C. and Jung H.C. (2002). An identity based signature from gap Diffie Hellman groups,

www.iacr.org

53. Jackson, W. A., Martin, K. M. and O' Keefe, C. M. (1995). Efficient secret sharing scheme

without mutually trusted party, Advance in Cryptology Eurocrypt - 95, Springer and Verlag,

p.p. 183-193.

54. Kim J. and Kim K. (2001). An efficient and provably secure threshold blind signature scheme,

www.caislab.icu.ac.kr.

55. Kobolitz N. (1987). Elliptic curve cryptosystem. Mathematics of Computation - 48, p.p.203-209.

56. Lal S. and Kumar M. (2003). A directed signature scheme and its application, Proceedings,

National Conference on Information Security, New Delhi -2003, p.p. 124 132.

57. Lal S. and Kumar M. (2003). Some applications of directed signature scheme, South East Asian

Journal of Mathematics and Mathematical Science - 1(2), p.p.13-26

58. Lal S. and Awasthi A.K. (2003). A scheme for obtaining warrant message from the digital

proxy signature scheme, Report No- 2003/73. http://www.eprint.iacr.org

59. Lal S. and Awasthi A.K. (2003). Proxy blind signature scheme. Report No- 2003/72.

http://www.eprint.iacr.org

60. Lanford S. K. (1995). Differential linear cryptanalysis and threshold signatures, Advance in

Cryptology Crypto - 94, Springer and Verlag, p.p.17-25.

61. Lanford S. K. (1995). Threshold DSS signatures without a trusted party, Advance in

Cryptology - Crypto 95, Springer-Verlag, p.p. 397-400.

62. Langford, S. K. (1996). Weaknesses in some threshold cryptosystems. Advance in Cryptology

Crypto - 96, Springer and Verlag, p.p. 74- 82.

63. Lee B. and Kim H. (2001). Strong proxy signature scheme and its applications, Proceeding of

SCIS, p.p.603- 608. http://www.citeseer.nj.nec.com

64. Lehmer D.H. and Powers R.E. (1931). On factoring large numbers, American Mathematical

Society, p.p.770-776.

65. Lidl R. and Mueller, W. B. (1983). Permutation polynomials in RSA cryptosystems, Advance

in Cryptology Crypto - 83, Springer and Verlag, p.p.293-301.


115/122

114

66. Lim C.H. and Lee P.J. (1993). Modified Maurer-Yacobis scheme and its Applications,

Advance in Cryptology Auscrypt - 93, Springer and Verlag, LNCS # 718, p.p. 308 323.

67. Lim C.H. and Lee P.J. (1996). A directed signature scheme and its application to threshold

cryptosystems, Security Protocol, In Proceedings of International Workshop, (Cambridge, United

Kingdom), Springer-Verlag, LNCS #1189, p.p. 131-138.

68. Luciano D.M. and Prichett G.D. (1987). Cryptology: From Caesar cipher to public key

cryptosystem, College Mathematics Journal - 18, p.p. 2-17.

69. Mambo M. Usuda K. and Okamoto E. (1996). Proxy signature: Delegating of the power to sign

messages, In IEICE Transaction E79-A (9), p.p. 1338 1356.

70. Maurer U. (1990). Fast generation of secure of RSA- moduli with almost maximal diversity,

Advances in Cryptology Eurocrypt - 89,Springer and Verlag, LNCS # 434, p.p. 636 - 647.

71. Merkle R.C. and Hellman M.E. (1978). Hiding information and signatures in trapdoor

knapsacks, IEEE transactions on the information theory, IT 24, p.p.525- 530.

72. Merker R. C. (1987). A digital signature based on convential encryption function, Advance in

Cryptology Crypto - 87, Springer and Verlag, LNCS # 293, p.p.369-378.

73. Michali S. (1992). Fair public key cryptosystems, Advance in Cryptology Crypto - 92,

Springer and Verlag, p.p.113-138.

74. Miller V.S. (1986). Use of Elliptic Curves in Cryptography, Advances in Cryptology Crypto -

86, Springer and Verlag, p.p. 417 - 426.

75. Mu Y. Varadharajan V. (2000). Distributed signcryption, Advance in Cryptology Indocrypt -

2000, Springer and Verlag,LNCS # 1977, p.p. 155- 164.

76. Naor M. and Shamir A. (1994). Visual cryptography, Advances in Cryptology Eurocrypt

94,Springer and Verlag, p.p. 1 12.

77. NIST. (1994). Digital signature standard, U.S Department of Commerence, FIPS PUB, 186.

78. Nyberg K. and Rueppel R.A. (1994). A new signature scheme based on the DLP giving

message recovery, Advances in Cryptology Eurocrypt 94,Springer and Verlag, p.p. 182 -193.

79. Odlyzko A.M. (1984). Discrete logs in a finite field and their cryptographic significance,

Advances in Cryptology Eurocrypt - 84, Springer and Verlag, LNCS # 209, p.p. 224 - 314.


116/122

115

80. Okamoto T. (1988). A digital Multi-signature scheme using bijective PKC, ACM transactions

on computer systems 6(8), p.p. 432-441.

81. Ohta K. and Okamoto T. (1991). A digital multi-signature scheme based on Fiat- Shamir

scheme, Advance in Cryptology - Asiacrypt 91, Springer and Verlag, p.p. 75 79.

82. Okamoto T. (1992). Provably secure and practical Identification schemes and corresponding

signature scheme, Advance in Cryptology Crypto - 92, Springer and Verlag, LNCS # 740, p.p.

31-53.

83. Okamoto T. (1994). Designated confirmer and public encryption are equivalent, Advance in

Cryptology Crypto 94, Springer and Verlag, LNCS # 839, p.p. 61-74.

84. Pedersen, T. P. (1991). A threshold cryptosystem without trusted party, Advances in

Cryptology Eurocrypt - 91, Springer and Verlag, p.p. 522- 526.

85. Pederson T. P. (1991). Distributed provers with application to undeniable signatures,

Advances in Cryptology Eurocrypt - 91, Springer and Verlag, p.p. 221- 242.

86. Pointcheval D. and Stern J. (2000). Security arguments for digital signature and blind

signatures, Journal of Cryptology -13 (3), Springer-Verlag, p.p. 361-390.

87. Pomerance C. (1981). Recent development in the primality testing, The mathematical

intelligencer - 3, p.p. 97-105.

88. Pomerance C. (1982). The search for the prime numbers, Scientific American - 247, p.p. 136-

147.

89. Poupard G. and Stern J. (1998). Security analysis of practical On the Fly authentication

and signature generation, Advance in Cryptology - Eurocrypt - 98, Springer Verlag, LNCS #

1403, p.p. 422 - 436.

90. Rabin T. (1998). A simplified approach to threshold and proactive RSA, Advances in

Cryptology Crypto - 98, Springer Verlag, p.p. 89-104.

91. Rivest, R., Shamir A. and Aldeman L. (1978). A method of obtaining digital signatures and

PKCS, Communication of ACM - 21(2), p.p. 120-126.

92. Rubin F. (1995). Message authentication using Quadric Residues, Cryptologia - 19(4), p.p.

397- 207.


117/122

116

93. Ruland.C. (1993). Realizing digital signature with one-way Hash function, Cryptologia

17(3), p.p. 285-300.

94. Schnorr C.P. (1990). Efficient identification and signature for smart cards, Advance in

Cryptology Crypto - 89, Springer-Verlag, LNCS # 435, p.p. 239-251.

95. Schnorr C.P. (1991). Efficient signature generation by Smart cards, Journal of Cryptology -

4(3), p.p. 161-174.

96. Schnorr C. P. and Jakobsson M. (2000). Security of signed ElGamal encryption, Advance in

Cryptology - AsiaCrypt - 2000, Springer-Verlag, LNCS # 1976, p.p. 73-89.

97. Shamir A. (1979). How to share a secret, Communications of the association for computing

machinery 22, p.p. 612 - 613.

98. Shamir A. (1982). A polynomial time algorithm for breaking the basic Merkle Hellman

Cryptosystem, Proceeding of th

manoj dr brau thesis

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