A New Signature Protocol Based on RSA and Elgamal Scheme

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.3, August 2016
DOI: 10.5121/ijitmc.2016.4302 11
A NEW SIGNATURE PROTOCOL BASED ON RSA
AND ELGAMAL SCHEME
J. Ettanfouhi and O. Khadir
Laboratory of Mathematics, Cryptography and Mechanics, Fstm,
University Hassan II of Casablanca, Morocco
ABSTRACT
In this paper, we present a new signature scheme based on factoring and discrete logarithm problems.
Derived from a variant of ElGamal signature protocol and the RSA algorithm, this method can be seen as
an alternative protocol if known systems are broken.
KEYWORDS
Factoring, DLP, PKC, ElGamal signature scheme, RSA.
MSC: 60
94A
1. INTRODUCTION
In 1977, Rivest, Shamir, and Adleman[6 ] described the famous RSA algorithm which is based
on the presumed difficulty of factoring large integers. In 1985, ElGamal [2] proposed a signature
digital protocol that uses the hardness of the discrete logarithm problem[5 p.116, 7 p. 213, 8
p. 228 ]. Since then, many similar schemes were elaborated and published[1,3].
Among them, a new variant was conceived in 2010 by the second author[ 4 ].In this work, we
apply a combination of the new variant of Elgamal and RSA algorithm to build a secure digital
signature. The efficiency of the method is discussed and its security analyzed.
The paper is organised as follows: In section 2, we describe the basic ElGamal digital signature
algorithm and its variant. Section 3 is devoted to our new digital signature method. We end with
the conclusion in section 4.
In the paper, we will respect ElGamal work notations [3]. N, Z are respectively the sets of
integers and non-negative integers. For every positive integer n , we denote by Z
Z n
/ the finite
ring of modular integers and by *
)
/
( Z
Z n the multiplicative group of its invertible elements. Let
a ,b , c be three integers. The GCD of a and b is written as )
,
( b
a
gcd . We write
]
[
0.1 c
cm
b
a ≡ if c divides b
a − , and c
b
a mod
= if a is the rest in the division of b by
c . The bit length of n is the number of bits in its binary model, with n an integer .We start by
presenting the basic ElGamal digital signature algorithm and its variant:

12
2. ELGAMAL SIGNATURE SCHEME
In this section we recall ElGamal signature scheme[2] and its variant[4].
1. Alice chooses three numbers:
- p , a large prime integer.
- α , a primitive root of the finite multiplicative group *
)
/
( Z
Z p
- x , a random element of {1,2,..., 1
−
p }
2. She computes x
y α
= mod p . Alice’s public key is )
,
,
( y
p α , and x is her private key.
3. To sign the document m , Alice must solve the problem:
(1)
where s
r, are the unknown variables.
Alice fixes arbitrary r to be k
r α
= mod p , where k is chosen randomly and invertible
modulo 1
−
p . Equation (1) is then equivalent to:
(2)
Since Alice has the secret key x , and as the number k is invertible modulo 1
−
p , she calculates
the other unknown variable s by
(3)
4. Bob can verify the signature by checking if congruence (1) is valid for the variables r and s
given by Alice.
3. VARIANT OF ELGAMAL SIGNATURE SCHEME
We present a variant of ElGamal digital signature system.
This variant Error! Reference source not found. is based on the equation:
(4)
t
s
r ,
, are the unknown parameters, and )
,
,
( y
p α are Alice public keys. p is an integer (a large
prime). α is a primitive root of *
p
Z . y is calculated by x
y α
= mod p . x is a random
element of {1,2,...,p-1}.

13
Let )
(
= M
h
m , where h is a hash function, and M the message to be signed by Alice.
To give the solution (3), she fixes randomly r as p
mod
r k
α
≡ , and s to be p
mod
s l
α
≡ ,
where l
k, are selected arbitrary in {1,2,...,p-1}.
Equation (3) is then equivalent to:
(5)
as Alice recognize the values of x
m
l
k
s
r ,
,
,
,
, , she is able to calculate the last unknown variable
t .
Bob verify the signature by verifying the congruence (4).
this system does not use the extended Euclidean algorithm for calculating 1)
(
1
−
−
p
mod
k .
We clarify the scheme by the example given by the creator of this alternative[4].
3.1 EXAMPLE
Let )
,
,
( y
p α be Alice public keys where: 509
=
p , 2
=
α and 482
=
y . We assert that we are
not confident if using a small value of α does not abate the protocol. The private key is
281
=
x . Suppose that Alice wants to generate a signature for the document M for which
432[508]
)
( ≡
≡ M
h
m with the exponents 208
=
k and 386
=
l are randomly taken. She
computes ]
332[
2208
p
r k
≡
≡
≡ α , ]
39[
2386
p
s l
≡
≡
≡ α and 1]
440[ −
≡
+
+
≡ p
lm
ks
rx
t .
Bob or anyone can verify the relation ]
[p
s
r
y m
s
r
t
≡
α . Indeed, we find that ]
436[p
≡
α and
]
436[p
s
r
y m
s
r
≡ .
4. OUR PROTOCOL
4.1 DESCRIPTION
In this section, we describe our new digital signature. The protocol is based simultaneously on
two hard problems.
We assume first that h is a public secure hash function like SHA1[5 p.348, 7 p. 242 , 8
p.133].
We suppose that Alice public keys are )
,
,
,
( e
y
P α where:
- 1
2
= +
pq
P , p , q are three primes.
- α , a primitive root of the multiplicative group *
)
/
( Z
Z p .
- x
y α
= mod P , where x is the private key of Alice, which is randomly taken in
1}
{1,2,..., −
P .

14
- Element e is the public exponent in the RSA cryptosystem.
We propose the following protocol:
If Alice wants to sign the message M , she must give a solution for the modular equation:
(6)
where )
(
= M
h
m mod p , and r , s , t are unknown.
To solve equation (5) , Alice starts by putting:
(7)
(8)
Equation (5) becomes:
(9)
Alice uses the new variant of Elgamal algorithm[4] to solve equation (9) and to get the values of
r′ , s′ and t .
Then with her RSA private key she solves equations (7) and (8) . The cupel r and s is her
signature for the message M .
Bob or anybody can check that the signature is valid by replacing r , s and t in relation (5) .
4.2 EXAMPLE
Let us illustrate the method by the following example.
Suppose that Alice’s public key is: 104543
=
1
313
*
167
*
2
= +
P , 5
=
α , 23292
=
y , 7
=
e .
The private keys for RSA and ElGamal systems are respectively: 9502
=
x , 7399
=
d .
Assume that 12345
=
)
(
= M
h
m is the hashed message that she likes to sign.
If she takes Randomly 845
=
k and 2561
=
l .
She will find from equation 8 that 17744
=
r′ , 31839
=
s′ .
Relation 4 implies 57764
=
t .
Alice uses (6) and (7) to obtain:

15
r ≡ 1]
[ −
′ P
r d
≡ 75282
s ≡ 1]
[ −
′ P
s d
≡ 19005.
To verify Bob puts 62833
=
mod
= P
A t
α , 79849
=
mod
= 1
mod
P
y
B P
e
r −
,
83421
=
mod
1)
mod
(
= 1)
mod
(
P
P
r
C P
e
s
e −
− and 212997
=
mod
1)
mod
(
= P
P
s
D m
e
− , and
checks if P
D
C
B
A mod
*
*
= .
4.3 SECURITY ANALYSIS
Now that we have presented the protocol, we will discuss some possible attacks. Assume that
Oscar is Alice’s opponent.
ATTACK 1: If the attacker try to imitate the computation made by Alice, he can find r and s ,
but to find t he needs the value of the private key x to solve equation 4 .
ATTACK 2: Suppose Oscar is capable to solve the discrete logarithm problem [2]. He cannot
calculate r and s from equation (7) and (8) he will be confronted to the factorisation of a
large composite modulus [5,8].
ATTACK 3: Suppose Oscar is capable to solve RSA equations (7) and (8) . Oscar cannot get t
from equation (9) since x is Alice’s secret key. If he tries to get t from equation (4) , he will be
stopped by the discrete logarithm problem.
4.4 COMPLEXITY OF OUR ALGORITHM
As in [1], let exp
T , mult
T and h
T be appropriately the time to calculate an exponentiation, a
multiplication and hash function of a document M . We neglect the time needed for modular
substraction, additions, comparisons and apply the conversion mult
exp T
T 240
= .
4.4.1 SIGNATURE COMPLEXITY
To sign the message M , Alice must compute the six parameters:
)
(
= M
h
m mod P , k
r α
≡
′ ]
[P , l
s α
≡
′ ]
[P , d
r
r ′
≡ 1]
[ −
P , d
s
s ′
≡ 1]
[ −
P ,
lm
s
k
r
x
t +
′
+
′
≡ 1]
[ −
P .
Alice needs to perform four modular exponentiations, three modular multiplications and one hash
function computation. So the global required time is :
h
mult
h
mult
exp T
T
T
T
T
T +
+
+ 963
=
3
4
=
1
4.4.2 VERIFICATION COMPLEXITY
Bob should calculate 4 exponentiations, 2 multiplications and one hash function. So the global
required time is :
h
mult
h
mult
exp T
T
T
T
T
T +
+
+ 962
=
2
4
=
2

16
5. CONCLUSION
In this work, we proposed a new signature protocol that can be an alternative if old systems are
broken. Our method is based simultanyously on RSA cryptosystem and DLP.
ACKNOWLEDGEMENTS
This work is supported by the MMS e-orientation project.
REFERENCES
[1] R. R. Ahmad, E. S. Ismail,and N. M. F. Tahat , A new digital signature scheme based on factoring
and discrete logarithms , J. of Mathematics and Statistics (4): (2008), pp. .
[2] T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithm problem,
IEEE Trans. Info. Theory, IT-31, (1985), pp. .
[3] L.C. Guillou, J.J. Quisquater, A Paradoxial Identity-based SIgnature Scheme Resulting from Zero-
Knowledge, Advances in cryptography, LNCS 403, (1990) pp. .
[4] O. Khadir, New variant of ElGamal signature scheme, Int. J. Contemp. Math. Sciences, Vol. 5, no.
34, (2010), pp. .
[5] A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of applied cryptography, CRC
Press, Boca Raton, Florida, 1997.
[6] R. Rivest, A. Shamir and L. Adeleman, A method for obtaining digital signatures and public key
cryptosystems, Communication of the ACM, Vol. no 21, (1978), pp. .
[7] J. Buchmann, Introduction to Cryptography,(Second Edition), Springer 2000.
[8] D. R. Stinson, Cryptography, theory and practice, second Edition, Chapman & Hall/CRC, 2006.
Authors
Jaouad Ettanfouhi holds an engineer degree in Computer Science from the University of
Hassan II of Casablanca (2011). Member of the laboratory of Mathematics, Cryptography
and Mechanics, he is preparing a thesis in public key cryptography.
Dr Omar Khadir received his Ph.D. degree in Computer Science from the University
of Rouen, France (1994). Co-founder of the Laboratory of Mathematics, Cryptography
and Mechanics at the University of Hassan II Casablanca, Morocco, where he is a
professor in the Department of Mathematics. He teaches cryptography for graduate
students preparing a degree in computer science. His current research interests include
public key cryptography, digital signature, primality, factorisation of large integers and
more generally, all subjects connected to the information technology.

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