J. Comp. & Math. Sci. Vol. 1(2), 193-196 (2010).
New Discrete Logarithm Problem over Finite Field by applying One Way Hash Function and Public Key Cryptography SUNIL KUMAR KASHYAP, BIRENDRA KUMAR SHARMA 1 and AMITABH BANERJEE2 Department of Mathematics Rungta College of Engineering and Technology, Chhattisgarh Swami Vivekanand Technical University, Bhilai, Chhattisgarh, 491024 (INDIA) 7sunilkumarkashyap@gmail.com 1 School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, Chhattisgarh, 492010 (INDIA) 2 Department of Mathematics, Govt. DBGA Post Graduate College, Pt. Ravishankar Shukla University, Raipur Raipur, Chhattisgarh 492001, (INDIA) ABSTRACT We found the new result as “New Discrete Logarithm Problem over Finite Field by applying One Way Hash Function and Public Key Cryptography”. In this paper, we found the distinct discrete logarithm problem as compare to the traditional exist discrete logarithm problem. We also applied the one way hash function in the sense of Cryptographic Applications. The main objective of this paper is to present a new theory of the Discrete Logarithm Problem over Finite Field by applying One Way Hash Function and design the New Public Key Cryptography, whose security is impossible to break, because we claim that there are no algorithm are in the existence which can be compute the proposed new mathematical problem. 2000 AMS Mathematics Subject Classification Number: 94A60. Key Words: Discrete Logarithm Problem, Finite Field, One Way Hash Function, Public Key Cryptography. Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
194 Sunil Kumar Kashyap et al., J.Comp.&Math.Sci. Vol.1(2), 193-196 (2010). 1. INTRODUCTION The Discrete Logarithm Problem (DLP) is well known and popular number theoretic problem, the importance of DLP is not only sense of pure mathematics but also in the sense of applied mathematics, especially for the cryptographic applications. The DLP is one of the hardest mathematical problem in different settings. Some variants of the DLP are also the same importance to the traditional DLP. Although DLP is the most natural number theoretic problem but this is only used in the cryptographic applications. We give the new result in this paper, which is very important and useful for the both mathematical phases, i.e. pure and applied mathematics. “New Discrete Logarithm Problem over Finite Field by applying One Way Hash Function and Public Key Cryptography” is the first research work, where The Discrete Logarithm Problem, The Finite Field and The Hash Function are used in the Compact form. We give this result with the help of the theorem. The main result of this work is not only given the new mathematical theory but also given the best application in the field of public key cryptography. We claim that this new public key cryptography, whose security is based on the proposed new mathematical problem is impossible to break, because there are so many reasons, but we would like to share the two most important reasons, the first is, This is the first mathematical theory of this new type of discrete logarithm problem thus
there are no possibilities of existence of any algorithm for computing this over the proposed problem. Second is, The discrete logarithm problem, the finite field and the one way hash function used in this work as the compact form, thus again this will be the not open solution in present. We motivated by the study of the literatures1-6. 2. Preliminaries. 2.1. The Discrete Logarithm Problem. The discrete logarithm problem applies to mathematical structures called groups. A group is a collection of elements with a binary operation, in this case multiplication. For a group element g and a number n, let g n refer to the element obtained by multiplying g by itself n times. The discrete logarithm problem is as follows: given an element g in a finite group G and another element h G , find an integer x such that g x h . Solving the discrete logarithmic problem is, in essence, computing the isomorphism between the cyclic group G of order n, and n . “x is the discrete logarithm problem of h at the base g” or, “the problem of computing the index x, is called the discrete logarithm problem of h to the base g”. 2.2. The One Way hash Function. The hash function is often called “the one way hash function” if the preimage can not be efficiently solved.
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
Sunil Kumar Kashyap et al., J.Comp.&Math.Sci. Vol.1(2), 193-196 (2010). 195 A hash function h : X Y and an element y Y. Then finding x X such that f(x) = y. 2.3. The Finite Field. The order of a finite field must be a power of a prime pn, where n is a positive integer. Prime is an integer whose only positive integer factors are itself and 1. The finite field of order pn is usually denoted by GF(pn), GF stands for Galois Field in the honor of the Great French Mathematician Evarist Galois (1811-1832).
3. The Proposed Theory. 3.1. Theorem. If the one way hash function be apply in F. Let F be an algebraic extension field of K, with char K = p 0 . If F is separable over K , then pn
[ # F # [ KF ] for each n 1. If [F:K] is finite and # F =#[ KF p ], then F is s e p ara b l e ove r K. I n p a rt i c u l a r, u F is separable over K if and only if #[K(u p)] =# [K(u)] . Then, the problem to find the value of p is the discrete logarithm problem.
2.3.1. Finite Fields of Order p
Proof:
For a given prime p,GF(p) is defined as the set Zp={0,1,..,p-1} of integers together with arithmetic operations modulo p. For such prime numbers, holds - Multiplicative inverse axiom.
First we apply the one way hash function on S. Let S is a separable extension field of K. If [F : K] is finite and, then # F=#[K(u1,…, um)]=# [S(u1,…, um)]. Since each ui is purely inseparable over S, there is an n 1 such that
Because elements w of Zp are relatively prime to p, if we multiply all the elements of Zp by w, the resulting residues are all of elements Zp, permuted. Thus, exactly one of the residues has the value 1, respective multiplier is just the inverse element for w, designated w-1. Now, equation (4.2) can be written without condition: If ab ac mod p then b c mod p Consequence is obtained by multiplication of the above both parts by a-1. The simplest finite field is GF(2): + 0 0 0 1 1
1 1 0
Addition
X 0 1
0 0 0
1 0 1
w -w w-1 0 0 1 1 1
Multiplication Inverses
ui
pn
very element S for every i. Since ev n
of S is purely inseparable over # F p , n and hence over # KF p . S is separable n over K, and hence over # KF p . n Therefore # S KF p . Use the fact that char K = p to show that for any t
t
t
y
t 1, # F p # [ K (u1,... , u m )] p # K p (u1 p ,..., t
p e u m ). consequently for any t 1, we have t
t
py
pt
# [ KF p ] # [ K ( K p (u1 ,..., u m )] # [ K (u1
py
pt
,..., u m )]. Note that this argu-ment works for any generators u1,... , um of F n over K. Now if F KF p , then #[K(u1,…, n
um)] = #F = #[ KF p ]=# S, whence F is separable over K. Conver-sely, if F is separable over K, than F is both
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
196 Sunil Kumar Kashyap et al., J.Comp.&Math.Sci. Vol.1(2), 193-196 (2010). separable and purely inseparable over n
# [ KF p ] (for any n 1). Therefore n
# F # [ KF p ] and the problem to find the value of p is the discrete logarithm problem. 4. Public Key Cryptography. In this section, we present the new public key cryptography based on the proposed mathematical theory, “New Discrete Logarithm Problem over Finite Field by applying One Way Hash Function”. 4.1. Key Generation. 4.1.1. (By using Theorem 3.1.) The Message be m, F be an algebraic extension field of K, with char K=p 0. pn and pm are the public and private keys respectively. 4.2. Encryption. 4.2.1. The Ciphertext be C, which can be defined by the following equation; n
C (C1, C 2 ) [C1 #{KF p }, m
n
C 2 # [{mKF p } p ]. Here, pn is the public key. 4.3. Decryption. 4.3.1. The Plaintext be P, which can be obtained by the following equation;
P C 2 C1
pm
m.
Here, pm is the private key. 4. CONCLUSION “New Discrete Logarithm Problem over Finite Field by applying One Way
Hash Function and Public Key Cryptography “ is the first original research work, where the three important mathematical problem in the sense of Public Key Cryptography, is presented as the compact form. This new theory concerns with the “new hard mathematical problem and the public key cryptography”. Both will be the very applicable in the development of the studies of mathematics and cryptography. REFERENCES 1. Cheng, Q., On the bounded sumof-digits discrete logarithm problem in finite fields, Advances in Cryptology, CRYPTO 2004, Springer-Verlag, LNCS, 3152, 2005, 201-212. 2. Diffie W., Hellman M.E., New directions in cryptography, Transactions on information theory, 22, 644-654 (1976). 3. Holden, J., Fixed points and two cycles of the discrete logarithm, Algorithmic number theory, LNCS, 2369, 405-415 (2002). 4. Muuller, V., Vanstone S., Zuccherato, R., Discrete logarithm based cryptosystems in quadratic function fields of characteristic two, Design, codes and cryptography, 14, No. 2, 159178 (1998). 5. Rivest R., Shamir A., Adleman L., A method for obtaining digital signatures and public key cryptosystems, Communication of the ACM, 21(2), 20-126 (1978). 6. Stinson D. R., Cryptography: Theory and Practice, CRC Press, Boca Raton, Florida (1995).
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)