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What is Discrete Logarithmic Problem in Information Security?
Let G be a finite cyclic set with n elements. It consider that the group is written multiplicatively. Let b be a generator of G and thus each element g of G can be written in the form g = bk for some integer k.
Moreover, any two such integers defining g will be congruent modulo n. It can represent a function logb: G → Zn(where Zn indicates the ring of integers modulo n) by creating to g the congruence class of k modulo n. This function is a group isomorphism known as the discrete algorithm to base b.
In mathematics, particularly in abstract algebra and its applications, discrete logarithms are set theoretic analogues of ordinary algorithms. In specific, an ordinary algorithm loga(b) is a solution of the equation ax = b over the real or complex number.
Equally if g and h are elements of a finite cyclic group G then a solution x of the equation gx = h is known as discrete logarithm to the base g of h in the group G.
Discrete logs have a large history in number theory. Originally, they were used basically in computations in finite area. However, they were rather ambiguous only like Integer Factorization Problem (IFP).
The foremost tool essential for the implementation of public-key cryptosystem is the Discrete Log Problem (DLP). There are some popular modern crypto-algorithms base their security on the DLP. It is based on the complexity of this problem. Diffie- Hellman suggested the well-known Diffie-Hellman key agreement scheme in 1976.
Example
Discrete logarithms are easiest to learn in the group (Zp). This is the group of congruence classes (1,…., p – 1) under multiplication modulo, the prime p.
If it is required to find the kth power of one of the numbers in this group, it can do so by discovering its kth power as an integer and then discovering the remainder after division by p.
This process is known as discrete exponentiation.
For instance, consider (Z17)x . It can compute 34 in this group, it can first calculate 34 = 81, and thus it can divide 81 by 17 acquiring a remainder of 13.
Hence, 34 = 13 in the group (Z17)x . Discrete logarithm is only the inverse operation. For instance, it can take the equation 3k = 13 (mod 17) for k.
In this k = 4 is a solution. Since 316 ≡ 1(mod 17), it also follows that if n is an integer then 34+16n ≡ 13 x 1n ≡ 13 (mod 17).
Therefore, the equation has infinitely some solutions of the form 4 + 16n. Furthermore, because 16 is the smallest positive integer m satisfying 3m ≡ 1 (mod 17), i. e. , 16 is the order of 3 in (Z17)x , there are the only solutions. Similarly, the solution can be defined as k ≡ 4 (mod)16.
There is no efficient algorithm for calculating general discrete logarithms logbg is known.
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