Discrete algorithm problem applies to the mathematical structures, and entails collection of different elements using a binary operation referred to as group multiplication. Given an element ‘g’ in group ‘G’ of the order ‘t’, and the second element ‘y’, of group ‘G’, and the problem seeks for the value of ‘x’, with the conditions set, then element ‘g’ typically generates all the ‘G’ elements, or at least a considerable number of elements through exponentiation, with all integers ranging from zero to t-1. For instance, in a group element ‘g’ having a number ‘n’, then let ‘gn’ denotes the element that is obtained as a product of ‘g’ by itself for ‘n’ times. Discrete logarithmic problem is therefore expressed as: given element ‘g’ in the finite group ‘G’ and element h Î G, then finding an integer ‘x’ to give ‘gx’ = ‘h’, the solution would be 3x º 13 (mod 17) which is 4, since 34 = 81 º 13 (mod 17). Element ‘g’ is therefore referred to as the generator; it generates all elements within the group. The discrete algorithm problems are difficult and hard in generating a one-way function. As a result, different public-key cryptosystems, including ElGamal system, are used. Discrete logarithm problems have similar relationship to these systems, and security of these systems is based on the fact that computation of discrete algorithms is quite tasking. Generally, discrete logarithm in arbitrary group of the size ‘n’ may be computed in the running time O (Ön). Cryptographic algorithms
Cryptographic algorithms refer to the sequence of processes used in enciphering and deciphering messages within the cryptographic system. They transform the information given so as to conceal the meaning, and enhance security and authentication of the data. On the other hand, discrete algorithm refers to group-theoretic analogues of the ordinary logarithms. Cryptography entails the encryption process that converts plain text to an...
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