Computing in Cryptography

Nowadays fast computing and lightweight cryptography play a crucial role in the field of cryptography. Whenever we concern about the cryptography, the aspect of discrete mathematics can't be omitted. Security in cryptography is completely depends upon the key and the computations. Generation of key is nothing but the implementation of graph theory, discrete logarithms, linear & abstract algebra etc. These branches of modern mathematics has been playing a great role for the implementation of algorithms such as elliptic curve cryptography, stream cipher, block cipher, wireless sensor network etc. This paper is written to literally represent the key concept of such branches of modern mathematics and their implementations in the field of cryptography. This paper mainly deals the applications of Galois Field, primitive polynomials, primitive polynomials over Galois Field, Number theoretic functions, Congruence Calculus or modular arithmetic's, Residue Class Rings and Prime Fields. The paper focus on these key topics to develop a mathematical tool, that are needed for the design and security analysis of a cryptosystems.