Saumik Barua

• Due to the intractability of the Discrete Logarithm Problem (DLP), it has been widely used in the field of cryptography and the security of several cryptosystems is based on the hardness of computation of DLP. In this paper, we start with the topics on Number Theory and Abstract Algebra as it will enable one to study the nature of discrete logarithms in a comprehensive way, and then, we concentrate on the application and computation of discrete logarithms. Application of discrete logarithms such as Diffie Hellman key exchange, ElGamal signature scheme, and several attacks over the DLP such as Baby-step Giant-step method, Silver Pohlig Hellman algorithm, etc have been analyzed. We also focus on the elliptic curve along with the discrete logarithm over the elliptic curve. Attacks for the elliptic curve discrete logarithm problem, ECDLP have been discussed. Moreover, the extension of several discrete logarithms-based protocols over the elliptic curve such as the elliptic curve digital signature algorithm, ECDSA have been discussed also.