Elliptic Curve Cryptography using Computational Intelligence

Abstract

Public-key cryptography is a fundamental component of modern electronic communication that can be constructed with many different mathematical processes. Presently, cryptosystems based on elliptic curves are becoming popular due to strong cryptographic strength per small key size. At the heart of these schemes is the complexity of the elliptic curve discrete logarithm problem (ECDLP). Pollard’s Rho algorithm is a well known method for solving the ECDLP and thereby breaking ciphers based on elliptic curves for reasonably small key sizes (up to approximately 100 bits in length). It has the same time complexity as other known methods but is advantageous due to smaller memory requirements. This study considers how to speed up the Rho process by modifying a key component: the iterating function, which is the part of the algorithm responsible for determining what point is considered next when looking for the solution to the ECDLP. It is replaced with an alternative that is found through an evolutionary process. This alternative consistently and significantly decreases the number of iterations required by Pollard’s Rho Algorithm to successfully find the sought after solution.