Performance Analysis of Fermat Factorization Algorithms

Abstract

The Rivest-Shamir-Adleman (RSA) cryptosystem is one of the strong encryption approaches currently being used for secure data transmission over an insecure channel. The difficulty encountered in breaking RSA derives from the difficulty in finding a polynomial time for integer factorization. In integer factorization for RSA, given an odd composite number n, the goal is to find two prime numbers p and q such that n = p q. In this paper, we study several integer factorization algorithms that are based on Fermat’s strategy, and do the following: First, we classify these algorithms into three groups: Fermat, Fermat with sieving, and Fermat without perfect square. Second, we conduct extensive experimental studies on nine different integer factorization algorithms and measure the performance of each algorithm based on two parameters: the number of bits for the odd composite number n, and the number of bits for the difference between two prime factors, p and q. The results obtained by the algorithms when applied to five different data sets for each factor reveal that the algorithm that showed the best performance is the algorithms based on (1) the sieving of odd and even numbers strategy, and (2) Euler’s theorem with percentage of improvement of 44% and 36%, respectively compared to the original Fermat factorization algorithm. Finally, the future directions of research and development are presented.