A New Search Pattern in Multiple Residue Method (MRM) and Its Importance in the Cryptanalysis of the RSA

Abstract

This paper presents a cryptanalysis attack on the RSA cryptosystem. The method, Multiple Residue Method (MRM), makes use of an algorithm which determines the value of ϕ(n) and hence, for a given modulus n where n = p×q, the prime factors can be uncovered. This algorithm calculates and stores all possible residues of p, q and (p + q) in different moduli. It then applies the Chinese Remainder Theorem (CRT) to different combinations of residues until the correct value is calculated, [6]. Further properties in relation to this structure show that improvements in the search process, within the residue of all parameters involved, can be effectively achieved. Besides, it has been established that the security of the RSA is no greater than the difficulty of factoring the modulus n into a product of two secret primes p and q. However, the MRM approaches the factorisation problem from a different angle. This method is aimed at finding towards the ϕ(n) in O(2^-j ×n), where j is the number of prime moduli. It may also be directed towards the computation of the sum (p + q) and, in the realistic case for the RSA, reduces to O(2^-j×√n).