A Fast Factorisation of Semi-Primes Using Sum of Squares

Abstract

For several centuries, prime factorisation of large numbers has drawn much attention due its practical applications and the associated challenges. In computing applications, encryption algorithms such as the Rivest–Shamir–Adleman (RSA) cryptosystems are widely used for information security, where the keys (public and private) of the encryption code are represented using large prime factors. Since prime factorisation of large numbers is extremely hard, RSA cryptosystems take advantage of this property to ensure information security. A semi-prime being, a product of two prime numbers, has wide applications in RSA algorithms and pseudo number generators. In this paper, we consider a semi-prime number whose construction consists of primes, N = p 1 p 2 , being Pythagorean and having a representation on the Cartesian plane such that, p = x 2 + y 2 . We prove that the product of two such primes can be represented as the sum of four squares, and further, that the sums of two squares can be derived. For such a semi-prime, if the original construction is unknown and the sum of four squares is known, by Euler’s factorisation the original construction p 1 p 2 can be found. By considering the parity of each of the squares, we propose a new method of factorisation of semi-primes. Our factorisation method provides a faster alternative to Euler’s method by exploiting the relationship between the four squares. The correctness of the new factorisation method is established with mathematical proofs and its practical value is demonstrated by generating RSA-768 efficiently.