Sampling Factorization Using Euclidean Division Representation of Factors

This paper investigates the canonical problem of factoring a product of two primes, when these primes are represented as Euclidean divisions with different modulo coefficients. Such a representation creates a structure having specific properties that can be exploited in devising algorithmic factorization strategies. In particular, two factorization algorithms are presented, and shown to have a smaller complexity, for many, but not all prime pairs, than a naive direct enumeration of primes as well as well-known Pollard's methods. These algorithms assume that the search over a small set of modulo coefficients can be parallelized on a single processor offering vector machine instructions. The complexity here is defined empirically as the number of prime candidates that need to be tested. These results point at possibly new direction of exploring division structures of integer products, and designing factorization algorithms.