Factoring RSA Moduli with Weak Prime Factors

In this paper, we study the problem of factoring an RSA modulus N = pq in polynomial time, when p is a weak prime, that is, p can be expressed as ap = u0 + M1u1 + . . . + Mkuk for some k integers M1, . . . ,Mk and k+2 suitably small parameters a, u0, . . . uk. We further compute a lower bound for the set of weak moduli, that is, moduli made of at least one weak prime, in the interval [2, 2] and show that this number is much larger than the set of RSA prime factors satisfying Coppersmith’s conditions, effectively extending the likelihood for factoring RSA moduli. We also prolong our findings to moduli composed of two weak primes.