How to Factor N 1 and N 2 When $p_1=p_2 \bmod 2^t$

Let N1 = p1q1 and N2 = p2q2 be two different RSA moduli. Suppose that p1 = p2 mod 2 t for some t, and q1 and q2 are α bit primes. Then May and Ritzenhofen showed that N1 and N2 can be factored in quadratic time if t ≥ 2α+ 3. In this paper, we improve this lower bound on t. Namely we prove that N1 and N2 can be factored in quadratic time if t ≥ 2α+ 1. Further our simulation result shows that our bound is tight.