Further results on implicit factoring in polynomial time

In PKC 2009, May and Ritzenhofen presented interesting problems related to factoring large integers with some implicit hints. One of the problems is as follows. Consider N1 = p1q1 and N2 = p2q2, where p1, p2, q1, q2 are large primes. The primes p1, p2 are of same bit-size with the constraint that certain amount of Least Significant Bits (LSBs) of p1, p2 are same. Further the primes q1, q2 are of same bit-size without any constraint. May and Ritzenhofen proposed a strategy to factorize both N1, N2 in poly(log N) time (N is an integer with same bit-size as N1, N2) with the implicit information that p1, p2 share certain amount of LSBs. We explore the same problem with a different lattice-based strategy. In a general framework, our method works when implicit information is available related to Least Significant as well as Most Significant Bits (MSBs). Given q1, q2 ≈ N , we show that one can factor N1, N2 simultaneously in poly(log N) time (under some assumption related to Gröbner Basis) when p1, p2 share certain amount of MSBs and/or LSBs. We also study the case when p1, p2 share some bits in the middle. Our strategy presents new and encouraging results in this direction. Moreover, some of the observations by May and Ritzenhofen get improved when we apply our ideas for the LSB case.