Further Improvement of Factoring RSA Moduli with Implicit Hint

Abstract

We investigate the problem of factoring RSA moduli with implicit hint, which was firstly proposed by May and Ritzenhofen in 2009 where unknown prime factors of several RSA moduli shared some number of least significant bits (LSBs) and was considered by Faug\(\grave{e}\)re et al. in 2010 where some most significant bits (MSBs) were shared between the primes. In this paper, we further consider this factorization with implicit hint problem, present a method to deal with the case when the number of shared LSBs or MSBs is not large enough to satisfy the bound proposed by May et al. and Faug\(\grave{e}\)re et al. by making use of a result from Herrmann and May for solving linear equations modulo unknown divisors, and finally get a better lower bound on the the number of shared LSBs or MSBs. To the best of our knowledge, our lower bound is better than all known results and we can theoretically deal with the implicit factorization for the case of balanced RSA moduli.