A Simple and Improved Algorithm for Integer Factorization with Implicit Hints

Abstract

Given two integers $N_1 = p_1q_1$ and $N_2 = p_2q_2$ with $\alpha$-bit primes $q_1,q_2$, suppose that the $t$ least significant bits of $p_1$ and $p_2$ are equal. May and Ritzenhofen (PKC 2009) developed a factoring algorithm for $N_1,N_2$ when $t \ge 2\alpha + 3$; Kurosawa and Ueda (IWSEC 2013) improved the bound to $t \ge 2\alpha + 1$. In this paper, we propose a polynomial-time algorithm in a parameter $\kappa$, with an improved bound $t = 2\alpha - O(\log \kappa )$; it is the first non-constant improvement of the bound. Both the construction and the proof of our algorithm are very simple; the worst-case complexity of our algorithm is evaluated by an easy argument. We also give some computer experimental results showing the efficiency of our algorithm for concrete parameters, and discuss potential applications of our result to security evaluations of existing factoring-based primitives.