Implicit Factoring with Shared Middle Discrete Bits

Abstract

We study the problem of implicit factoring presented by May and Ritzenhofen in 2009 and apply it to more general settings, where prime factors of both integers are only known by implicit information of middle discrete bits. Consider two integers N ₁ = p ₁ q ₁ and N ₂ = p ₂ q ₂ where p ₁, p ₂, q ₁, and q ₂ are primes and q ₁, q ₂ ≈ N ^α. In the case of tlog₂ N bits shared in one consecutive middle block, we describe a novel lattice-based method that leads to the factorization of two integers in polynomial time as soon as t > 4α. Moreover, we use much lower lattice dimensions and obtain a great speedup. Subsequently, we heuristically generalize the method to an arbitrary number n of shared blocks. The experimental results show that the constructed lattices work well in practical attacks.