Paper 2007/374

On Factoring Arbitrary Integers with Known Bits

Mathias Herrmann and Alexander May

Abstract

We study the {\em factoring with known bits problem}, where we are given a composite integer $N=p_1p_2\dots p_r$ and oracle access to the bits of the prime factors $p_i$, $i=1, \dots, r$. Our goal is to find the full factorization of $N$ in polynomial time with a minimal number of calls to the oracle. We present a rigorous algorithm that efficiently factors $N$ given $(1-\frac{1}{r}H_r)\log N$ bits, where $H_r$ denotes the $r^{th}$ harmonic number.