Factoring a Multiprime Modulus N with Random Bits

In 2009, Heninger and Shacham presented an algorithm using the Hensel's lemma for reconstructing the prime factors of the modulus N = r1r2. This algorithm computes the prime factors of N in polynomial time, with high probability, assuming that a fraction greater than or equal to 59% random bits of its primes r1 and r2 is given. In this paper, we present the analysis of Hensel's lemma for a multi-prime modulus N = ∏u i=1 ri (for u ≥ 2) and we generalise the Heninger and Shacham's algorithm to determine the minimum fraction of random bits of its prime factors that is su cient to factor N in polynomial time with high probability.