We study the factoring with known bits problem, where we are given a composite integer N = p1p2 . . . pr and oracle access to the bits of the prime factors pi, i = 1, . . . , 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 - 1 H r r) log N bits, where Hr denotes the rth harmonic number.