Solving Linear Equations Modulo Divisors: On Factoring Given Any Bits

We study the problem of finding solutions to linear equations modulo an unknown divisor p of a known composite integer N. An important application of this problem is factorization of N with given bits of p. It is well-known that this problem is polynomial-time solvable if at most half of the bits of p are unknown and if the unknown bits are located in one consecutive block. We introduce an heuristic algorithm that extends factoring with known bits to an arbitrary number n of blocks. Surprisingly, we are able to show that ln(2) 70% of the bits are sufficient for any n in order to find the factorization. The algorithm's running time is however exponential in the parameter n. Thus, our algorithm is polynomial time only for n = O(log log N) blocks.

• Type: Conference
• Year: 2008
• Where: ASIACRYPT
• Authors: Mathias Herrmann, Alexander May
• Keywords: Algorithm's Running Time, ASIACRYPT 2008, Cryptology, Unknown Bits, Unknown Divisor