A heuristic search based factoring tool

Factoring is believed to be a diflcult task. Although factoring is of interest in its own right, the security of RSA cryptography, among other cryptographic systems, is dependent on the diflculty of factoring the product of two large primes. We show how to cast the problem of factoring integers as a state-based search to which the techniques of AI may be applied. Using small primes as operators, goal states can be characterized by integers close to a multiple kN of N, where N is the number to be factored. For a given value of k we formulate a heuristic formula for the resulting search which is universally optimistic. From a basic platform of depth first search we then make modifications, both with standard AI techniques such as pruning and with techniques which are specific for the particular problem. In the process we note the properties of a novel sub-class of integers, pseudo-smooth integers, which are useful in construction of an expanded search. We review empirical evidence of the improvements our various modifications make. Finally, we extend the factoring tool in various ways to deal effectively with the search task when the factor base is relatively small.