Factoring Integers Using the Web

This note provides background on the www-factoring project, which was started in the fall of 1995. Factoring a positive integer ri means finding two positive integers u and v such that the product of u and v equals ri, and such that both u a.nd v are greater than 1. Such u and v are called factors (or divisors) of ii, and n = u v is called a factorization of n. Positive integers that ca.n be factored are called composites. Positive integers greater than 1 that cannot be factored are called primes. For example, n = 15 can be factored as the product of the primes u = 3 and v = 5, and n = 105 can be factored as the product of the prime u = 7 and the composite v 15. There are efficient methods to distinguish primes from composites that do not require factoring the composites (cf. [9], [20], and Appendix). These methods can be used to establish beyond doubt that a certain number is composite without, however, giving any information about its factors. Factoring a composite integer is believed to be a hard problem. This is, of course, not the case for all composites—composites with small factors are easy to factor—but, in general, the problem seems to be difficult. Currently the limits of our factoring capabilities lie around 130 decimal digits. Factoring hard integers in that range requires enormous amounts of computing power. One way to get the computing power needed is to distribute the computation over the Internet. The www-factoring effort is intended to be a convenient way to divide the factoring work among volunteers on the Internet. Factoring on the Internet is not new. The approach described in [12] was first used in 1988 to factor a 100-digit integer; since then, to factor many integers in the 100 to 120 digit range; and most recently (1993-1994), to factor the famous 129-digit RSA-challenge number (cf. [1]).’ This note is intended for contributors to the www-factoring effort who want to understand how modern factoring algorithms work. Using simple examples