A Tale of Two Sieves

I t is the best of times for the game of factoring large numbers into their prime factors. In 1970 it was barely possible to factor “hard” 20-digit numbers. In 1980, in the heyday of the Brillhart-Morrison continued fraction factoring algorithm, factoring of 50-digit numbers was becoming commonplace. In 1990 my own quadratic sieve factoring algorithm had doubled the length of the numbers that could be factored, the record having 116 digits. By 1994 the quadratic sieve had factored the famous 129-digit RSA challenge number that had been estimated in Martin Gardner’s 1976 Scientific American column to be safe for 40 quadrillion years (though other estimates around then were more modest). But the quadratic sieve is no longer the champion. It was replaced by Pollard’s number field sieve in the spring of 1996, when that method successfully split a 130-digit RSA challenge number in about 15% of the time the quadratic sieve would have taken. In this article we shall briefly meet these factorization algorithms—these two sieves—and some of the many people who helped to develop them. In the middle part of this century, computational issues seemed to be out of fashion. In most books the problem of factoring big numbers was largely ignored, since it was considered trivial. After all, it was doable in principle, so what else was there to discuss? A few researchers ignored the fashions of the time and continued to try to find fast ways to factor. To these few it was a basic and fundamental problem, one that should not be shunted to the side. But times change. In the last few decades we have seen the advent of accessible and fast computing power, and we have seen the rise of cryptographic systems that base their security on our supposed inability to factor quickly (and on other number theoretic problems). Today there are many people interested in factoring, recognizing it not only as a benchmark for the security of cryptographic systems, but for computing itself. In 1984 the Association for Computing Machinery presented a plaque to the Institute for Electrical and Electronics Engineers (IEEE) on the occasion of the IEEE centennial. It was inscribed with the prime factorization of the number 2251 − 1 that was completed that year with the quadratic sieve. The president of the ACM made the following remarks: