Department of Mathematics
Server

In general, a generalization in mathematics must not only yield all the special cases from which the generalization started, but also give in addition more than that is contained in all of those special instances. For example, projective planes over division rings is one such generalization which characterizes the validity of Desargues theorem and gives new examples of non-Pappus projective planes. However, the generalization suggested here - cubic curves over rings - is not such a generalization. In fact, this fails to yield a meaningful group law or even a loop law over the curve unless the ring happens to be a field. However, in 1986, H.W. Lenstra discovered that it is precisely this failure that contributes to the success of the factorization of large composite numbers. This technique has found enormous applications in modern cryptography. In this talk, we will give the algebra and geometry behind ECM - the Lenstra algorithm for integer factorization and will go through some simple examples.

Reference
H.W. Lenstra, Factoring integers with elliptic curves, Annals of Mathematics, (2), 126(3) 1987, 649-673.