William Stein
Date: Math 124 HARVARD UNIVERSITY Fall 2001
In 1987, Hendrik Lenstra published the landmark paper Factoring Integers with Elliptic Curves, Annals of Mathematics, 126, 649-673, which you can download from the Math 124 web page. Lenstra's method is also described in §IV.4 of Silverman and Tate's Rational Points on Elliptic Curves, §VIII.5 of [Davenport], and in §10.3 of Cohen's A Course in Computational Algebraic Number Theory.
In this lecture and the next, I will tell you about Lenstra's clever
algorithm. It shines at finding ``medium sized'' factors of an
integer , which these days means to decimal digits but
probaby not decimal digits. The ECM method is thus not useful
for earning money by factoring RSA challenge numbers, but is
essential when factoring most integers. It also has small storage
requirements. Lenstra writes:
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Lenstra's discover of the elliptic curve method was inspired by Pollard's -method. I will spend most of the rest of this lecture introducing you to it.