Lenstra’s Factorisation Method Based on Elliptic Curves

The purpose of this exposition is to explain the method due to H.W. Lenstra, Jr. [1] of determining a non-trivial factor, p, of a composite number, n. The method uses the theory of elliptic curves and has a n expected running time of L (p)√2 where L(p)=exp (√log p log log p). The aim of the exposition is to be completely elementary. with an introduction to the arithmetic of elliptic curves sufficient to enable the reader to follow the later section explaining the method. The paper ends with a few remarks on techniques for the practical implementation of the algorithm.