Continued Fractions, Quadratic Fields, and Factoring: Some Computational Aspects

TL;DR

This paper explores how continued fraction expansions of square roots relate to factoring integers and sums of squares, proposing a modified Shanks' method that improves computational efficiency in factoring.

Contribution

It introduces a variation of Shanks' infrastructural method leveraging continued fractions to enhance factoring efficiency for composite numbers.

Findings

Continued fractions of √N with even period can produce factors of N.

The proposed method reduces asymptotic computational complexity.

Application of the method improves factoring performance over existing techniques.

Abstract

Legendre discovered that the continued fraction expansion of $\sqrt N$ having odd period leads directly to an explicit representation of $N$ as the sum of two squares. In this vein, it was recently observed that the continued fraction expansion of $\sqrt N$ having even period directly produces a factor of composite $N$. It is proved here that these apparently fortuitous occurrences allow us to propose and apply a variation of Shanks' infrastructural method which significantly reduces the asymptotic computational burden with respect to currently used factoring techniques.