Integer Factorization via Continued Fractions and Quadratic Forms

TL;DR

This paper introduces a new integer factorization algorithm based on continued fractions and quadratic forms, offering improved efficiency over classical methods and potential polynomial-time performance with additional number theoretic information.

Contribution

The paper presents a novel factorization algorithm that enhances existing methods by incorporating quadratic forms and infrastructure concepts, with a detailed complexity analysis.

Findings

Algorithm has complexity $O( ext{exp}(rac{3}{ oot{8} ext{}} oot{ ext{ln} N ext{ln} ext{ln} N})$

More efficient than SQUFOF and CFRAC algorithms

Potential polynomial-time performance with known multiple of regulator

Abstract

We propose a novel factorization algorithm that leverages the theory underlying the SQUFOF method, including reduced quadratic forms, infrastructural distance, and Gauss composition. We also present an analysis of our method, which has a computational complexity of $O \left( \exp \left( \frac{3}{\sqrt{8}} \sqrt{\ln N \ln \ln N} \right) \right)$, making it more efficient than the classical SQUFOF and CFRAC algorithms. Additionally, our algorithm is polynomial-time, provided knowledge of a (not too large) multiple of the regulator of $\mathbb{Q}(\sqrt{N})$.