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Integer Factorization via Continued Fractions and Quadratic Forms

Nadir Murru, Giulia Salvatori

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 $\mathbb{Q}(\sqrt{N})$.

Integer Factorization via Continued Fractions and Quadratic Forms

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 , 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 .
Paper Structure (6 sections, 30 theorems, 83 equations, 1 table, 2 algorithms)

This paper contains 6 sections, 30 theorems, 83 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Sequences from continued fractions of quadratic irrationals
  3. Even period and nontrivial factor
  4. Quadratic forms
  5. The factorization algorithm
  6. Conclusions

Key Result

Proposition 2.2

The sequence $\left \{ \mathfrak{c}_n \right \}_{n \ge -1}$ satisfies the relation

Theorems & Definitions (70)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4: elia2019
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • ...and 60 more