A time-space tradeoff for Lehman's deterministic integer factorization method

TL;DR

This paper presents a new deterministic integer factorization algorithm that improves the runtime complexity significantly by leveraging a time-space tradeoff based on Lehman's method and Lawrence's generalization, achieving the first exponential improvement since 1977.

Contribution

It introduces a novel time-space tradeoff for Lawrence's generalization and combines it with Lehman's approach to develop a faster deterministic factorization algorithm.

Findings

Achieves a runtime complexity of O(N^{2/9+o(1)})

First exponential improvement since 1977

Demonstrates the effectiveness of the time-space tradeoff in factorization algorithms

Abstract

Fermat's well-known factorization algorithm is based on finding a representation of natural numbers $N$ as the difference of squares. In 1895, Lawrence generalized this idea and applied it to multiples $kN$ of the original number. A systematic approach to choose suitable values for $k$ was introduced by Lehman in 1974, which resulted in the first deterministic factorization algorithm considerably faster than trial division. In this paper, we construct a time-space tradeoff for Lawrence's generalization and apply it together with Lehman's result to obtain a deterministic integer factorization algorithm with runtime complexity $O(N^{2/9+o(1)})$. This is the first exponential improvement since the establishment of the $O(N^{1/4+o(1)})$ bound in 1977.