A babystep-giantstep method for faster deterministic integer factorization

TL;DR

This paper introduces an improved deterministic integer factorization algorithm that combines Strassen's polynomial-based approach with a babystep-giantstep method, achieving a superpolynomial speedup over previous methods.

Contribution

The paper presents a novel algorithm that significantly accelerates deterministic integer factorization by integrating babystep-giantstep techniques with Strassen's classical approach.

Findings

Achieves a superpolynomial speedup over previous deterministic methods.

Reduces the runtime complexity to (N^{1/4} \, ext{exp}(-C \, rac{\, ext{log} N}{\, ext{log} \, ext{log} N}))

Demonstrates the effectiveness of combining classical polynomial methods with number-theoretic techniques.

Abstract

In 1977, Strassen presented a deterministic and rigorous algorithm for solving the problem of computing the prime factorization of natural numbers $N$. His method is based on fast polynomial arithmetic techniques and runs in time $\widetilde{O}(N^{1/4})$, which has been state of the art for the last forty years. In this paper, we will combine Strassen's approach with a babystep-giantstep method to improve the currently best known bound by a superpolynomial factor. The runtime complexity of our algorithm is of the form \[ \widetilde{O}\left(N^{1/4}\exp(-C\log N/\log\log N)\right). \]