Deterministic Integer Factorization Algorithms

TL;DR

This paper introduces a new deterministic integer factorization algorithm with improved complexity, capable of factoring integers more efficiently than previous methods, especially when partial information about factors is available.

Contribution

The paper presents a novel deterministic factorization algorithm with complexity $O(N^{1/6+ ext{ε}})$, improving upon the previous $O(N^{1/5+ ext{ε}})$ algorithm.

Findings

Achieves factorization in $O(N^{1/6+ ext{ε}})$ arithmetic operations.

Can factor integers in polynomial time given partial bits of a factor.

Provides a more efficient deterministic approach compared to prior algorithms.

Abstract

A new integer deterministic factorization algorithm, rated at arithmetic operations to $O(N^{1/6+\varepsilon})$ arithmetic operations, is presented in this note. Equivalently, given the least $(\log N)/6$ bits of a factor of the balanced integer $N = pq$, where $p$ and $q$ are primes, the algorithm factors the integer in polynomial time $O(\log(N)^c)$, with $c \geq 0$ constant, and $\varepsilon > 0$ an arbitrarily small number. It improves the current deterministic factorization algorithm, rated at arithmetic operations to $O(N^{1/5+\varepsilon})$ arithmetic operations.