Deterministic factorization of sums and differences of powers

TL;DR

This paper presents a deterministic algorithm for factoring numbers of the form a^n ± b^n efficiently, improving upon existing general bounds for integer factorization.

Contribution

It generalizes previous results to a broader class of numbers and achieves a faster deterministic factorization method with complexity depending on N^{1/4}.

Findings

Factorization of a^n ± b^n can be done in al(M_{int}(N^{1/4}\u221a(\u221a N))) time.

The method improves the known bounds for deterministic integer factorization.

Applicable to numbers of the form a^n ^n with fixed coprime a, b.

Abstract

Let $a,b\in \mathbb{N}$ be fixed and coprime such that $a>b$, and let $N$ be any number of the form $a^n\pm b^n$, $n\in\mathbb{N}$. We will generalize a result of Bostan, Gaudry and Schost and prove that we may compute the prime factorization of $N$ in \[ \mathcal{O}(\text{M}_{\text{int}}(N^{1/4}\sqrt{\log N})), \] $\text{M}_{\text{int}}(k)$ denoting the cost for multiplying two $k$-bit integers. This result is better than the currently best known general bound for the runtime complexity for deterministic integer factorization.