Divisors in Residue Classes Revisited

TL;DR

This paper generalizes Lenstra's polynomial-time algorithm for finding divisors in residue classes from integers to broader euclidean rings and polynomial rings, with implementation confirming its efficiency.

Contribution

It introduces a new method extending Lenstra's algorithm to euclidean rings and polynomial rings, overcoming previous limitations.

Findings

Algorithm runs in polynomial time in tested cases.

Successfully generalized to larger euclidean rings and polynomial rings.

Implementation confirms theoretical polynomial runtime.

Abstract

In 1984, H. W. Lenstra described an algorithm finding divisors of $N$ congruent to $r \mod S$. When $S^3 > N$, this algorithm runs in polynomial time and hence factors $N$ in time $N^{1/3+o(1)}$. Lenstra's algorithm relies on a sign change in a constructed sequence and so cannot be adapted directly to larger euclidean number rings. We present a new method that generalizes to a larger class of euclidean rings and the polynomial ring $\mathbb{Z}[x]$. The algorithm is implemented and timed confirming its polynomial run time.