On nonlinear polynomial selection for the number field sieve

TL;DR

This paper develops two new nonlinear polynomial selection algorithms for the number field sieve, improving the construction of polynomials with small coefficients by extending methods for creating modular geometric progressions.

Contribution

It introduces extended methods for constructing modular geometric progressions, leading to novel nonlinear polynomial selection algorithms for the number field sieve.

Findings

Two new nonlinear polynomial selection algorithms developed

Enhanced methods for constructing modular geometric progressions

Potential improvements in polynomial coefficient minimization

Abstract

Nonlinear polynomial selection algorithms for the number field sieve address the problem of constructing polynomials with small coefficients by reducing to instances of the well-studied problem of finding short vectors in lattices. The reduction rests upon the construction of modular geometric progressions with small terms. In this paper, the methods used to construct the geometric progressions are extended, resulting in the development of two nonlinear polynomial selection algorithms.