On computing factors of cyclotomic polynomials

TL;DR

This paper presents efficient algorithms with quadratic complexity for computing the integer coefficients of polynomials related to cyclotomic identities, providing explicit formulas and applications to integer factorization.

Contribution

It introduces simple, quadratic-time algorithms for calculating coefficients of cyclotomic-related polynomials, expanding on classical identities.

Findings

Algorithms require O(n^2) operations

Explicit formulas and generating functions derived

Applications demonstrated through numerical examples

Abstract

For odd square-free n > 1 the n-th cyclotomic polynomial satisfies an identity of Gauss. There are similar identity of Aurifeuille, Le Lasseur and Lucas. These identities all involve certain polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O(n^2) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for the polynomials, and illustrate the application to integer factorization with some numerical examples.