Efficient indexing of necklaces and irreducible polynomials over finite fields

TL;DR

This paper introduces the first efficient algorithms for indexing irreducible polynomials over finite fields and necklaces, with applications in pseudorandomness and solving an open problem in computational algebra.

Contribution

It presents the first poly(n, log q)-size circuit for bijectively indexing irreducible polynomials and necklaces over finite fields, connecting these concepts for the first time.

Findings

Efficient algorithm for indexing necklaces of given length and alphabet.

Poly(n, log q)-size circuits for bijection between indices and irreducible polynomials.

Answers an open question in the computational algebra of finite fields.

Abstract

We study the problem of indexing irreducible polynomials over finite fields, and give the first efficient algorithm for this problem. Specifically, we show the existence of poly(n, log q)-size circuits that compute a bijection between {1, ... , |S|} and the set S of all irreducible, monic, univariate polynomials of degree n over a finite field F_q. This has applications in pseudorandomness, and answers an open question of Alon, Goldreich, H{\aa}stad and Peralta[AGHP]. Our approach uses a connection between irreducible polynomials and necklaces ( equivalence classes of strings under cyclic rotation). Along the way, we give the first efficient algorithm for indexing necklaces of a given length over a given alphabet, which may be of independent interest.