Efficient indexing of necklaces and irreducible polynomials over finite fields

Abstract

We study the problem of indexing necklaces, and give the first polynomial time algorithm for this problem. Specifically, we give a poly(n, log|∑|)-time computable bijection between {1,...,|N|} and the set N of all necklaces of length n over a finite alphabet ∑. Our main application is to give an explicit indexing of all irreducible polynomials of degree n over the finite field double-struck Fq in time poly(n, logq) (with n logq bits of advice). This has applications in pseudorandomness, and answers an open question of Alon, Goldreich, Håstad and Peralta [2]. © 2014 Springer-Verlag.