Efficient Indexing of Necklaces and Irreducible Polynomials over Finite Fields

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 Fq. This has applications in pseudorandomness, and answers an open question of Alon, Goldreich, H̊astad and Peralta [AGHP92]. 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.