Efficient indexing of necklaces and irreducible polynomials over finite fields

Swastik Kopparty; Mrinal Kumar; Michael Saks

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arXiv · 1504.00572

Efficient indexing of necklaces and irreducible polynomials over finite fields

Swastik Kopparty·Mrinal Kumar·Michael Saks

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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.

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Swastik Kopparty, Mrinal Kumar, Michael Saks. 2015-04-02. Efficient indexing of necklaces and irreducible polynomials over finite fields. https://arxiv.org/abs/1504.00572

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