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Let q be an odd prime power and D be the set of monic irreducible polynomials in Fq[x] which can be written as a composition of monic degree two polynomials. In this paper we prove that D has a natural regular structure by showing that there exists a finite automaton having D as accepted language. Our method is constructive.2010 Mathematics Subject Classification. 11T06, 68Q45.

Let Fq be a finite field, F/Fq be a function field of genus g having full constant field Fq, S a set of places of F and H the holomorphy ring of S. In this paper, we compute the density of coprime m-tuples of elements of H. As a side result, we obtain that whenever the complement of S is finite, the computation of the density can be reduced to the computation of the L-polynomial of the function field. In the genus zero case, classical results for the density of coprime m-tuples of polynomials are obtained as corollaries.

Abstract. Let S be a set of monic degree 2 polynomials over a finite field and let C be the compositional semigroup generated by S. In this paper we establish a necessary and sufficient condition for C to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set S. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree 2 polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions.

In this work we provide a suite of protocols for group key management based on general semigroup actions. Construction of the key is made in a distributed and collaborative way. Examples are provided that may in some cases enhance the security level and communication overheads of previous existing protocols. Security against passive attacks is considered and depends on the hardness of the semigroup action problem in any particular scenario.