2 research outputs found

    Baby-Step Giant-Step Algorithms for the Symmetric Group

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    We study discrete logarithms in the setting of group actions. Suppose that GG is a group that acts on a set SS. When r,s∈Sr,s \in S, a solution g∈Gg \in G to rg=sr^g = s can be thought of as a kind of logarithm. In this paper, we study the case where G=SnG = S_n, and develop analogs to the Shanks baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two sets A,B⊆SnA, B \subseteq S_n such that every permutation of SnS_n can be written as a product abab of elements a∈Aa \in A and b∈Bb \in B. Our deterministic procedure is optimal up to constant factors, in the sense that AA and BB can be computed in optimal asymptotic complexity, and ∣A∣|A| and ∣B∣|B| are a small constant from n!\sqrt{n!} in size. We also analyze randomized "collision" algorithms for the same problem

    Computation Schemes for Splitting Fields of Polynomials

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    International audienceIn this article, we present new results about the computation of a general shape of a triangular basis generating the splitting ideal of an irreducible polynomial given with the permutation representation of its Galois group G. We provide some theoretical results and a new general algorithm based on the study of the non redundant bases of permutation groups. These new results deeply increase the efficiency of the computation of the splitting field of a polynomial
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