Structure computation and discrete logarithms in finite abelian 𝑝-groups

Sutherland, Andrew

doi:10.1090/S0025-5718-10-02356-2

Structure computation and discrete logarithms in finite abelian $p$-groups
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by Andrew V. Sutherland PDF

Math. Comp. 80 (2011), 477-500

Abstract:

We present a generic algorithm for computing discrete logarithms in a finite abelian $p$-group $H$, improving the Pohlig–Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for $H$ without using a relation matrix. The problem of computing a basis for some or all of the Sylow $p$-subgroups of an arbitrary finite abelian group $G$ is addressed, yielding a Monte Carlo algorithm to compute the structure of $G$ using $O(|G|^{1/2})$ group operations. These results also improve generic algorithms for extracting $p$th roots in $G$.

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