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Mathematics of Computation

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Structure computation and discrete logarithms in finite abelian $ p$-groups

Author: Andrew V. Sutherland
Journal: Math. Comp. 80 (2011), 477-500
MSC (2010): Primary 11Y16; Secondary 20K01, 12Y05
Published electronically: April 16, 2010
MathSciNet review: 2728991
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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(\vert G\vert^{1/2})$ group operations. These results also improve generic algorithms for extracting $ p$th roots in $ G$.

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Additional Information

Andrew V. Sutherland
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): September 19, 2008
Received by editor(s) in revised form: July 27, 2009, and August 29, 2009
Published electronically: April 16, 2010
Article copyright: © Copyright 2010 by the author

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