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Smooth ideals in hyperelliptic function fields

Authors: Andreas Enge and Andreas Stein
Journal: Math. Comp. 71 (2002), 1219-1230
MSC (2000): Primary 11R58, 11Y16, 11R44, 14H40, 68Q25
Published electronically: October 4, 2001
MathSciNet review: 1898752
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Abstract: Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of high-genus hyperelliptic curves over finite fields. Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense. We explicitly show how these density results can be derived. All proofs are purely combinatorial and do not exploit analytic properties of generating functions.

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Additional Information

Andreas Enge
Affiliation: Lehrstuhl für Diskrete Mathematik, Optimierung und Operations Research, Universität Augsburg, 86135 Augsburg, Germany

Andreas Stein
Affiliation: University of Illinois at Urbana-Champaign, Department of Mathematics, 1409 West Green Street, Urbana, Illinois 61801

Keywords: Distribution of prime ideals, smooth ideal, hyperelliptic function field, subexponential algorithm
Received by editor(s): January 30, 2000
Received by editor(s) in revised form: October 3, 2000
Published electronically: October 4, 2001
Article copyright: © Copyright 2001 American Mathematical Society