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

DOI:
https://doi.org/10.1090/S0025-5718-01-01352-7

Published electronically:
October 4, 2001

MathSciNet review:
1898752

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Abstract | References | Similar Articles | Additional Information

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

Email:
enge@math.uni-augsburg.de

**Andreas Stein**

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

Email:
andreas@math.uiuc.edu

DOI:
https://doi.org/10.1090/S0025-5718-01-01352-7

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