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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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The arithmetic of certain cubic function fields
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by Mark L. Bauer;
Math. Comp. 73 (2004), 387-413
DOI: https://doi.org/10.1090/S0025-5718-03-01559-X
Published electronically: June 17, 2003

Abstract:

In this paper, we discuss the properties of curves of the form $y^3=f(x)$ over a given field K of characteristic different from 3. If $f(x)$ satisfies certain properties, then the Jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field. We seek to make this connection concrete and then use it to develop an explicit arithmetic for the Jacobian of such curves. From a purely mathematical perspective, this provides explicit and efficient techniques for performing arithmetic in certain ideal class groups which are of fundamental interest in algebraic number theory. At the same time, it provides another source of groups which are suitable for Diffie-Hellman type protocols in cryptographic applications.
References
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Bibliographic Information
  • Mark L. Bauer
  • Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
  • Address at time of publication: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2G 3L1 Canada
  • Email: m-bauer@math.uiuc.edu, mbauer@math.uwaterloo.ca
  • Received by editor(s): April 10, 2001
  • Received by editor(s) in revised form: July 18, 2002
  • Published electronically: June 17, 2003
  • © Copyright 2003 American Mathematical Society
  • Journal: Math. Comp. 73 (2004), 387-413
  • MSC (2000): Primary 11R58, 94A60
  • DOI: https://doi.org/10.1090/S0025-5718-03-01559-X
  • MathSciNet review: 2034129