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Picard Groups and Refined Discrete Logarithms

Published online by Cambridge University Press:  01 February 2010

W. Bley
Affiliation:
Institut für Mathematik, Universität Augsburg, Universitätsstrasse 8, D-86135 Augsburg, Germany, bley@math.uni-augsburg.de
M. Endres
Affiliation:
Institut für Mathematik, Universität Augsburg, Universitätsstrasse 8, D-86135 Augsburg, Germany, me@mendres.org

Abstract

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Let K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by $/cal{O}_K$, and $/cal{A}$ denotes any $/cal{O}_K$-order in K[G]. The paper describes an algorithm that explicitly computes the Picard group Pic($/cal{A}$), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of $/cal{O}_L$ in Pic($/cal{O}_K[G]$) can be numerically determined.

Type
Research Article
Copyright
Copyright © London Mathematical Society 2005

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