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Reducing number field defining polynomials: an application to class group computations

Part of: Computational number theory Algebraic number theory: global fields

Published online by Cambridge University Press:  26 August 2016

Alexandre Gélin  and
Antoine Joux
Alexandre Gélin
Affiliation:
Sorbonne Universités, UPMC Paris 6, UMR 7606, LIP6, 75005, Paris, France email alexandre.gelin@lip6.fr
Antoine Joux
Affiliation:
Chaire de Cryptologie, Fondation UPMC, Sorbonne Universités, UPMC Paris 6, UMR 7606, LIP6, 75005, Paris, France email antoine.joux@m4x.org

Abstract

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In this paper we describe how to compute smallest monic polynomials that define a given number field $\mathbb{K}$. We make use of the one-to-one correspondence between monic defining polynomials of $\mathbb{K}$ and algebraic integers that generate $\mathbb{K}$. Thus, a smallest polynomial corresponds to a vector in the lattice of integers of $\mathbb{K}$ and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of $\mathbb{K}$. This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.

MSC classification

Primary: 11Y40: Algebraic number theory computations
Secondary: 11R21: Other number fields 11R29: Class numbers, class groups, discriminants
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Special Issue A: Algorithmic Number Theory Symposium XII , 2016 , pp. 315 - 331
Copyright
© The Author(s) 2016 

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