Improved CRT Algorithm for Class Polynomials in Genus $2$
Language
en
Communication dans un congrès
This item was published in
ANTS X - Algorithmic Number Theory 2012, 2012-07-09, San Diego. 2013-11-14, vol. 1, p. 437-461
Mathematical Sciences Publisher
English Abstract
We present a generalization to genus~2 of the probabilistic algorithm of Sutherland for computing Hilbert class polynomials. The improvement over the Br{ö}ker-Gruenewald-Lauter algorithm for the genus~2 case is that we do ...Read more >
We present a generalization to genus~2 of the probabilistic algorithm of Sutherland for computing Hilbert class polynomials. The improvement over the Br{ö}ker-Gruenewald-Lauter algorithm for the genus~2 case is that we do not need to find a curve in the isogeny class whose endomorphism ring is the maximal order; rather, we present a probabilistic algorithm for ''going up'' to a maximal curve (a curve with maximal endomorphism ring), once we find any curve in the right isogeny class. Then we use the structure of the Shimura class group and the computation of $(\ell,\ell)$-isogenies to compute all isogenous maximal curves from an initial one. This is an extended version of the article published at ANTS~X.Read less <
English Keywords
Class polynomials
European Project
Algorithmic Number Theory in Computer Science
ANR Project
Espaces de paramètres pour une arithmétique efficace et une évaluation de la sécurité des courbes - ANR-12-BS01-0010
Origin
Hal imported