Improved CRT Algorithm for Class Polynomials in Genus $2$

Kristin Lauter 1 Damien Robert 2
1 Cryptography group
Microsoft Research [Redmond]
2 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
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 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.
Keywords : Class polynomials
Type de document :
Communication dans un congrès
Everett W. Howe and Kiran S. Kedlaya. ANTS X - Algorithmic Number Theory 2012, Jul 2012, San Diego, United States. Mathematical Sciences Publisher, 1, pp.437-461, 2013, The Open Book Series. 〈10.2140/obs.2013.1.437〉
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Kristin Lauter, Damien Robert. Improved CRT Algorithm for Class Polynomials in Genus $2$. Everett W. Howe and Kiran S. Kedlaya. ANTS X - Algorithmic Number Theory 2012, Jul 2012, San Diego, United States. Mathematical Sciences Publisher, 1, pp.437-461, 2013, The Open Book Series. 〈10.2140/obs.2013.1.437〉. 〈hal-00734450v2〉

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