### Computing endomorphism rings of Jacobians of genus 2 curves over finite fields

David Freeman and Kristin Lauter

##### Abstract

We present probabilistic algorithms which, given a genus 2 curve C defined over a finite field and a quartic CM field K, determine whether the endomorphism ring of the Jacobian J of C is the full ring of integers in K. In particular, we present algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups J[l^d] for prime powers l^d. We use these algorithms to create the first implementation of Eisentrager and Lauter's algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem, and we demonstrate the algorithm for a few small examples. We observe that in practice the running time of the CRT algorithm is dominated not by the endomorphism ring computation but rather by the need to compute p^3 curves for many small primes p.

Note: Revised version, incorporating reader feedback.

Available format(s)
Category
Implementation
Publication info
Published elsewhere. Proceedings of SAGA 2007, Tahiti (to appear)
Keywords
CM methodhyperelliptic curvesjacobiansgenus 2point counting
Contact author(s)
dfreeman @ math berkeley edu
History
2007-05-30: last of 3 revisions
See all versions
Short URL
https://ia.cr/2007/010

CC BY

BibTeX

@misc{cryptoeprint:2007/010,
author = {David Freeman and Kristin Lauter},
title = {Computing endomorphism rings of Jacobians of genus 2 curves over finite fields},
howpublished = {Cryptology ePrint Archive, Paper 2007/010},
year = {2007},
note = {\url{https://eprint.iacr.org/2007/010}},
url = {https://eprint.iacr.org/2007/010}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.