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Computing endomorphism rings of Jacobians of genus 2 curves over finite fields
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| 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. |
BibTeX
@misc{eprint-2007-13292,
title={Computing endomorphism rings of Jacobians of genus 2 curves over finite fields},
booktitle={IACR Eprint archive},
keywords={implementation / CM method, hyperelliptic curves, jacobians, genus 2, point counting},
url={http://eprint.iacr.org/2007/010},
note={Proceedings of SAGA 2007, Tahiti (to appear) dfreeman@math.berkeley.edu 13663 received 10 Jan 2007, last revised 30 May 2007},
author={David Freeman and Kristin E. Lauter},
year=2007
}