Paper 2007/010

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

David Freeman and Kristin Lauter


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.

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Publication info
Published elsewhere. Proceedings of SAGA 2007, Tahiti (to appear)
CM methodhyperelliptic curvesjacobiansgenus 2point counting
Contact author(s)
dfreeman @ math berkeley edu
2007-05-30: last of 3 revisions
2007-01-19: received
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      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{}},
      url = {}
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