IACR News item: 01 April 2014
Sorina Ionica, Emmanuel Thomé
ePrint Reportabelian varieties and whose edges are isogenies between these varieties. In
his thesis, Kohel described the structure of isogeny graphs for elliptic
curves and showed that one may compute the endomorphism ring of an elliptic
curve defined over a finite field by using a depth first search algorithm
in the graph. In dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive.
Our setting considers genus 2 jacobians with complex multiplication,
with the assumptions that the real multiplication subring is maximal and
has class number one. We fully describe the isogeny graphs in that
case.
Over finite fields, we derive a depth first search algorithm for computing endomorphism rings locally at prime numbers, if the real multiplication is maximal. To the best of our knowledge, this is the first DFS-based algorithm in genus 2.
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