**Isogeny graphs with maximal real multiplication**

*Sorina Ionica and Emmanuel Thomé*

**Abstract: **An isogeny graph is a graph whose vertices are principally polarized abelian 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.

**Category / Keywords: **public-key cryptography / genus 2 curve, isogeny graph, endomorphism ring, computation

**Date: **received 30 Mar 2014, last revised 23 Jun 2014

**Contact author: **sorina ionica at m4x org

**Available format(s): **PDF | BibTeX Citation

**Version: **20140623:230901 (All versions of this report)

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