Cryptology ePrint Archive: Report 2014/230

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 describes the structure of isogeny graphs for elliptic curves and shows that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth-first search (DFS) 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. In this article, we show that, under certain circumstances, the problem of determining the endomorphism ring can also be solved in genus~2 with a DFS-based algorithm. We consider the case of genus-2 Jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number one. We describe the isogeny graphs in that case, locally at prime numbers which split in the real multiplication subfield. The resulting algorithm is implemented over finite fields, and examples are provided. To the best of our knowledge, this is the first DFS-based algorithm in genus~2.

Category / Keywords: isogeny graphs, abelian varieties, genus 2

Date: received 30 Mar 2014, last revised 19 Jun 2015

Contact author: sorina ionica at m4x org

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Version: 20150619:094222 (All versions of this report)

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