Counting points on genus 2 curves with real multiplication

Authors:
Pierrick Gaudry;David Kohel
Affiliations:
LORIA, CNRS / INRIA / Nancy Université, Vandoeuvre lès Nancy, France;Institut de Mathématiques de Luminy, Université de la Méditerranée, Marseille Cedex 9, France
Venue:
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
Year:
2011

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Abstract

We present an accelerated Schoof-type point-counting algorithm for curves of genus 2 equipped with an efficiently computable real multiplication endomorphism. Our new algorithm reduces the complexity of genus 2 point counting over a finite field $\mathbb{F}_{q}$ of large characteristic from ${\widetilde{O}}(\log^8 q)$ to ${\widetilde{O}}(\log^5 q)$ . Using our algorithm we compute a 256-bit prime-order Jacobian, suitable for cryptographic applications, and also the order of a 1024-bit Jacobian.