Cryptology ePrint Archive: Report 2008/491

CM Construction of Genus 2 Curves with p-rank 1

Laura Hitt O'Connor and Gary McGuire and Michael Naehrig and Marco Streng

Abstract: We construct Weil numbers corresponding to genus-$2$ curves with $p$-rank~$1$ over the finite field $\F_{p^2}$ of $p^2$ elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of $\F_{p^2}$-valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over $\F_{p^2}$ out of necessity: we show that curves of $p$-rank $1$ over $\F_p$ for large~$p$ cannot be efficiently constructed using explicit CM constructions.

Category / Keywords: foundations / hyperelliptic curve cryptography

Date: received 20 Nov 2008, last revised 12 Jan 2010

Contact author: gary mcguire at ucd ie

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Version: 20100112:123544 (All versions of this report)

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