## Cryptology ePrint Archive: Report 2009/103

Constructing pairing-friendly hyperelliptic curves using Weil restriction

David Mandell Freeman and Takakazu Satoh

Abstract: A pairing-friendly curve is a curve over a finite field whose Jacobian has small embedding degree with respect to a large prime-order subgroup. In this paper we construct pairing-friendly genus 2 curves over finite fields $\mathbb{F}_q$ whose Jacobians are ordinary and simple, but not absolutely simple. We show that constructing such curves is equivalent to constructing elliptic curves over $\mathbb{F}_q$ that become pairing-friendly over a finite extension of $\mathbb{F}_q$. Our main proof technique is Weil restriction of elliptic curves. We describe adaptations of the Cocks-Pinch and Brezing-Weng methods that produce genus 2 curves with the desired properties. Our examples include a parametric family of genus 2 curves whose Jacobians have the smallest recorded $\rho$-value for simple, non-supersingular abelian surfaces.

Category / Keywords: Public-key cryptography / pairing-friendly curves, Weil restriction

Publication Info: (None)

Date: received 1 Mar 2009, last revised 26 Nov 2009

Contact author: satohaar at mathpc-satoh math titech ac jp

Available format(s): PDF | BibTeX Citation

Note: Submission version.

Short URL: ia.cr/2009/103

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