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Title:
Genus 2 Hyperelliptic Curve Families with Explicit Jacobian Order Evaluation and Pairing-Friendly Constructions
Authors: Aurore Guillevic, Damien Vergnaud
Abstract:The use of (hyper)elliptic curves in cryptography relies on the ability to compute the Jacobian order of a given curve. Recently, Satoh proposed a probabilistic polynomial time algorithm to test whether the Jacobian – over a finite field \mathbb{F}_q – of a hyperelliptic curve of the form Y^2 = X^5 + aX^3 + bX (with a,b \in \mathbb{F}_q^*) has a large prime factor. His approach is to obtain candidates for the zeta function of the Jacobian over \mathbb{F}_q^* from its zeta function over an extension field where the Jacobian splits. We extend and generalize Satoh’s idea to provide \emph{explicit} formulas for the zeta function of the Jacobian of genus 2 hyperelliptic curves of the form Y^2 = X^5 + aX^3 + bX and Y^2 = X^6 + aX^3 + b (with a,b \in \mathbb{F}_q^*). Our results are proved by elementary (but intricate) polynomial root-finding techniques. Hyperelliptic curves with small embedding degree and large prime-order subgroup are key ingredients for implementing pairing-based cryptographic systems. Using our closed formulas for the Jacobian order, we present several algorithms to obtain so-called \emph{pairing-friendly} genus 2 hyperelliptic curves. Our method relies on techniques initially proposed to produce pairing-friendly elliptic curves (namely, the Cocks-Pinch method and the Brezing-Weng method). We demonstrate this method by constructing several interesting curves with \rho-values around 3. We found for each embedding degree 5 \leqslant k \leqslant 35 a family of curves of \rho-value between 2.25 and 4.
ePrint: https://eprint.iacr.org/2011/604
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