Paper 2011/604

Genus 2 Hyperelliptic Curve Families with Explicit Jacobian Order Evaluation and Pairing-Friendly Constructions

Aurore Guillevic and Damien Vergnaud


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 $4$.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Unknown where it was published
Hyperelliptic CurvesGenus 2Order ComputationOrdinary CurvesPairing-Friendly ConstructionsCocks-Pinch MethodBrezing-Weng Method.
Contact author(s)
guillevi @ di ens fr
2012-05-12: revised
2011-11-10: received
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      author = {Aurore Guillevic and Damien Vergnaud},
      title = {Genus 2 Hyperelliptic Curve Families with Explicit Jacobian Order Evaluation and Pairing-Friendly Constructions},
      howpublished = {Cryptology ePrint Archive, Paper 2011/604},
      year = {2011},
      note = {\url{}},
      url = {}
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