**Families of genus 2 curves with small embedding degree**

*Laura Hitt*

**Abstract: **Hyperelliptic curves of small genus have the advantage of
providing a group of comparable size as that of elliptic curves,
while working over a field of smaller size. Pairing-friendly
hyperelliptic curves are those whose order of the Jacobian is
divisible by a large prime, whose embedding degree is small enough
for computations to be feasible, and whose minimal embedding field
is large enough for the discrete logarithm problem in it to be
difficult. We give a sequence of $\F_q$-isogeny classes for a family
of Jacobians of genus two curves over $\F_{q}$, for $q=2^m$, and
their corresponding small embedding degrees. We give examples of
the parameters for such curves with embedding degree $k<(\log q)^2$,
such as $k=8,13,16,23,26,37,46,52$.

For secure and efficient implementation of pairing-based cryptography on genus g curves over $\F_q$, it is desirable that the ratio $\rho=\frac{g\log_2 q}{\log_2N}$ be approximately 1, where $N$ is the order of the subgroup with embedding degree $k$. We show that for our family of curves, $\rho$ is often near 1 and never more than 2.

We also give a sequence of $\F_q$-isogeny classes for a family of Jacobians of genus 2 curves over $\F_{q}$ whose minimal embedding field is much smaller than the finite field indicated by the embedding degree $k$. That is, the extension degrees in this example differ by a factor of $m$, where $q=2^m$, demonstrating that the embedding degree can be a far from accurate measure of security. As a result, we use an indicator $k'=\frac{\ord_N2}{m}$ to examine the cryptographic security of our family of curves.

**Category / Keywords: **embedding degree, genus 2, hyperelliptic curves, binary curves, pairing-based cryptosystems

**Date: **received 30 Dec 2006, last revised 13 Feb 2009

**Contact author: **hitt36 at gmail com

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**Note: **updated details in proofs

**Version: **20090213:170832 (All versions of this report)

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