**Supersingular Hyperelliptic Curve of Genus 2 over Finite Fields**

*Y. Choie and E. Jeong and E. Lee*

**Abstract: **In this paper we describe an elementary criterion to determine
supersingular hyperelliptic curve of genus $2$, using
only the given Weierstrass equation.
Furthermore, we show that the discrete logarithm problem defined on
any
supersingular abelian variety of dimension $2$ over
${\mathbb F}_p, p>16,$ can be embedded to that over the extension field
${\mathbb F}_{p^{k}}$, with $k \leq 6.$
This implies that
supersingular hyperelliptic curves are cryptographically
weaker than the general case due to
the Frey-R$\ddot{u}$ck attack.
A family of the hyperelliptic
curve $H/{\mathbb F}_p$ of the type $v^2=u^5+a$ and $v^2 = u^5 +
au$ have been studied and further examples are listed.

**Category / Keywords: **public-key cryptography / hyperelliptic curves, supersingular, discrete logarithm problem

**Date: **received 11 Mar 2002, last revised 12 Mar 2002

**Contact author: **ejlee at postech ac kr

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | BibTeX Citation

**Version: **20020313:012606 (All versions of this report)

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