Cryptology ePrint Archive: Report 2002/032

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

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

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