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**2002/032**

**Title:**

Supersingular Hyperelliptic Curve of Genus 2 over Finite Fields

**Authors:**
Y. Choie, E. Jeong, 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.

**ePrint:**
https://eprint.iacr.org/2002/032

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