We present an infinite family of hyperelliptic curves of genus two over a finite field of even characteristic which are suitable for the vector decomposition problem.

Intractable mathematical problems such as the integer factorization problem, the discrete logarithm problem (DLP), and the computational Diffie-Hellman problem (CDHP) are being used to provide secure protocols for cryptosystems. A new hard problem which is called the vector decomposition problem (VDP) was proposed by Yoshida et al. [

We state the definition of VDP and the conditions for the VDP on a two-dimensional vector space to be at least as hard as the CDHP on a one-dimensional subspace given by Yoshida [

The VDP on

The CDHP on

The vector decomposition problem on

For any

There are

The proof is in [

The VDP is hard in general but for certain bases the VDP can be solved in polynomial time even if it satisfies Yoshida’s conditions [

We give the definition of eigenvector base and distortion eigenvector base.

Let

An eigenvector base

The VDP with respect to an eigenvector base is solvable in polynomial time.

Two applications of the VDP are watermarking scheme designed for cryptographic date given in [

By Theorem

We state a theorem given by Kani and Rosen [

Given a curve

The Jacobian of the hyperelliptic curve

Let

For ease of computation we transform the elliptic curves

Elliptic function field.

In Figure

The modular equation of level three is

Let

In order to find a 3-isogeny using Theorem

Let

Let

In this section, we set up the VDP on the hyperelliptic curve

Choose

The following is a summary of the VDP setting:

For any element

Let

We need to show that

For an element

We begin by showing that

We have shown that the basis

Yoshida and Fujiwara introduced a new watermarking scheme for cryptographic data which is based on VDP. Duursma and Kiyavash showed that elliptic curves are not suitable for VDP and presented an infinite family of genus two hyperelliptic curves suitable for the VDP defined over a finite field of odd characteristic. In this paper, we introduce an infinite family of genus two hyperelliptic curves suitable for the VDP defined over a finite field of even characteristic.

The author declares that there is no conflict of interests regarding the publication of this paper.

This research was supported by the Sookmyung Women’s University Research Grants (1-1103-0682).