## CryptoDB

### Iwan M. Duursma

#### Publications

Year
Venue
Title
2006
EPRINT
We generalize the ElGamal signature scheme for cyclic groups to a signature scheme for n-dimensional vector spaces. The higher dimensional version is based on the untractability of the vector decomposition problem (VDP). Yoshida has shown that under certain conditions, the VDP on a two-dimensional vector space is at least as hard as the computational Diffie-Hellman problem (CDHP) on a one-dimensional subspace. (Added November 19: Steven Galbraith recently showed that for the examples that are used in the paper, the VDP is at most as hard as the Discrete Logarithm problem (DLP) on a one-dimensional subspace. This has as a consequence for the proposed signature scheme that the given examples provide the same security as (ordinary) Elliptic Curve DLP based signature schemes.)
2005
EPRINT
The group of m-torsion points on an elliptic curve, for a prime number m, forms a two-dimensional vector space. It was suggested and proven by Yoshida that under certain conditions the vector decomposition problem (VDP) on a two-dimensional vector space is at least as hard as the computational Diffie-Hellman problem (CDHP) on a one-dimensional subspace. In this work we show that even though this assessment is true, it applies to the VDP for m-torsion points on an elliptic curve only if the curve is supersingular. But in that case the CDHP on the one-dimensional subspace has a known sub-exponential solution. Furthermore, we present a family of hyperelliptic curves of genus two that are suitable for the VDP.
2003
ASIACRYPT
2003
EPRINT
We give a closed formula for the Tate-pairing on the hyperelliptic curve $y^2 = x^p - x + d$ in characteristic $p$. This improves recent implementations by Barreto et.al. and by Galbraith et.al. for the special case $p=3$. As an application, we propose a $n$-round key agreement protocol for up to $3^n$ participants by extending Joux's pairing-based protocol to $n$ rounds.
1999
ASIACRYPT

#### Coauthors

Pierrick Gaudry (1)
Negar Kiyavash (1)
Hyang-Sook Lee (2)
François Morain (1)
SeungKook Park (1)