**An Elliptic Curve Trapdoor System**

*Edlyn Teske*

**Abstract: **We propose an elliptic curve trapdoor system which is of interest in
key escrow applications. In this system, a pair
($E_{\rm s}, E_{\rm pb}$) of elliptic curves over $\F_{2^{161}}$ is constructed with the following properties: (i) the Gaudry-Hess-Smart Weil descent attack reduces the elliptic curve discrete logarithm problem (ECDLP) in $E_{\rm s}(\F_{2^{161}})$ to a hyperelliptic curve DLP in the Jacobian of a curve of genus 7 or 8, which is computationally feasible, but by far not trivial; (ii) $E_{\rm pb}$ is isogenous to $E_{\rm s}$; (iii) the best attack on the
ECDLP in $E_{\rm pb}(\F_{2^{161}})$ is the parallelized Pollard rho method.\\
The curve $E_{\rm pb}$ is used just as usual in elliptic curve cryptosystems. The curve $E_{\rm s} is submitted to a trusted authorityfor the purpose of key escrow. The crucial difference from other key escrow scenarios is that the trusted authority has to invest a considerable amount of computation to compromise a user's
private key, which makes applications such as widespread wire-tapping
impossible.

**Category / Keywords: **public-key cryptography / elliptic curve cryptosystem, key escrow

**Date: **received 31 Mar 2003

**Contact author: **eteske at math uwaterloo ca

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**Version: **20030401:012139 (All versions of this report)

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