Paper 2001/084

Analysis of the GHS Weil Descent Attack on the ECDLP over Characteristic Two Finite Fields of Composite Degree

Markus Maurer, Alfred Menezes, and Edlyn Teske

Abstract

In this paper, we analyze the Gaudry-Hess-Smart (GHS) Weil descent attack on the elliptic curve discrete logarithm problem (ECDLP) for elliptic curves defined over characteristic two finite fields of composite extension degree. For each such field $F_{2^N}$, $N \in [100,600]$, we identify elliptic curve parameters such that (i) there should exist a cryptographically interesting elliptic curve $E$ over $F_{2^N}$ with these parameters; and (ii) the GHS attack is more efficient for solving the ECDLP in $E(F_{2^N})$ than for solving the ECDLP on any other cryptographically interesting elliptic curve over $F_{2^N}$. We examine the feasibility of the GHS attack on the specific elliptic curves over $F_{2^{176}}$, $F_{2^{208}}$, $F_{2^{272}}$, $F_{2^{304}}$, and $F_{2^{368}}$ that are provided as examples inthe ANSI X9.62 standard for the elliptic curve signature scheme ECDSA. Finally, we provide several concrete instances of the ECDLP over $F_{2^N}$, $N$ composite, of increasing difficulty which resist all previously known attacks but which are within reach of the GHS attack.