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 F₂ ^N, N ∈ [160, 600], we identify elliptic curve parameters such that (i)there should exist a cryptographically interesting elliptic curve E over F₂ ^N with these parameters; and (ii)the GHS attack is more efficient for solving the ECDLP in E(F ₂ ^N )than for any other cryptographically interesting elliptic curve over F₂ ^N.