Cover attacks for elliptic curves with prime order

Abstract

We give a new approach to the elliptic curve discrete logarithm problem over cubic extension fields $\mathbb{F}_{q^3}$. It is based on a transfer: First an $\mathbb{F}_q$-rational $(\ell,\ell,\ell)$-isogeny from the Weil restriction of the elliptic curve under consideration with respect to $\mathbb{F}_{q^3}/\mathbb{F}_q$ to the Jacobian variety of a genus three curve over $\mathbb{F}_q$ is applied and then the problem is solved in the Jacobian via the index-calculus attacks. Although using no covering maps in the construction of the desired homomorphism, this method is, in a sense, a kind of cover attack. As a result, it is possible to solve the discrete logarithm problem in some elliptic curve groups of prime order over $\mathbb{F}_{q^3}$ in a time of $\tilde{O}(q)$.