  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 $\tilde{O}(q)$.
