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We give a new approach to the elliptic curve discrete logarithm problem overcubic 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 ofthe elliptic curve under consideration with respect to$\mathbb{F}_{q^3}/\mathbb{F}_q$ to the Jacobian variety of a genus three curveover $\mathbb{F}_q$ is applied and then the problem is solved in the Jacobianvia the index-calculus attacks. Although using no covering maps in theconstruction of the desired homomorphism, this method is, in a sense, a kind ofcover attack. As a result, it is possible to solve the discrete logarithmproblem in some elliptic curve groups of prime order over $\mathbb{F}_{q^3}$ ina time of $\tilde{O}(q)$.

Cover attacks for elliptic curves with prime order