On implementation of GHS attack against elliptic curve cryptosystems over cubic extension fields of odd characteristic

In this paper, we present algorithms implementing the GHS attack against Elliptic curve cryptosystems (ECC). In particular, we consider two large classes of elliptic curves over cubic extension fields of odd characteristic which have weak covering curves against GHS attack, whose existence have been shown recently [17], [18], [19], [20]. We give algorithms to compute the defining equation of the covering curve and to transfer the DLP from the elliptic curve to the Jacobian of the covering curve. An algorithm to test if the covering curve is hyperelliptic is also given in the appendix.