On Implementation of GHS Attack against Elliptic Curve Cryptosystems over Cubic Extension Fields of Odd Characteristics

In this paper, we present algorithms for implementation of the GHS attack to Elliptic curve cryptosystems (ECC). In particular, we consider two large classes of elliptic curves over cubic extension fields of odd characteristics which have weak covering curves against GHS attack, whose existence have been shown recently [16][17][18]. We show an algorithm to find definition equation of the covering curve and an algorithm to transfer DLP of the elliptic curve to Jacobian of the covering curve. An algorithm to test if the covering curve is hyperelliptic is also shown in the appendix. keywords Elliptic curve cryptosystems, Discrete logarithm problem, GHS attack