A Weil pairing on the p-torsion of ordinary elliptic curves over K[ϵ]K[ϵ]☆

For an elliptic curve E over any field K, the Weil pairing en is a bilinear map on n-torsion. In this paper, we consider E over the dual numbers K[ϵ] and define a non-degenerate “Weil pairing on p-torsion” which shares many of the same properties of the Weil pairing. We also show that the discrete logarithm attacks on p-torsion subgroups of Semaev and Rück may be viewed as Weil-pairing-based attacks, just like the MOV attack. Finally, we describe an attack on the discrete logarithm problem on anomalous curves, analogous to that of Smart, using a lift of E over Fp[ϵ].