Section: New Results

Pairings

Participants : Andreas Enge, Damien Robert.

In [27] , A. Enge gives an elementary and self-contained introduction to pairings on elliptic curves over finite fields. For the first time in the literature, the three different definitions of the Weil pairing are stated correctly and proved to be equivalent using Weil reciprocity. Pairings with shorter loops, such as the ate, ate${}_i$, R-ate and optimal pairings, together with their twisted variants, are presented with proofs of their bilinearity and non-degeneracy. Finally, different types of pairings are reviewed in a cryptographic context. The article can be seen as an update chapter to [40] .

With D. Lubicz, D. Robert has worked on extending the algorithm to compute Weil and Tate pairings using theta functions from [42] to the ate and optimal ate pairings in [29] . The result includes how to compute the Miller functions with theta functions, but also how to generalise ate and optimal ate pairings to Kummer varieties. In contrast to preceding algorithms using Miller functions which needed a geometric interpretation of the addition law and worked with Jacobians, this new algorithm uses only the algebraic Riemann relations and works on any abelian variety (provided with a theta structure). This algorithm has been implemented using AVIsogenies .