Laurent Poinsot

Nadia El Mrabet, Laurent Poinsot

Pairings are particular bilinear maps, and as any bilinear maps they factor through the tensor product as group homomorphisms. Besides, nothing seems to prevent us to construct pairings on other abelian groups than elliptic curves or more general abelian varieties. The point of view adopted in this contribution is based on these two observations. Thus we present an elliptic curve free study of pairings which is essentially based on tensor products of abelian groups (or modules). Tensor products of abelian groups are even explicitly computed under finiteness conditions. We reveal that the existence of pairings depends on the non-degeneracy of some universal bilinear map, called the canonical bilinear map. In particular it is shown that the construction of a pairing on $A\times A$ is always possible whatever a finite abelian group $A$ is. We also propose some new constructions of pairings, one of them being based on the notion of group duality which is related to the concept of non-degeneracy.

Laurent Poinsot

It is well-known that degree two finite field extensions can be equipped with a Hermitian-like structure similar to the extension of the complex field over the reals. In this contribution, using this structure, we develop a modular character theory and the appropriate Fourier transform for some particular kind of finite Abelian groups. Moreover we introduce the notion of bent functions for finite field valued functions rather than usual complex-valued functions, and we study several of their properties. In particular we prove that this bentness notion is a consequence of that of Logachev, Salnikov and Yashchenko, introduced in "Bent functions on a finite Abelian group" (1997). In addition this new bentness notion is also generalized to a vectorial setting.