Pairings from a tensor product point of view

TL;DR

This paper offers an elliptic curve free, tensor product-based framework for understanding pairings on abelian groups, revealing their universal properties and proposing new constructions beyond elliptic curves.

Contribution

It introduces a tensor product perspective on pairings, generalizes their construction to arbitrary finite abelian groups, and proposes novel pairing constructions using group duality.

Findings

Pairings depend on the non-degeneracy of the canonical bilinear map.

Construction of pairings on any finite abelian group is always possible.

New pairing constructions based on group duality are proposed.

Abstract

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.