# Pairings from a tensor product point of view

1 liasd
LIASD - Laboratoire d'Informatique Avancée de Saint-Denis
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.
Keywords :
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-00816318
Contributor : Laurent Poinsot <>
Submitted on : Friday, May 10, 2013 - 7:51:12 PM
Last modification on : Wednesday, February 6, 2019 - 1:22:11 AM
Long-term archiving on : Sunday, August 11, 2013 - 4:08:54 AM

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Pairing_and_tensor_10_05_2013....
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### Identifiers

• HAL Id : hal-00816318, version 2
• ARXIV : 1304.5779

### Citation

Nadia El Mrabet, Laurent Poinsot. Pairings from a tensor product point of view. 2013. ⟨hal-00816318v2⟩

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