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Bilinear pairings on elliptic curves

Andreas Enge 1, 2 
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : We give 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, we review different types of pairings in a cryptographic context. This article can be seen as an update chapter to A. Enge, Elliptic Curves and Their Applications to Cryptography - An Introduction, Kluwer Academic Publishers 1999.
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Submitted on : Friday, February 14, 2014 - 6:59:58 PM
Last modification on : Thursday, January 20, 2022 - 5:31:35 PM
Long-term archiving on: : Thursday, May 15, 2014 - 8:20:11 AM


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  • HAL Id : hal-00767404, version 2
  • ARXIV : 1301.5520



Andreas Enge. Bilinear pairings on elliptic curves. L'Enseignement Mathématique , 2015, 61 (2), pp.211-243. ⟨hal-00767404v2⟩



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