Efficient pairing computation with theta functions

David Lubicz 1 Damien Robert 2
1 IRMAR - Institut de Recherche Mathématique de Rennes
2 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : In this paper, we present a new approach based on theta functions to compute Weil and Tate pairings. A benefit of our method, which does not rely on the classical Miller's algorithm, is its generality since it extends to all abelian varieties the classical Weil and Tate pairing formulas. In the case of dimension $1$ and $2$ abelian varieties our algorithms lead to implementations which are efficient and naturally deterministic. We also introduce symmetric Weil and Tate pairings on Kummer varieties and explain how to compute them efficiently. We exhibit a nice algorithmic compatibility between some algebraic groups quotiented by the action of the automorphism $-1$, where the $\Z$-action can be computed efficiently with a Montgomery ladder type algorithm.
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Communication dans un congrès
Guillaume Hanrot and François Morain and Emmanuel Thomé. ANTS IX - Algorithmic Number Theory 2010, Jul 2010, Nancy, France. Springer-Verlag, 6197, pp.251-269, 2010, Lecture Notes in Computer Science. 〈10.1007/978-3-642-14518-6_21〉
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INRIA | INSMI | IRMAR | IRMAR-GAE | UNIV-LORRAINE | UNAM | LORIA-ALGO | LORIA | UNIV-RENNES2 | UR2-HB | UNIV-RENNES

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David Lubicz, Damien Robert. Efficient pairing computation with theta functions. Guillaume Hanrot and François Morain and Emmanuel Thomé. ANTS IX - Algorithmic Number Theory 2010, Jul 2010, Nancy, France. Springer-Verlag, 6197, pp.251-269, 2010, Lecture Notes in Computer Science. 〈10.1007/978-3-642-14518-6_21〉. 〈hal-00528944〉

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