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Efficient pairing computation with Theta functions
 ANTS IX. LNCS
"... 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 cas ..."
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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 Zaction can be computed efficiently with a Montgomery ladder type algorithm.
A generalisation of Miller’s algorithm and applications to pairing computations on abelian varieties
, 2013
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Miller’s algorithm Pairings on abelian varieties Theta functions Pairings with theta functions Performance Outline
"... Miller’s algorithm Pairings on abelian varieties Theta functions Pairings with theta functions Performance The Weil pairing on elliptic curves Let E: y2 = x3+ax+b be an elliptic curve over a field k (chark 6 = 2,3, 4a3+27b2 6 = 0.) Let P,Q E[`] be points of `torsion. Let fP be a function associate ..."
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Miller’s algorithm Pairings on abelian varieties Theta functions Pairings with theta functions Performance The Weil pairing on elliptic curves Let E: y2 = x3+ax+b be an elliptic curve over a field k (chark 6 = 2,3, 4a3+27b2 6 = 0.) Let P,Q E[`] be points of `torsion. Let fP be a function associated to the principal divisor `(P) − `(0), and fQ to `(Q) − `(0). We define: eW,`(P,Q)= fP((Q) − (0)) fQ((P) − (0)). The application eW, ` : E[`]×E[`]→µ`(k) is a non degenerate pairing: the Weil pairing. Definition (Embedding degree) If E is defined over a finite field Fq, the Weil pairing has image in µ`(Fq)⊂F∗qd where d is the embedding degree, the smallest number such that `  qd − 1. Miller’s algorithm Pairings on abelian varieties Theta functions Pairings with theta functions Performance The Tate pairing on elliptic curves over Fq Definition The Tate pairing is a non degenerate bilinear application given by eT: E0[`]×E(Fq)/`E(Fq) − → F∗qd/F∗qd `