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In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual. It was introduced by André Weil (1940) for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function. Choose an elliptic curve E defined over a field K, and an integer n > 0 (we require n to be coprime to char(K) if char(K) > 0) such that K contains a primitive nth root of unity. Then the n-torsion on is known to be a Cartesian product of two cyclic groups of order n. The Weil pairing produces an n-th root of unity by means of Kummer theory, for any two points , where and . A down-to-earth construction of the Weil pairing is as follows. Choose a function F in the function field of E over the algebraic closure of K with divisor So F has a simple zero at each point P + kQ, and a simple pole at each point kQ if these points are all distinct. Then F is well-defined up to multiplication by a constant. If G is the translation of F by Q, then by construction G has the same divisor, so the function G/F is constant. Therefore if we define we shall have an n-th root of unity (as translating n times must give 1) other than 1. With this definition it can be shown that w is alternating and bilinear, giving rise to a non-degenerate pairing on the n-torsion. The Weil pairing does not extend to a pairing on all the torsion points (the direct limit of n-torsion points) because the pairings for different n are not the same. However they do fit together to give a pairing Tl(E) × Tl(E) → Tl(μ) on the Tate module Tl(E) of the elliptic curve E (the inverse limit of the ln-torsion points) to the Tate module Tl(μ) of the multiplicative group (the inverse limit of ln roots of unity).

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Weil pairing

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Isogeny graphs of ordinary abelian varieties

Dimitar Petkov Jetchev, Benjamin Pierre Charles Wesolowski, Ernest Hunter Brooks

Fix a prime number l. Graphs of isogenies of degree a power of l are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called l-isogenies, resolving that, in arbitrary dimension, their structure is similar, but not identical, to the "volcanoes" occurring as graphs of isogenies of elliptic curves. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as (l, l)-isogenies: those whose kernels are maximal isotropic subgroups of the l-torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.

Springer Heidelberg2017

Efficient multistage secret sharing scheme using bilinear map

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In a multistage secret sharing (MSSS) scheme, the authorised subsets of participants could recover a number of secrets in different stages. A one-stage multisecret sharing (OSMSS) scheme is a special case of MSSS schemes in which all the secrets are recovered simultaneously. In these schemes, in addition to the individual shares, the dealer should provide the participants with a number of public values associated with the secrets. The less the number of public values, the more efficient is the scheme. It is desired that the MSSS and OSMSS schemes provide computational security. In this study, the authors show that in the OSMSS schemes any unauthorised coalition of the participants can reduce the uncertainty of the secrets. In addition, in MSSS schemes recovering a secret causes reducing uncertainty of the unrecovered secrets. Furthermore, by introducing a new multi-use MSSS scheme based on weil pairing, they reduce the number of public values comparing with the previous schemes.

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