Weil pairing
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 "n"th 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. Formulation. 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 formula_1 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 formula_3, where formula_4 and formula_5. 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 "T"("E") × "T"("E") → "T"(μ) on the Tate module "T"("E") of the elliptic curve "E" (the inverse limit of the ℓ-torsion points) to the Tate module "T"(μ) of the multiplicative group (the inverse limit of ℓ roots of unity). Generalisation to abelian varieties. For abelian varieties over an algebraically closed field "K , the Weil pairing is a nondegenerate pairing for all n" prime to the characteristic of " K". Here formula_9 denotes the dual abelian variety of "A". This is the so-called "Weil pairing" for higher dimensions. If "A" is equipped with a polarisation then composition gives a (possibly degenerate) pairing If "C" is a projective, nonsingular curve of genus ≥ 0 over "k", and "J" its Jacobian, then the theta-divisor of "J" induces a principal polarisation of "J", which in this particular case happens to be an isomorphism (see autoduality of Jacobians). Hence, composing the Weil pairing for "J" with  the polarisation gives a nondegenerate pairing for all "n" prime to the characteristic of "k". As in the case of elliptic curves, explicit formulae for this pairing can be given in terms of divisors of "C". Applications. The pairing is used in number theory and algebraic geometry, and has also been applied in elliptic curve cryptography and identity based encryption.