Weil pairing

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Description:
In mathematics, the Weil pairing is a construction of roots of unity by means of functions on an elliptic curve E, in such a way as to constitute a pairing (bilinear form, though with multiplicative notation) on the torsion subgroup of E. The name is for André Weil, 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

Suppose E is defined over a field K. Given an integer n > 0 (we require n to be prime to char(K) if char(K)> 0) such that K contains a primitive nth root of unity, then the n-torsion on <math>E(overline)</math> has known structure, as a Cartesian product of two cyclic groups of order n. The basis of the construction is of an n-th root of unity

<math>w(P,Q) in mu_n</math>


for given points <math>P,Q in E(K)</math>, where <math>E(K)= </math> and <math>mu_n = </math>, by means of Kummer theory.

By a direct argument one can define a function F in the function field of E over the algebraic closure of K, by its divisor:

<math> (F)= sum(P+kcdot Q) - sum (kcdot Q) </math>


with sums for 0 ≤ k &lt; n. In words F has a simple zero at each point P + kQ, and a simple pole at each point kQ. 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. One can show...... ...
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