Weil Pairing

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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

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:

with sums for 0 ≤*k* < *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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- <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

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

with sums for 0 ≤

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