 Weil Pairing

# 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 [itex]E(overline)[/itex] 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

[itex]w(P,Q) in mu_n[/itex]

for given points [itex]P,Q in E(K)[/itex], where [itex]E(K)= [/itex] and [itex]mu_n = [/itex], 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:

[itex] (F)= sum(P+kcdot Q) - sum (kcdot Q) [/itex]

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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