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This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed. (February 2008) |
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.
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) and suppose that K contains a primitive nth root of unity. Then the n-torsion on E 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 ![P,Q \in E[n]](http://upload.wikimedia.org/math/c/1/9/c19489d4ef1b8d68b7e4dbe63f126c8b.png)
, where ![E[n]=\{T \in E \mid n \cdot T = O \}](http://upload.wikimedia.org/math/6/f/3/6f33e5bb7d1fd1cabfd9781ff048642f.png)
and 
, 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 that
In fact then G/F would yield a function on the isogenous curve E/C where C is the cyclic subgroup generated by Q, having just one simple pole. Such a function cannot exist, as follows by proving the residue at the pole is zero, a contradiction.
Therefore if we define
we shall have an n-th root of unity (translating n times must give 1) other than 1. With this definition it can be shown that w is antisymmetric and bilinear, giving rise to a non-degenerate pairing on the n-torsion.
The Weil pairing is used in number theory and algebraic geometry, and has also been applied in elliptic curve cryptography and identity based encryption.
