Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Weil pairing
Summary
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 nth 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.
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 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 , where and .
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 Tl(E) × Tl(E) → Tl(μ) on the Tate module Tl(E) of the elliptic curve E (the inverse limit of the ln-torsion points) to the Tate module Tl(μ) of the multiplicative group (the inverse limit of ln roots of unity).
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.