Weil pairing

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 prime to char(K) if char(K) > 0) such that K contains a primitive nth root of unity. Then the n-torsion on ${\displaystyle E({\overline {K}})}$ 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 ${\displaystyle P,Q\in E(K)[n]}$, where ${\displaystyle E(K)[n]=\{T\in E(K)\mid n\cdot T=O\}}$ and ${\displaystyle \mu _{n}=\{x\in K\mid x^{n}=1\}}$.

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