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.
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
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,[1] giving rise to a non-degenerate pairing on the n-torsion.
If A is equipped with a polarisation then composition gives a (possibly degenerate) pairing If C is a projective, nonsingular curve of genus ≥ 0 over k, and J its Jacobian, then the theta-divisor of J induces a principal polarisation of J, which in this particular case happens to be an isomorphism (see autoduality of Jacobians).
Hence, composing the Weil pairing for J with the polarisation gives a nondegenerate pairing for all n prime to the characteristic of k. As in the case of elliptic curves, explicit formulae for this pairing can be given in terms of divisors of C. The pairing is used in number theory and algebraic geometry, and has also been applied in elliptic curve cryptography and identity based encryption.