n prime to the characteristic of K. Here denotes the dual abelian variety of A. This is the so-called Weil pairing for higher dimensions. 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. Applications The pairing is used in number theory and algebraic geometry, and has also been applied in elliptic curve cryptography and identity based encryption. See also Tate pairing Pairing-based cryptography Boneh–Franklin scheme Homomorphic Signatures for Network Coding References External links The Weil pairing on elliptic curves over C (PDF) Elliptic curves Abelian varieties Pairing-based cryptography