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 (Weil|1940}}|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.

## Contents

* 1 Formulation
* 2 Generalisation to abelian varieties
* 3 Applications
* 4 See also
* 5 References
* 6 External links

## Formulation

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 [math]\displaystyle{ E(\overline{K}) }[/math] is known to be a Cartesian product of two cyclic groups of order n. The Weil pairing produces an n-th root of unity

[math]\displaystyle{ w(P,Q) \in \mu_n }[/math]

by means of Kummer theory, for any two points [math]\displaystyle{ P,Q \in E(K)[n] }[/math], where [math]\displaystyle{ E(K)[n]=\\{T \in E(K) \mid n \cdot T = O \\} }[/math] and [math]\displaystyle{ \mu_n = \\{x\in K \mid x^n =1 \\} }[/math].

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

[math]\displaystyle{ \mathrm{div}(F)= \sum_{0 \leq k \lt n}[P+k\cdot Q] - \sum_{0 \leq k \lt n} [k\cdot Q]. }[/math]

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

[math]\displaystyle{ w(P,Q):=\frac{G}{F} }[/math]

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.

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).

## Generalisation to abelian varieties

For abelian varieties over an algebraically closed field K, the Weil pairing is a nondegenerate pairing

[math]\displaystyle{ A[n] \times A^\vee[n] \longrightarrow \mu_n }[/math]

for all n prime to the characteristic of  K.[2] Here [math]\displaystyle{ A^\vee }[/math] denotes the dual abelian variety of A. This is the so-called Weil pairing for higher dimensions. If A is equipped with a polarisation

[math]\displaystyle{ \lambda: A \longrightarrow A^\vee }[/math],

then composition gives a (possibly degenerate) pairing

[math]\displaystyle{ A[n] \times A[n] \longrightarrow \mu_n. }[/math]

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

[math]\displaystyle{ J[n]\times J[n] \longrightarrow \mu_n }[/math]

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

1. ↑ Silverman, Joseph (1986). The Arithmetic of Elliptic Curves. New York: Springer-Verlag. ISBN 0-387-96203-4.
2. ↑ James Milne, Abelian Varieties, available at www.jmilne.org/math/

* Weil, André (1940), "Sur les fonctions algébriques à corps de constantes fini", Les Comptes rendus de l'Académie des sciences 210: 592–594

## External links

* The Weil pairing on elliptic curves over C (PDF)

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