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.

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

${\displaystyle w(P,Q)\in \mu _{n}}$

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 ${\displaystyle \mu _{n}=\{x\in K\mid x^{n}=1\}}$.

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

${\displaystyle \mathrm {div} (F)=\sum _{0\leq k<n}[P+k\cdot Q]-\sum _{0\leq k<n}[k\cdot Q].}$

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

${\displaystyle w(P,Q):={\frac {G}{F}}}$

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 T_ℓ(E) × T_ℓ(E) → T_ℓ(μ) on the Tate module T_ℓ(E) of the elliptic curve E (the inverse limit of the ℓⁿ-torsion points) to the Tate module T_ℓ(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

${\displaystyle A[n]\times A^{\vee }[n]\longrightarrow \mu _{n}}$

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

${\displaystyle \lambda :A\longrightarrow A^{\vee }}$,

then composition gives a (possibly degenerate) pairing

${\displaystyle A[n]\times A[n]\longrightarrow \mu _{n}.}$

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

${\displaystyle J[n]\times J[n]\longrightarrow \mu _{n}}$

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[]

• Weil, André (1940), "Sur les fonctions algébriques à corps de constantes fini", Les Comptes rendus de l'Académie des sciences, 210: 592–594, MR 0002863

External links[]

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