Weil pairing

Weil pairing. In mathematics, particularly in algebraic geometry and number theory, the Weil pairing is a bilinear, alternating, and non-degenerate map defined on the torsion subgroups of an elliptic curve. It was introduced by André Weil as part of his foundational work on abelian varieties, providing a crucial tool for proving the Weil conjectures and analyzing the structure of divisor class groups. The pairing has become central to modern elliptic curve cryptography, enabling protocols like identity-based encryption and pairing-based cryptography by facilitating efficient computations on discrete logarithm problems.

Definition and basic properties

The Weil pairing is defined for an elliptical curve \(E\) over a finite field \(\mathbb{F}_q\) and a positive integer \(n\) coprime to the characteristic of the field. It takes two points \(P\) and \(Q\) from the \(n\)-torsion subgroup \(E[n]\) and outputs an \(n\)-th root of unity in the algebraic closure \(\overline{\mathbb{F}_q}\). Formally, it is a map \(e_n: E[n] \times E[n] \to \mu_n\), where \(\mu_n\) denotes the group of \(n\)-th roots of unity. Key properties include bilinearity, meaning \(e_n(aP, bQ) = e_n(P, Q)^{ab}\) for integers \(a, b\); alternation, implying \(e_n(P, P) = 1\); and non-degeneracy, ensuring if \(e_n(P, Q) = 1\) for all \(Q\), then \(P\) is the identity element. These properties are instrumental in proving the Weil conjectures and analyzing the Galois representation associated with the curve.

Algebraic construction

The construction of the Weil pairing utilizes the theory of divisors and functions on elliptic curves. For points \(P, Q \in E[n]\), one selects rational functions \(f_P\) and \(f_Q\) whose divisors are \(n(P) - n(O)\) and \(n(Q) - n(O)\), respectively, where \(O\) is the point at infinity. The pairing is then defined as \(e_n(P, Q) = f_P(D_Q) / f_Q(D_P)\), where \(D_P\) and \(D_Q\) are divisors equivalent to \((P) - (O)\) and \((Q) - (O)\). This quotient is independent of the choices made and lies in \(\mu_n\). The construction relies on theorems from algebraic geometry, such as the Riemann-Roch theorem, and connects to concepts like the Tate module and étale cohomology. Efficient computation often employs Miller's algorithm, which is foundational for pairing-based cryptography.

Cryptographic applications

In cryptography, the Weil pairing enables the construction of novel cryptographic protocols that are infeasible with traditional discrete logarithm systems. It is a core component of pairing-based cryptography, facilitating three-party Diffie-Hellman key exchange and identity-based encryption schemes like the Boneh-Franklin scheme. The pairing allows the translation of the discrete logarithm problem on an elliptic curve to a related problem in a finite field, which can be weaker, necessitating careful selection of curve parameters for security. Protocols such as short signatures and group signatures leverage its bilinearity, and its implementation relies on efficient pairings like the Tate pairing or Ate pairing over Barreto-Naehrig curves. The National Institute of Standards and Technology has explored standards for post-quantum cryptography involving these techniques.

Generalizations and related pairings

The Weil pairing has been generalized and adapted in various mathematical and cryptographic contexts. The Tate pairing, introduced by John Tate, is a more computationally efficient variant defined on abelian varieties over local fields, often used in cryptographic implementations. Further refinements include the Ate pairing and optimal pairings, which reduce computational complexity for pairing-friendly curves. In algebraic geometry, similar pairings arise in the study of Jacobian varieties and principal homogeneous spaces, connecting to the Weil conjectures and l-adic cohomology. Related concepts include the Lichtenbaum pairing on K-groups and the Cassels-Tate pairing on Selmer groups, which play roles in arithmetic geometry and the Birch and Swinnerton-Dyer conjecture.

Historical context and development

The Weil pairing was introduced by André Weil in the 1940s as part of his seminal work on abelian varieties and the proof of the Riemann hypothesis for curves over finite fields, a key case of the Weil conjectures. Its development was intertwined with advances in algebraic geometry, including the theories of divisors and sheaf cohomology by Jean-Pierre Serre and Alexander Grothendieck. In the late 20th century, applications in cryptography emerged with the advent of elliptic curve cryptography, notably through the work of Victor Miller and Neal Koblitz. The pairing's utility in identity-based encryption was popularized by Dan Boneh and Matthew Franklin, leading to ongoing research in post-quantum cryptography and isogeny-based cryptography at institutions like the Massachusetts Institute of Technology and Microsoft Research.

Category:Algebraic geometry Category:Number theory Category:Cryptography