Weil pairing

In mathematics, the Weil pairing is a construction of roots of unity by means of functions on an elliptic curve "E", in such a way as to constitute a pairing (bilinear form, though with multiplicative notation) on the torsion subgroup of "E". The name is for André Weil, 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.

Suppose "E" is defined over a field "K". Given an integer "n" > 0 (We require "n" to be prime to char("K") if char("K")> 0) and suppose that "K" contains a primitive nth root of unity. Then the "n"-torsion on "E" has known structure, as a Cartesian product of two cyclic groups of order "n". The basis of the construction is of an "n"-th root of unity

:$w(P,Q)\; in\; mu\_n$

for given points $P,Q\; in\; E\; [n]$, where $E\; [n]\; =\{T\; in\; E\; mid\; n\; cdot\; T\; =\; O\; \}$ and $mu\_n\; =\; \{xin\; K\; mid\; x^n\; =1\; \}$, by means of Kummer theory.

By a direct argument one can define a function "F" in the function field of "E" over the algebraic closure of "K", by its divisor:

:$(F)=\; sum(P+kcdot\; Q)\; -\; sum\; (kcdot\; Q)$

with sums for 0 ≤ "k" < "n". In words "F" has a simple zero at each point "P" + "kQ", and a simple pole at each point "kQ". 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. One can show that

:$frac\{G\}\{F\}\; e\; 1$

In fact then "G"/"F" would yield a function on the isogenous curve "E"/"C" where "C" is the cyclic subgroup generated by "Q", having just one simple pole. Such a function cannot exist, as follows by proving the residue at the pole is zero, a contradiction.

Therefore if we define

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

we shall have an "n"-th root of unity (translating "n" times must give 1) other than 1. With this definition it can be shown that "w" is antisymmetric and bilinear, giving rise to a non-degenerate pairing on the "n"-torsion.

The Weil pairing is used in number theory and algebraic geometry, and has also been applied in elliptic curve cryptography and identity based encryption.