FAQs for Weil Pairing Cryptography

What is the Weil pairing of a curve?

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 n th roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual.

Does Weil pairing exist?

In the following theorem the existence of the Weil pairing is stated along with some of its properties. First we will introduce divisors, then the Weil pairing is constructed on elliptic curves and some of the properties of the Weil pairing are proved. Then we will use Victor Miller’s algorithm for eciently computing the Weil pairing.

Does the Weil pairing algorithm run linearly in Sage?

We will implement the Weil pairing in Sage with Miller’s algorithm and show that it runs linearly in the number of bits of its input points order n. 28 The Weil pairing Theorem 3.1. Let Ebe an elliptic curve dened over a eld K, let nbe a positive integer and let  nbe the set of n’th roots of unity.

How do you find the Weil pairing of two arbitrary points?

Thus the Weil pairing of two arbitrary points P;Q2E(F 27) will be trivial byRemark3.2. To get a non-trivial Weil pairing we want to useTheorem3.24, but then we will need to determine the smallest k>1 (embedding degree) such that 113 j(27k1), i.e the smallest ksuch that the whole torsion group E[113] ˆE(F 27k).

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