15 General rules of elliptic curve pairing explained

This chapter presents the point-pairing mathematics over field extension elliptic curves. I’ll explain the general mathematical rules of pairings at a high level. The rules are mostly symbolic manipulation, so the meanings will be explained before we get into details.

With the background mathematics under our belts, we can now begin to look at the mathematics of pairing points on an elliptic curve. Pairing points gives us a one-way trap door function that is efficient and secure. It also gives us efficient ways to compute algorithms that would be horribly complicated otherwise. Point pairing is a very deep subject, and I will do my best to ignore most of it. The fundamental reason for ignoring this depth is that the underlying mathematics works, and fully understanding why it works is not critical to writing functional code.

15.1 Mathematical rules of elliptic curve pairings

15.1.1 Elliptic curve point pairing rule of bilinearity

15.1.2 Nondegeneracy rule with the point at infinity

15.2 Algorithms for pairing

15.2.1 Function \(h_{P,Q}(R)\)

15.2.2 Miller’s algorithm

15.3 Subroutine \(h_{P,Q}\)

15.4 Miller’s algorithm code

Answers to exercises

Summary