Part 2: Interlude

In chapters 7 to 12, I go into the background subroutines required to implement elliptic curve pairings used in blockchain technology. It is essential to dig through these chapters to understand what is described in chapters 13 to 19.

The pairing of points on an elliptic curve requires the curve to have special properties. The majority of curves covered in part 1 do not have those properties. Our goal is to find pairing-friendly curves and use pairings to create cryptographic protocols. Rather than attempt a direct assault on the concepts of pairings, I’ll first cover the mathematics of finite fields over polynomials, also called extension fields.

This mathematical interlude includes discussions on basic polynomials, multiplication of polynomials, taking polynomials to an exponential power, division of polynomials, and computing square roots. All these operations are done modulo a prime polynomial. There is a very real connection between prime numbers and prime polynomials..

An equivalent term for prime polynomial is irreducible polynomial. An irreducible polynomial has no other factors than itself and 1. That is why it is similar to a prime number. As we will see in detail during this part, factoring a polynomial depends on the field prime, which is the modulus for the polynomial coefficients.