# Chapter 1: Introduction

Since the construction of Miller's algorithm [Mil], the cryptography community has started to use elliptic curves and their pairing extensively; by now; many publicly available code libraries allow one to efficiently compute these mathematical objects.

Compared to Machine Learning, where the mathematical pre-requisites consist of Linear Algebra, Calculus, and basic Statistics, elliptic curves require substantially more background and are usually taught at a master level in pure Mathematics. This state of affairs poses a challenge to engineers and others who wish to understand the mathematical building blocks.

This notes aim to give a self-contained, rigorous and elementary account of most of the maths required for pairing-based cryptography. I sometimes formulated elementary arguments to replace non-elementary ones. I completely avoid relying on Galois theory or algebraic gemotery andeven ring theory is mostly skipped.

Footnotes:
- Mil (Miller's algorithm): Miller, V.S., 2004. The Weil pairing, and its efficient calculation. Journal of cryptology, 17(4), pp.235-261