1. Introduction
Group, field and elliptic curve theories make a regular appearance in the study of crypto-assets including but not limited to cryptocurrencies. For example, the security strength of a number of crypto-specific primitives relies on the math of elliptic curve groups over finite fields. These groups constitute a robust infrastructure to generate adequate public keys from private ones.
Groups and fields are foundational pillars of modern algebra. While in elementary algebra we rely on common arithmetic operations (e.g., addition and multiplication of real numbers), in modern algebra we raise further the level of abstraction. In particular, we introduce more general counterparts to real number addition and multiplication and define them over more general sets. An important objective is to study the common properties of all sets on which a fixed number of operations are defined. These operations tend to be interrelated in some definite way (e.g., distributivity of multiplication over addition).
In this post, we provide a concise (but by no means comprehensive) introduction to group and finite-field theory at the level needed to better appreciate the mathematical foundation of crypto assets. In a subequent post we build on this material to introduce elliptic curve groups defined over finite fields. The interested reader could consult e.g., [1] for a deeper dive on the theory of finite fields and its applications.