7.6 Finite fields
Fields are mathematical structures in which the basic arithmetic rules that we are used to hold true: in a field, we can add and multiply things together, and we can also reverse these operations, if we like. If we choose to combine addition and multiplication in a single expression, the so-called distributive law holds, which tells us we can factor common factors out of a sum. We are introducing finite fields here because they provide the stage for another important public-key cryptosystem, the RSA algorithm, which we will cover in Section 7.7, The RSA algorithm. Moreover, we can build other interesting cyclic groups out of finite fields, namely elliptic curves, which will be the subject of Chapter 8, Elliptic Curves.
To begin with, here are the formal requirements for a set M being a field:
(F1) There is an operation + on M so that (M,+) forms an abelian group with neutral element 0
(F2) There is an operation ⋅ on M so that (M ∖{0},⋅...
