Introduction to Cryptography with Open-Source Software

Chapter 9

Finite fields

Finite fields are mathematical structures that have enormous applications in

cryptography. In fact, finite fields have been used already, but without inves-

tigating their algebraic underpinnings. This chapter provides the formal defi-

nitions of finite fields, and provides the basis for discussions of cryptosystems

that require them, such as the Advanced Encryption Standard and elliptic

curve cr yptosystems. The theory of fields will be developed slowly, starting

with a simpler structure: the group.

In particular, this chapter will ex plore:

• The definition of gr oups, rings and fields, with s ome of their most basic

algebraic properties. ...