Groups, rings, and fields

Abstract

Several different algebraic systems will often share common structural properties. Accordingly, it is efficient and productive to gather together all algebraic systems with a similar structure, and to study them collectively as a single category. Any properties that follow directly from the common structure will hold for all algebraic systems with that same structure, and these properties can be developed and studied for a given category, and then applied to all algebraic systems in that same category. The three most important algebraic categories are the category of groups , the category of rings , and the category of fields . We will discuss each of these in turn. Each of these algebraic categories plays a different but powerful role in the development of the subject of cryptography. The most familiar example of a group is the set of integers Z under the usual operation of addition. The set of integers is a group with an infinite number of elements. An example of a group with a finite number of elements is the group of nonnegative integers smaller than n with addition modulo n , and denoted Z n . The most familiar example of a ring is again the set of integers Z , but now with two operations. These are the two usual operations of addition and multiplication of integers. Many other rings are also important. Much of the mathematical structure of the ring of integers is mimicked by a similar mathematical structure within the ring of univariate polynomials over a field F .