## Finite Fields

Simply said, a field is a place where you can add, substract, multiply and divide, without leaving the set.

A field has several properties:

1. The rules of addition apply, and the field contains an additive identitive element

2. The rules of multiplication apply, and the field contains a multiplicative identity element

3. Every element in a field has an inverse

The set of integers is not a field, because integers don’t include fractions and so do not have multiplicative inverses.

The underlying set of a field determines whether a field is finite or infinite. If the set F is finite, then the field is said to be finite.

Infinite fields are not of particular interest in cryptographic applications, yet finite fields play a crucial role in many cryptographic algorithm.

Examples of infinite fields includes the real number, the rational numbers, the complex numbers and rational functions over a field.

The simplest finite field is modulo prime arithmetic.

Zp = {0, 1, …, p-1}, arithmetic mod p, where p is a prime, is a (finite) field

Examples:

Notice that Z4 (arithmetic mod 4) is not a field, since 2 has no inverse (look at the division table), there is no element x such that 2x = 1 (mod 4).

[Will post more on Finite Fields for Cryptographic Applications]