3.1.1.6.1 Finite Field Arithmetic

A finite field is a finite set of numbers. All arithmetic operations performed on this field will yield a result that belongs to the same finite field. For example, a finite field of size 256 with numbers from 0 to 255 is defined. All the arithmetic operations (addition, subtraction, multiplication, and division) on this field will yield a result in the range of 0 to 255, thus belonging to the original finite field itself. Conventional arithmetic differs from finite field arithmetic as it operates on an infinite set of real numbers. For more details on finite fields, see [Lidl].

All binary numbers belonging to a finite field (also known as a Galois field, GF(pⁿ)), where p is a prime number and n is a positive integer, can be represented in a polynomial form and in a finite field with binary numbers (for example in GF(256)=GF(2⁸)), where a is the coefficient of this equation with a value equal to zero or 1.

Figure 5: Galois field and binary representation example