GF(pm)

Important

If the modulus g(x) is an irreducible polynomial of degree m over GF(p), then the finite field GF(pm) can be constructed by the set of polynomials over GF(p) whose degree is at most m-1, where addition and multiplication are done modulo g(x).

For example, the finite field GF(32) can be constructed as the set of polynomials whose degrees are at most 1, with addition and multiplication done modulo the irreducible polynomial x2+1 (you can also choose another modulus, as long as it is irreducible and has degree 2).

Table 4.1. Addition modulo x2+1

x+1 x+2 2x 2x+1 2x+2 
0012xx+1x+22x2x+12x+2
1120x+1x+2x2x+12x+22x
2201x+2xx+12x+22x2x+1
xxx+1x+22x2x+12x+2012
x+1x+1x+2x2x+12x+22x120
x+2x+2xx+12x+22x2x+1201
2x2x2x+12x+2012xx+1x+2
2x+12x+12x+22x120xx+2x+2
2x+22x+22x2x+1201x+2xx+1

Table 4.2. Multiplication modulo x2+1

x+1 x+2 2x 2x+1 2x+2 
0000000000
1012xx+1x+22x2x+12x+2
20212x2x+22x+1xx+2x+1
x0x2x2x+22x+21x+12x+1
x+10x+12x+2x+22x12x+12x
x+20x+22x+12x+21xx+12x2
2x02xx12x+1x+122x+2x+2
2x+102x+1x+2x+122x2x+2x1
2x+202x+2x+12x+1x2x+212x