The
mod function can also be combined with polynomials:
Evaluate
x5 +9x4 - x3 +7x - 2mod5 = x5 +4x4 +4x3 +2x + 3
Thus Evaluate reduces each of the coefficients modulo 5.
Given a prime p, the set of polynomials with coefficients reduced modulo p is a ring, denoted by GFp[x].
To calculate a product of polynomials a(x) and b(x) in GFp[x]
To calculate the product of 4x5 + 5x + 3 and 6x4 + x3 + 3 in GF7[x], do the following steps.
Expand
4x5 + 5x + 3
6x4 + x3 + 3
= 24x9 +4x8 +42x5 +23x4 +3x3 +15x + 9
Evaluate
24x9 +4x8 +42x5 +23x4 +3x3 +15x + 9mod7 = 3x9 +4x8 +2x4 +3x3 + x + 2
The sum of 4x5 + 5x + 3 and 6x4 + x3 + 3 in GF7[x] is slightly simpler.
Evaluate
4x5 + 5x + 3
+
6x4 + x3 + 3
mod7 = 4x5 +6x4 + x3 + 5x + 6