Implement Multiplier in GF(2^m) using VHDL Matlab / Octave example code a = gf(,4,19) The results do not belong to the GF(16) so we need to perform the division by the primitive polynomial The “*” operator is omitted for sake of visualization. (a3*x3 + a2*x^2 + a1*x1 + a0) * (b3*x3 + b2*x^2 + b1*x1 + b0) =Īs clear is the classical multiplication algorithm where the second row is shifted right by 1, the second by 2 and so on. The bitwise multiplication is performed using AND operator.įor example: f(x) = (a3*x3 + a2*x^2 + a1*x1 + a0) The result of a multiplication is the reminder of the multiplication result f(x)*g(x) divided by the primitive polynomial of the field. The multiplication is a little bit more complicated since the multiplication of two polynomial g(x) and f(x) is defined as follow: m(x) = (g(x) * f(x)) mod(p(x)) In the previous section, we saw that the add/sub operation in the Galois field is very simple since we can perform it as bitwise XOR. We will obtain the same result for G1(x) -G2(x) Here we will consider P=2.Ī primitive polynomial p(X) is defined to be an irreducible binary polynomial of degree m which divides X^ n+1 for n = P^m-1 = 2^m-1 and which does not divide X^i+1 for i 1101 GF(P m) represents a Galois field of P m element. I put a lot of references at the bottom of this post that will help you to go deeper in the Galois field theory if you are really interested in it.īefore starting let’s review some arithmetic notation: This post wishes to give you a practical implementation starting point for your VHDL design using Galois arithmetic. It might put noses out of joint after reading this post because I will not be very rigorous in the representation of the theory of the Galois fields. The third point maybe is the most difficult to understand. the Galois arithmetic notation, just to understand how to interpret the equation.In this post we want to address the galois field theory from the practical application point of view. If you never deal with Galois field, at the beginning the topic could seem very hard to understand.

The Galois fields are mainly used in cryptography and error correction algorithm.

Galois lfsr