calculating reminder in modulus of a polynomials

I want to create a substitution box using mobius transformation over the Galois field of order 256 with irreducuible polynomial

p(x)=x^8+x^4+x^3+x+1. for this i neede to calculate a.b mod(p(x)) where a and b are the elements of given Galois field . Please help me in this regard.

Answer 1

[Sameer]

Hi Rashad

From my understanding, you want to perform multiplication of two elements within the “Galois Field GF(256)” using the irreducible polynomial “p(x) = x^8 + x^4 + x^3 + x + 1 )” .

To perform multiplication in a “Galois Field” of order 256 using an irreducible polynomial, MATLAB has “Communications Toolbox” which provides functions for working with “Galois fields”.

Here's how to calculate the product of two elements ‘a’ and ‘b’ in “GF(256)” using the given irreducible polynomial “p(x) = x^8 + x^4 + x^3 + x + 1”:

% Define the irreducible polynomial as a decimal number
prim_poly = 283; % Corresponds to x^8 + x^4 + x^3 + x + 1

% Create the Galois Field GF(256) using the specified polynomial
gf256 = gf(0:255, 8, prim_poly);

% Define the elements a and b that you want to multiply
% For example, let's take a = 3 and b = 5
a = gf(3, 8, prim_poly);
b = gf(5, 8, prim_poly);

% Multiply the elements a and b in GF(256)
result = a .* b;

% Display the result
disp('The result of a * b in GF(256) is:');
disp(result.x);

Please refer to the below MathWorks documentation link:

https://www.mathworks.com/help/comm/ref/gf.html

Hope this helps!