I have a binary matrix A (only 1 and 0), and a vector D in Galois field (256). The vector C is calculated as:
C = (A^^-1)*D
where A^^-1 denotes the inverse matrix of matrix A in GF(2), * is multiply operation. The result vector C must be in GF(256). I tried to do it in Matlab.
A= [ 1 0 0 1 1 0 0 0 0 0 0 0 0 0;
1 1 0 0 0 1 0 0 0 0 0 0 0 0;
1 1 1 0 0 0 1 0 0 0 0 0 0 0;
0 1 1 1 0 0 0 1 0 0 0 0 0 0;
0 0 1 1 0 0 0 0 1 0 0 0 0 0;
1 1 0 1 1 0 0 1 0 1 0 0 0 0;
1 0 1 1 0 1 0 0 1 0 1 0 0 0;
1 1 1 0 0 0 1 1 1 0 0 1 0 0;
0 1 1 1 1 1 1 0 0 0 0 0 1 0;
0 0 0 0 1 1 1 1 1 0 0 0 0 1;
0 1 1 1 1 0 1 1 1 0 1 1 1 0;
0 0 0 1 0 0 0 1 0 0 0 0 0 0;
0 0 1 0 0 0 0 1 0 0 0 0 0 0;
1 1 1 1 0 0 0 0 0 0 0 0 0 0]
D=[0;0;0;0;0;0;0;0;0;0;103;198;105;115]
A=gf(A,1);
D=gf(D,8); %%2^8=256
C=inv(A)*D
%% The corrected result must be
%%C=[103;187;125;210;181;220;161;20;175;175;187;187;220;115]
However, for above code, I cannot achieved as my expected result
C=[103;187;125;210;181;220;161;20;175;175;187;187;220;115]
It produces an error as
Error using * (line 14)
Orders must match.
Could you help me achieve my expected result?
The error
Error using * (line 14)
Orders must match.
arises because Matlab does not support applying standard mathematical operations (+, *, .*, .^, \, etc.) to Galois Fields of different order, GF(2) and GF(256). The inverse operation, inv(A) is a valid operation and if you perform it on its own as shown by @Ander Biguri it will succeed and generate the inverse of A in GF(2). Your calculation breaks down when you attempt to multiply inv(A) and D as the orders of these Galois Fields do not match.
In this example it is unnecessary to explicitly define A as a Galois Field of order 2, GF(2) as multiplication in GF(256) takes place in GF(2). That is 1 + 1 = 0.
If we thus modify the code that creates the Galois Field for A to
A = gf(A, 8);
we can then perform
C = inv(A)*D
which produces
C = GF(2^8) array. Primitive polynomial = D^8+D^4+D^3+D^2+1 (285 decimal)
Array elements =
103
187
125
210
181
220
161
20
175
175
187
187
220
115
C is thus in GF(256) and produces the expected result.
It seems there is either
A) a theoretical fault in your computation or
B) something that MATLAB doesn't support.
As per the documentation in Galois field arithmetic (emphasis mine):
Section Overview. You can perform arithmetic operations on Galois arrays by using familiar MATLAB operators, listed in the table below. Whenever you operate on a pair of Galois arrays, both arrays must be in the same Galois field.
And your matrices aren't.
I cant know which one is it as I have no idea how Galois fields work
来源：https://stackoverflow.com/questions/33527625/how-to-perform-inverse-in-gf2-and-multiply-in-gf256-in-matlab