Communications Toolbox

Example: Exponentiation

The examples below illustrate how to compute integer powers of a Galois array. To perform matrix exponentiation on a Galois array, you must use a square Galois array as the base and an ordinary (not Galois) integer scalar as the exponent.

Elementwise Exponentiation

This example computes powers of a primitive element, A, of a Galois field. It then uses these separately computed powers to evaluate the default primitive polynomial at A. The answer of zero shows that A is a root of the primitive polynomial. Notice that the .^ operator exponentiates each array element independently.

m = 3;
av = gf(2*ones(1,m+1),m); % Row containing primitive element
expa = av .^ [0:m]; % Raise element to different powers.
evp = expa(4)+expa(2)+expa(1) % Evaluate D^3 + D + 1.

The output is below.

evp = GF(2^3) array. Primitive polynomial = D^3+D+1 (11 decimal)
 
Array elements = 
 
     0

Matrix Exponentiation

This example computes the inverse of a square matrix by raising the matrix to the power -1. It also raises the square matrix to the powers 2 and -2.

m = 5;
mat = gf([1 2 3; 4 5 6; 7 8 9],m);
minvs = mat ^ (-1); % Matrix inverse
matsq = mat^2; % Same as mat * mat
matinvssq = mat^(-2); % Same as minvs * minvs

Example: Division

Example: Elementwise Logarithm

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