| Communications Toolbox | |
The code below adds two Galois arrays to create an addition table for GF(8). Addition uses the ordinary + operator. The code below also shows how to index into the array addtb to find the result of adding 1 to the elements of GF(8).
m = 3; e = repmat([0:2^m-1],2^m,1); f = gf(e,m); % Create a Galois array. addtb = f + f' % Add f to its own matrix transpose. addone = addtb(2,:); % Assign 2nd row to the Galois vector addone.
The output is below.
addtb = GF(2^3) array. Primitive polynomial = D^3+D+1 (11 decimal)
Array elements =
0 1 2 3 4 5 6 7
1 0 3 2 5 4 7 6
2 3 0 1 6 7 4 5
3 2 1 0 7 6 5 4
4 5 6 7 0 1 2 3
5 4 7 6 1 0 3 2
6 7 4 5 2 3 0 1
7 6 5 4 3 2 1 0
As an example of reading this addition table, the (7,4) entry in the addtb array shows that gf(6,3) plus gf(3,3) equals gf(5,3). Equivalently, the element A2+A plus the element A+1 equals the element A2+1. The equivalence arises from the binary representation of 6 as 110, 3 as 011, and 5 as 101.
The subtraction table, which you can obtain by replacing + by -, would be the same as addtb. This is because subtraction and addition are identical operations in a field of characteristic two. In fact, the zeros along the main diagonal of addtb illustrate this fact for GF(8).
The code below illustrates scalar expansion and the implicit creation of a Galois array from an ordinary MATLAB array. The Galois arrays h and h1 are identical, but the creation of h uses a simpler syntax.
g = gf(ones(2,3),4); % Create a Galois array explicitly. h = g + 5; % Add gf(5,4) to each element of g. h1 = g + gf(5*ones(2,3),4) % Same as h.
The output is below.
h1 = GF(2^4) array. Primitive polynomial = D^4+D+1 (19 decimal)
Array elements =
4 4 4
4 4 4
Notice that 1+5 is reported as 4 in the Galois field. This is true because the 5 represents the polynomial expression A2+1, and 1+(A2+1) in GF(16) is A2. Furthermore, the integer that represents the polynomial expression A2 is 4.
| | Arithmetic in Galois Fields | Example: Multiplication | |
© 1994-2005 The MathWorks, Inc.