Communications Toolbox

Arithmetic in Galois Fields

You can add, subtract, multiply, and divide elements of Galois fields using the functions gfadd, gfsub, gfmul, and gfdiv, respectively. Each of these functions has a mode for prime fields and a mode for extension fields.

Arithmetic in Prime Fields

Arithmetic in GF(p) is the same as arithmetic modulo p. The functions gfadd, gfmul, gfsub, and gfdiv accept two arguments that represent elements of GF(p) as integers between 0 and p-1. The third argument specifies p.

Example: Addition Table for GF(5)

The code below constructs an addition table for GF(5). If a and b are between 0 and 4, then the element gfp_add(a+1,b+1) represents the sum a+b in GF(5). For example, gfp_add(3,5) = 1 because 2+4 is 1 modulo 5.

• p = 5;
  row = 0:p-1;
  table = ones(p,1)*row;
  gfp_add = gfadd(table,table',p)
  
  gfp_add =
  
       0     1     2     3     4
       1     2     3     4     0
       2     3     4     0     1
       3     4     0     1     2
       4     0     1     2     3
  

Other values of p produce tables for different prime fields GF(p). Replacing gfadd by gfmul, gfsub, or gfdiv produces a table for the corresponding arithmetic operation in GF(p).

Converting to Simplest Exponential Format

Arithmetic in Extension Fields