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c = gfadd(a,b)
c = gfadd(a,b,p)
c = gfadd(a,b,p,len)
c = gfadd(a,b,field)
Note: This function performs computations in GF(p^{m}) where p is prime. To work in GF(2^{m}), apply the + operator to Galois arrays of equal size. For details, see Example: Addition and Subtraction. |
c = gfadd(a,b) adds two GF(2) polynomials, a and b. If a and b are vectors of the same orientation but different lengths, then the shorter vector is zero-padded. If a and b are matrices they must be of the same size.
c = gfadd(a,b,p) adds two GF(p) polynomials, where p is a prime number. a, b, and c are row vectors that give the coefficients of the corresponding polynomials in order of ascending powers. Each coefficient is between 0 and p-1. If a and b are matrices of the same size, the function treats each row independently.
c = gfadd(a,b,p,len) adds row vectors a and b as in the previous syntax, except that it returns a row vector of length len. The output c is a truncated or extended representation of the sum. If the row vector corresponding to the sum has fewer than len entries (including zeros), extra zeros are added at the end; if it has more than len entries, entries from the end are removed.
c = gfadd(a,b,field) adds two GF(p^{m}) elements, where m is a positive integer. a and b are the exponential format of the two elements, relative to some primitive element of GF(p^{m}). field is the matrix listing all elements of GF(p^{m}), arranged relative to the same primitive element. c is the exponential format of the sum, relative to the same primitive element. See Representing Elements of Galois Fields for an explanation of these formats. If a and b are matrices of the same size, the function treats each element independently.
In the code below, sum5 is the sum of 2 + 3x + x^{2} and 4 + 2x + 3x^{2} over GF(5), and linpart is the degree-one part of sum5.
sum5 = gfadd([2 3 1],[4 2 3],5) linpart = gfadd([2 3 1],[4 2 3],5,2)
The output is
sum5 = 1 0 4 linpart = 1 0
The code below shows that A^{2} + A^{4} = A^{1}, where A is a root of the primitive polynomial 2 + 2x + x^{2} for GF(9).
p = 3; m = 2; prim_poly = [2 2 1]; field = gftuple([-1:p^m-2]',prim_poly,p); g = gfadd(2,4,field)
The output is
g = 1
Other examples are in Arithmetic in Galois Fields.