gfadd

Add polynomials over Galois field

Syntax

c = gfadd(a,b)

c = gfadd(a,b,p)

c = gfadd(a,b,p,len)

c = gfadd(a,b,field)

Description

c = gfadd(a,b) adds two GF(2) polynomials, a and b.

c = gfadd(a,b,p) adds two GF(p) polynomials, a and b.

c = gfadd(a,b,p,len) adds row vectors a and b and returns a row vector of length len.

c = gfadd(a,b,field) adds two GF(p^m) elements.

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.

Examples

Add Two GF Arrays

Sum $$2+3x+x^2$$ and $$4+2x+3x^2$$ over GF(5).

x = gfadd([2 3 1],[4 2 3],5)

x = 1×3

     1     0     4

Add the two polynomials and display the first two elements.

y = gfadd([2 3 1],[4 2 3],5,2)

y = 1×2

     1     0

For prime number p and exponent m, create a matrix listing all elements of GF(p^m) given primitive polynomial $$2+2x+x^2$$.

p = 3;
m = 2;
primpoly = [2 2 1];
field = gftuple((-1:p^m-2)',primpoly,p);

Sum $$A^2$$ and $$A^4$$. The result is $$A$$.

g = gfadd(2,4,field)

g = 
1

Input Arguments

a — Galois field polynomial

row vector | character vector | string

Galois field polynomial, specified as a row vector, character vector, or string. a can be either a Representation of Polynomials in Communications Toolbox or numeric vector.

a and b must both be GF(p) polynomials or GF(p^m) polynomials, where p is prime. The value of p is as specified when included, 2 when omitted, or implied when field is specified.

Example: [1 2 3 4] is the polynomial 1+2x+3x²+4x³ in GF(5) expressed as a row vector.

Data Types: double | char | string

b — Galois field polynomial

row vector | character vector | string

Galois field polynomial, specified as a row vector, character vector, or string. b can be either a Representation of Polynomials in Communications Toolbox or numeric vector.

a and b must both be GF(p) polynomials or GF(p^m) polynomials, where p is prime. The value of p is as specified when included, 2 when omitted, or implied when field is specified.

Example: '1 + x' is a polynomial in GF(2⁴) expressed as a character vector.

Data Types: double | char | string

m — Order of primitive polynomial

positive integer

Order of primitive polynomial, specified as a positive integer from 1 through 16. The function uses this value to calculate the distinct number of elements in the GF.

Data Types: double

p — Prime number

2 (default) | scalar prime number

Prime number, specified as a scalar prime number.

Data Types: double

field — m-tuple of all elements in GF(p^m)

matrix

m-tuple of all elements in GF(p^m), specified as a matrix. field is the matrix listing all elements of GF(p^m), arranged relative to the same primitive element. To generate the m-tuple of all elements in GF(p^m), use

field =gftuple([-1:p^m-2]',m,p)

where m is a positive integer. The coefficients, specified in exponential format, represent the field elements in GF(p^m). For an explanation of these formats, see Representing Elements of Galois Fields.

Data Types: double

Output Arguments

c — Galois field polynomial

row vector

Galois field polynomial, returned as a row vector of the polynomial coefficients in order of ascending powers.

Version History

Introduced before R2006a