gfsub

R2026b

Subtract polynomials over Galois field

Syntax

c = gfsub(a,b,p)

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

c = gfsub(a,b,field)

Description

c = gfsub(a,b,p) calculates a minus b, subtracting two GF(2) polynomials.

c = gfsub(a,b,p,len) calculates a minus b, subtracting two GF(p) polynomials and returns a row vector representing the answer output in c that is truncated or extended to length len.

c = gfsub(a,b,field) calculates a minus b, where a and b are the exponential format of two elements of GF(p^m), relative to some primitive element of GF(p^m). p is a prime number and m is a positive integer. 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 answer, relative to the same primitive element. If a and b are matrices of the same size, the function treats each element independently.

Examples

Subtract Two GF Arrays

Calculate $$(2+3x+x^2)-(4+2x+3x^2)$$ over GF(5).

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

x = 1×3

     3     1     3

Subtract the two polynomials and display the first two elements.

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

y = 1×2

     3     1

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);

Subtract $$A^4$$ from $$A^2$$. The result is $$A^7$$.

g = gfsub(2,4,field)

g = 
7

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 a prime number. 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

p — Prime number

2 (default) | scalar prime number

Prime number, specified as a scalar prime number.

Data Types: double

len — Output length

positive integer

Output length, specified as a positive integer.

Data Types: double

field — All elements of GF(p^m)

matrix

All elements of GF(p^m) relative to a primitive element, specified as a matrix.

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. If the row vector corresponding to the answer 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.

Tips

• For a description of the various formats that gfsub uses, see Representing Elements of Galois Fields.

• This function performs computations in GF(p^m) where p is a prime number. To work in GF(2^m), you can also apply the – operator to Galois arrays of equal size. For details, see Example: Addition and Subtraction.

Version History

Introduced before R2006a