gfminpol

Find minimal polynomial of Galois field element

Syntax

pol = gfminpol(k,m)

pol = gfminpol(k,m,p)

pol = gfminpol(k,prim_poly,p)

Description

pol = gfminpol(k,m) produces a minimal polynomial for each entry in k.

pol = gfminpol(k,m,p) finds the minimal polynomial of A^k over GF(p), where A is a root of the default primitive polynomial for GF(p^m). The format of the output is as follows:

• If k is a nonnegative integer, pol is a row vector that gives the coefficients of the minimal polynomial in order of ascending powers.

• If k is a vector of length len all of whose entries are nonnegative integers, pol is a matrix having len rows; the rth row of pol gives the coefficients of the minimal polynomial of A^k(r) in order of ascending powers.

pol = gfminpol(k,prim_poly,p) where A is a root of the primitive polynomial for GF(p^m) specified by prim_poly.

Examples

Find Minimal Polynomial of Galois Field Element

Consider GF element, GF$\left(3^4\right)$.

p = 3;
m = 4;

Compute the minimum polynomial of ${\alpha }^1$, where $\alpha$ is the primitive element of GF$\left(3^4\right)$.

k=1

k = 
1

prim_poly = gfminpol(k,m,p)

prim_poly = 1×5

     2     1     0     0     1

The numbers in the output are coefficients in GF(3).

Input Arguments

k — Element of Galois field polynomial

scalar | column vector

Element of Galois field polynomial, specified as a scalar or a column vector.

Data Types: double

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

prim_poly — Primitive polynomial

primitive polynomial in GF(2^m) (default) | binary row vector | character vector | string scalar | positive integer

Primitive polynomial, specified as one of these options:

• Binary row vector — This vector specifies coefficients of prim_poly in the order of ascending powers.

• Character vector or a string scalar — This value defines prim_poly in a textual representation. For more details, refer to polynomial character vector.

• Positive integer — This value defines prim_poly in the range [(2^m + 1), (2^m+1 – 1)].

If prim_poly is not specified, see Default Primitive Polynomials for the list of default primitive polynomial used for each Galois field array GF(2^m).

Data Types: double | char | string

Output Arguments

pol — Minimal polynomial of Galois field element

row vector | matrix

Minimal polynomial of Galois field element, returned as a row vector or matrix.

Version History

Introduced before R2006a