gfmul

R2026b

Multiply elements of Galois field

Syntax

c = gfmul(a,b)

c = gfmul(a,b,p)

c = gfmul(a,b,field)

Description

c = gfmul(a,b) returns the Galois field resulting from element-by-element multiplication of input Galois fields a and b. Each entry of a and b is an element in GF(2). For more information, see Galois Field Entries.

c = gfmul(a,b,p) returns the Galois field resulting from element-by-element multiplication of input Galois fields a and b in GF(p), where p is a prime number. For more information, see Galois Field Entries.

c = gfmul(a,b,field) returns the Galois field resulting from element-by-element multiplication of input Galois fields a and b in GF(p^m) in exponential form, where p is a prime number and m is a positive integer. For more information, see Galois Field Entries.

Examples

Perform Galois Field Multiplication

Use gfmul to multiply powers of a primitive polynomial root. If$A$ is a root of the primitive polynomial $2+2x+x^2$ for gf(9), then $A^2\cdot A^4=A^6$.

p = 3;
m = 2;
prim_poly = [2 2 1];
field = gftuple([-1:p^m-2]',prim_poly,p);
a = gfmul(2,4,field)

a = 
6

Input Arguments

a — Galois field

scalar | vector | matrix

Galois field, specified as a scalar, vector, or matrix.

b — Galois field

scalar | vector | matrix

Galois field, specified as a scalar, vector, or matrix. a and b must be the same size.

p — Prime number

2 (default) | scalar

Prime number, specified as a scalar. Each entry in a and b represents an element of GF(p).

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

matrix

List of all elements in GF(p^m), specified as a matrix arranged relative to the same primitive element. a and b must be the same size.

To generate field, use

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

where m is a positive integer and p is a prime number. 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.

Output Arguments

c — Galois field

scalar | vector | matrix

Galois field, returned as a scalar, vector, or matrix. c is the exponential format of the product, relative to the same primitive element. For more information, see Galois Field Entries.

More About

Galois Field Entries

When each entry of a and b is an element in a Galois field of degree:

• GF(2) — a and b are matrices of the same size and contain values 0 or 1.

• GF(p) — Each entry of a and b is in the range [0, (p – 1)].

• GF(p^m) in exponential format — Each entry a and b is an integer in the range [–∞, p^m – 2].

For more information on Galois formats, see Representing Elements of Galois Fields.

Tips

• This function multiplies elements of a Galois field.

• For polynomial multiplication in GF(p) or GF(p^m, use gfconv instead.

• This function performs computations in GF(p^m), where p is prime.

• To work in GF(2^m), you can also apply the .* operator to Galois arrays. For more information, see Example: Multiplication.

Version History

Introduced before R2006a