# gfdiv

Divide elements of Galois field

## Syntax

```quot = gfdiv(b,a) quot = gfdiv(b,a,p) quot = gfdiv(b,a,field) ```

## Description

Note

This function performs computations in GF(pm), where p is prime. To work in GF(2m), apply the `./` operator to Galois arrays. For details, see Example: Division.

The `gfdiv` function divides elements of a Galois field. (To divide polynomials over a Galois field, use `gfdeconv` instead.)

`quot = gfdiv(b,a)` divides `b` by `a` in GF(2) element-by-element. `a` and `b` are scalars, vectors or matrices of the same size. Each entry in `a` and `b` represents an element of GF(2). The entries of `a` and `b` are either 0 or 1.

`quot = gfdiv(b,a,p)` divides b by a in GF(`p`) and returns the quotient. `p` is a prime number. If `a` and `b` are matrices of the same size, the function treats each element independently. All entries of `b`, `a`, and `quot` are between 0 and `p`-1.

`quot = gfdiv(b,a,field)` divides `b` by `a` in GF(pm) and returns the quotient. p is a prime number and m is a positive integer. If `a` and `b` are matrices of the same size, then the function treats each element independently. All entries of `b`, `a`, and `quot` are the exponential formats of elements of GF(pm) relative to some primitive element of GF(pm). `field` is the matrix listing all elements of GF(pm), arranged relative to the same primitive element. See Representing Elements of Galois Fields for an explanation of these formats.

In all cases, an attempt to divide by the zero element of the field results in a “quotient” of `NaN`.

## Examples

The code below displays lists of multiplicative inverses in GF(5) and GF(25). It uses column vectors as inputs to `gfdiv`.

```% Find inverses of nonzero elements of GF(5). p = 5; b = ones(p-1,1); a = [1:p-1]'; quot1 = gfdiv(b,a,p); disp('Inverses in GF(5):') disp('element inverse') disp([a, quot1]) % Find inverses of nonzero elements of GF(25). m = 2; field = gftuple([-1:p^m-2]',m,p); b = zeros(p^m-1,1); % Numerator is zero since 1 = alpha^0. a = [0:p^m-2]'; quot2 = gfdiv(b,a,field); disp('Inverses in GF(25), expressed in EXPONENTIAL FORMAT with') disp('respect to a root of the default primitive polynomial:') disp('element inverse') disp([a, quot2])```

## Version History

Introduced before R2006a