# Documentation

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# gfsub

Subtract polynomials over Galois field

## Syntax

```c = gfsub(a,b,p) c = gfsub(a,b,p,len) c = gfsub(a,b,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 of equal size. For details, see Example: Addition and Subtraction.

`c = gfsub(a,b,p) ` calculates `a` minus `b`, where `a` and `b` represent polynomials over GF(`p`) and `p` is a prime number. `a`, `b`, and `c` are row vectors that give the coefficients of the corresponding polynomials in order of ascending powers. Each coefficient is between 0 and `p`-1. If `a` and `b` are matrices of the same size, the function treats each row independently. Alternatively, `a` and `b` can be represented as polynomial character vectors.

`c = gfsub(a,b,p,len) ` subtracts row vectors as in the syntax above, except that it returns a row vector of length `len`. The output `c` is a truncated or extended representation of the answer. 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.

`c = gfsub(a,b,field) ` calculates `a` minus `b`, where `a` and `b` are the exponential format of two elements of GF(pm), relative to some primitive element of GF(pm). p is a prime number and m is a positive integer. `field` is the matrix listing all elements of GF(pm), arranged relative to the same primitive element. `c` is the exponential format of the answer, relative to the same primitive element. See Representing Elements of Galois Fields for an explanation of these formats. If `a` and `b` are matrices of the same size, the function treats each element independently.

## Examples

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Calculate over GF(5).

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

Subtract the two polynomials and display the first two elements.

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

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

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

Subtract from . The result is .

`g = gfsub(2,4,field)`
```g = 7 ```