NAME

Math::GF - Galois Fields arithmetics

VERSION

This document describes Math::GF version {{[ version ]}}.

SYNOPSIS

use Math::GF;

# prime orders leverage on Math::GF::Zn
my $GF5 = Math::GF->new(order => 5);

# prints "yes!" because 5 is prime
say 'yes!' if $GF5->order_is_prime;
# prints "order 5 = 5^1"
say 'order ', $GF5->order, ' = ', $GF5->p, '^', $GF5->n;

# generate some elements
my $zero_gf5 = $GF5->additive_neutral;
my $one_gf5  = $GF5->multiplicative_neutral;
my $four_gf5 = $GF5->e(4);                # scalar context
my ($two_gf5, $three_gf5) = $GF5->(2, 3); # list context

# use some operations, both print "yes!"
say 'yes!' if $two_gf5   == $one_gf5  + $one_gf5;
say 'yes!' if $three_gf5 == $four_gf5 * $two_gf5;

# non-prime orders leverage on Math::GF::Extension
my $GF8 = Math::GF->new(order => 8);

# prints "order not prime!"
say 'order not prime!' unless $GF8->order_is_prime;

# prints "order 8 = 2^3"
say 'order ', $GF8->order, ' = ', $GF8->p, '^', $GF8->n;

# same operations as before anyway, no change in API
my $zero_gf8 = $GF8->additive_neutral;
my $one_gf8  = $GF8->multiplicative_neutral;
my ($three_gf8, $five_gf8) = $GF8->e(3, 5);

# use some operations... no more modulo operations in GF(2^3)
say 'yes!' if $three_gf8 * $five_gf8 == $GF8->e(4);

# import a factory function for building elements
Math::GF->import_builder(81, name => 'GF81'); # GF(3^4)
say 'yes!' if GF81(5) * GF81(8) == GF81(19);

# Need all elements? No problem
my @all_gf27 = Math::GF->new(order => 27)->all;

DESCRIPTION

This module allows you to generate and handle operations inside a Galois Field (GF) of any allowed order:

• orders that are too big are likely to explode
• orders that aren't prime number powers do not have associated Galois Fields.

It's easy to generate a new GF of a given order:

my $GF5 = Math::GF->new(order => 5); # GF(5)
my $GF8 = Math::GF->new(order => 8); # GF(2^3)

Since a GF of order N has exactly N elements, it's easy to refer to them with integers from 0 to N - 1. If you want to actually generate the associated element you can use the "e" method:

my $e5_gf8 = $GF8->e(5);

If you're planning to work extensively with a specific GF, or just want some syntactic sugar, you can import a factory function in your package that will generate elements in the specific GF:

# by default, import a function named GF_p_n for GF(p^n)
Math::GF->import_builder(8);
my $e5 = GF_2_3(5);

# you can give your name though
Math::GF->import_builder(8, name => 'GF8');
my $e5_gf8 = GF8(5);

If you need all elements, look at the "all" method. It's the same as doing this:

my @all = map { $GF8->e($_) } 0 .. 8 - 1;

but easier to type and possibly a bit quicker.

Elements associated to 0 and 1 have the usual meaning of the additive and multiplicative neutral elements, respectively. You can also get them with "additive_neutral" and "multiplicative_neutral".

METHODS

In the following, $GF is supposed to be a Math::GF object.

additive_neutral

my $zero = $GF->additive_neutral;

the neutral element of the Galois Field with respect to the addition operation. Same as $GF->e(0).

all

my @all_elements = $GF->all;

generate all elements of the Galois Field.

e

my $e5 = $GF->e(5);
my @some = $GF->e(2, 3, 5, 7);

factory method to generate one or more elements in the field. When called in scalar context it always operate on the first provided argument only.

element_class

my $class_name = $GF->element_class;

the underlying class for generating elements. It defaults to Math::GF::Zn when the "order" is a prime number and Math::GF::Extension when it is not; there is probably little motivation for you to fiddle with this.

import_builder

Math::GF->import_builder($order, %args);

import a factory function in the caller's package for easier generation of elements in the GF of the specified $order.

By default, the name of the imported function is GF_p or GF_p_n where p is a prime and n is the power of the prime such that $order = p ** n (the n part is omitted if it is equal to 1). For example:

Math::GF->import_builder(5); # imports GF_5()
Math::GF->import_builder(8); # imports GF_2_3()

You can pass your own name in the %args though:

Math::GF->import_builder(8, name => 'GF8'); # imports GF8()

The imported function is a wrapper around "e":

my $one = GF_2_3(1);
my @some = GF_5(1, 3, 4);

Allowed keys in %args:

• level

  by default the function is imported in the caller's package. This allows you to alter which level in the call stack you want to peek for importing the sub.

• name

  the name of the method, see above for the default.

multiplicative_neutral

my $one = $GF->multiplicative_neutral;

the neutral element of the Galois Field with respect to the multiplication operation. Same as $GF>e(1).

n

my $power = $GF->n;

the "order" of a Galois Field must be a power of a prime "p", this method provides the value of the power. E.g. if the order is 8, the prime is 2 and the power is 3.

order

my $order = $GF->order;

the order of the Galois Field. Only powers of a single prime are allowed.

order_is_prime

my $boolean = $GF->order_is_prime;

the "order" of a Galois Field can only be a power of a prime, with the special case in which this power is 1, i.e. the order itself is a prime number. This method provided a true value in this case, false otherwise.

p

my $prime = $GF->p;

the "order" of a Galois Field must be a power of a prime, this method provides the value of the prime number. E.g. if the order is 8, the prime is 2 and the power is 3. See also "n".

prod_table

my $pt = $GF->prod_table;

a table that can be used to evaluate the product of two elements in the field.

The table is provided as a reference to an Array of Arrays. The elements in the field are associated to indexes from 0 to order - 1; table element $pt->[$A][$B] represents the result of the product between element associated to $A and element associated to $B.

You shouldn't in general need to fiddle with this table, as it is used behind the scenes by Math::GF::Extension, where all operations are overloaded.

sum_table

my $st = $GF->sum_table;

a table that can be used to evaluate the product of two elements in the field.

The table is provided as a reference to an Array of Arrays. The elements in the field are associated to indexes from 0 to order - 1; table element $pt->[$A][$B] represents the result of the addition between element associated to $A and element associated to $B.

You shouldn't in general need to fiddle with this table, as it is used behind the scenes by Math::GF::Extension, where all operations are overloaded.

BUGS AND LIMITATIONS

Report bugs through GitHub (patches welcome).

SEE ALSO

Math::Polynomial is used behind the scenes to generate the tables in case the order is not a prime.

Math::GF::Zn is used for generating elements in the field and handling operations between them in an easy way in case of prime "order". Math::GF::Extension is used for elements in the field in case of non-prime "order"s.

AUTHOR

Flavio Poletti polettix@cpan.org

COPYRIGHT AND LICENSE

Copyright (C) 2017, 2018 by Flavio Poletti polettix@cpan.org

This module is free software. You can redistribute it and/or modify it under the terms of the Artistic License 2.0.

This program is distributed in the hope that it will be useful, but without any warranty; without even the implied warranty of merchantability or fitness for a particular purpose.