NAME
Math::GF - Galois Fields arithmetics
VERSION
This document describes Math::GF version 0.004.
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
COPYRIGHT AND LICENSE
Copyright (C) 2017, 2018 by Flavio Poletti
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.