Flavio Poletti

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 <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.