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.