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Math::String - Arbitrary sized integers having arbitrary charsets to calculate with key rooms


    use Math::String;
    use Math::String::Charset;

    $a = new Math::String 'cafebabe';   # default a-z
    $b = new Math::String 'deadbeef';   # a-z
    print $a + $b;                      # Math::String ""

    $a = new Math::String 'aa';         # default a-z
    $b = $a;
    print "$b > $a" if ($b > $a);       # prove that ++ makes it greater
    print "$b == $a" if ($b == $a);     # and that ++ and -- are reverse

    $d = Math::String->bzero( ['0'...'9'] );    # like Math::Bigint
    $d += Math::String->new ( '9999', [ '0'..'9' ] );
                                        # Math::String "9999"

    print "$d\n";                       # string       "00000\n"
    print $d->as_number(),"\n";         # Math::BigInt "+11111"
    print $d->last(5),"\n";             # string       "99999"
    print $d->first(3),"\n";            # string       "111"
    print $d->length(),"\n";            # faster than length("$d");

    $d = Math::String->new ( '', Math::String::Charset->new ( {
      minlen => 2, start => [ 'a'..'z' ], } );

    print $d->minlen(),"\n";            # print 2
    print ++$d,"\n";                    # print 'aa'


perl5.005, Exporter, Math::BigInt, Math::String::Charset


Exports nothing on default, but can export as_number(), string(), first(), digits(), from_number, bzero() and last().


This module lets you calculate with strings (specifically passwords, but not limited to) as if they were big integers. The strings can have arbitrary length and charsets. Please see Math::String::Charset for full documentation on possible character sets.

You can thus quickly determine the number of passwords for brute force attacks, divide key spaces etc.

Default charset

The default charset is the set containing "abcdefghijklmnopqrstuvwxyz" (thus producing always lower case output).


Uses internally Math::BigInt to do the math, all with overloaded operators. For the character sets, Math::String::Charset is used.

Actually, the 'numbers' created by this module are NOT equal to plain numbers. It works more than a counting sequence. Oh, well, example coming:

Imagine a charset from a-z (26 letters). The number 0 is defined as '', the number one is therefore 'a' and two becomes 'b' and so on. And when you reach 'z' and increment it, you will get 'aa'. 'ab' is next and so on forever.

That works a little bit like the automagic in ++, but more consistent and flexible. The following example 'breaks' (no, >= instead of gt won't help ;)

        $a = 'z'; $b = $a; $a++; print ($a gt $b ? 'greater' : 'lower');

With Math::String, it does work as intended, you just have to use '<' or '>' etc for comparing. That was also the main reason for this module ;o)

incidentily, '--' as well most other mathematical operations work as you expected them to work on big integers.

Compare a Math::String of charset '0-9' sequence to that of a 'normal' number:

    ''   0                       0
    '0'  1                       1
    '1'  2                       2
    '2'  3                       3
    '3'  4                       4
    '4'  5                       5
    '5'  6                       6
    '6'  7                       7
    '7'  8                       8
    '8'  9                       9
    '9'  10                     10
   '00'  11                1*10+ 1
   '01'  12                1*10+ 2
   '98'  109               9*10+ 9
   '99'  110               9*10+10
  '000'  111         1*100+1*10+ 1
  '001'  112         1*100+1*10+ 2
 '0000'  1111  1*1000+1*100+1*10+1
 '1234'  2345  2*1000+3*100+4*10+5

And so on. Here is another example that shows how it works with a number having 4 digits in each place (named "a","b","c", and "d"):

     a    1           1
     b    2           2
     c    3           3
     d    4           4
    aa    5       1*4+1
    ab    6       1*4+2
    ac    7       1*4+3
    ad    8       1*4+4
    ba    9       2*4+1
    bb   10       2*4+2
    bc   11       2*4+3
    bd   12       2*4+4
    ca   13       3*4+1
    cb   14       3*4+2
    cc   15       3*4+3
    cd   16       3*4+4
    da   17       4*4+1
    db   18       4*4+2
    dc   19       4*4+3
    dd   20       4*4+4
   aaa   21  1*16+1*4+1

Here is one with a charset containing 'characters' longer than one, namely the words 'foo', 'bar' and 'fud':

           foo           1
           bar           2
           fud           3
        foofoo           4
        foobar           5
        foofud           6
        barfoo           7
        barbar           8
        barfud           9
        fudfoo          10
        fudbar          11
        fudfud          12
     foofoofoo          13 etc

The number sequences are symmetrical to 0, e.g. 'a' is both 1 and -1. Internally the sign is stored and honoured, only on conversation to string it is lost.

The caveat is that you can NOT use Math::String to work, let's say with hexadecimal numbers. If you do calculate with Math::String like you would with 'normal' hexadecimal numbers (any base would or rather, would not do), the result may not mean anything and can not nesseccarily compared to plain hexadecimal math.

The charset given upon creation need not be a 'simple' set consisting of all the letters. You can, actually, give a set consisting of bi-, tri- or higher grams.

See Math::String::Charset for examples of higher order charsets and charsets with more than one character per, well, character.



Create a new Math::String object. Arguments are the value, and optional charset. The charset is set to 'a'..'z' as default.

Since the charset caches some things, it is much better to give an already existing Math::String::Charset object to the contructor, instead of creating a new one for each Math::String. This will save you memory and computing power. See for details, and Math::String::Charset for how to construct charsets.


Return the last error message or ''. The error message stems primarily from the underlying charset, and is created when you create an illegal charset.


Return the order of the string derived from the underlying charset. 1 for SIMPLE (or order 1), 2 for bi-grams etc.


Return the type of the string derived from the underlying charset. 0 for simple and nested charsets, 1 for grouped ones.


It is a bit tricky to get the first string of a certain length, because you need to consider the charsets at each digit. This method sets the given Math::String object to the first possible string of the given length. The length defaults to 1.


It is a bit tricky to get the last string of a certain length, because you need to consider the charsets at each digit. This method sets the given Math::String object to the last possible string of the given length. The length defaults to 1.


Return internal number as normalized string including sign.

        $string = Math::String::from_number(1234,$charset);

Create a Math::String from a given integer value and a charset.

If you want to use big integers as input, quote them:

        $string = Math::String::from_number('12345678901234567890',$set);

This avoids loosing precision due to intermidiate storage of the number as Perl scalar.

        $scale = $string->scale();

Get/set the (optional) scale of the characterset (thus setting it for all strings of that set from this point onwards). A scale is an integer factor that will be applied to each as_number() output as well as each from_number() input. E.g. for a scale of 3, the string to number mapping would be changed from the left to the right column:

        string form             normal number   scaled number
        ''                      0               0
        'a'                     1               3
        'b'                     2               6
        'c'                     3               9

And so on. Input like 8 will be divided by 3, which results in 2 due to rounding down to the nearest integer. So:

        $string = Math::String->new( 'a' );             # a..z
        print $string->as_number();                     # 1
        print $string->as_number();                     # 3
        $string = Math::String->from_number(9,3);       # 9/3 => 3
        $string = Math::String->bzero($charset);

Create a Math::String with the number value 0 (evaluates to ''). The following would set $x to '':

        $x = Math::String->new('cafebabe');
        $string = Math::String->bone($charset);

Create a Math::String with the number value 1 and the given charset

The following would set $x to the number 1 (and it's respective string):

        $x = Math::String->new('cafebabe');
        $string = Math::String->binf($sign);

Create a Math::String with the number infinity.

The following would set $x to -infinity (and it's respective string):

        $x = Math::String->new('deadbeef');
        $string = Math::String->bnan();

Create a Math::String as a NotANumber.

The following would set $x to NaN (and it's respective string):

        $x = Math::String->new('deadbeef');
        print $string->error(),"\n" if !$string->is_valid();

Returns 0 if the string is valid (according to it's charset and string representation) and the cached string value matches the string's internal number represantation. Costly operation, but usefull for tests.

        $count = $string->class($length);

Returns the number of possible strings with the given length, aka so many characters (not bytes or chars!).

        $count = $string->class(3);     # how many strings with len 3

Return the minimum length of a valid string as defined by it's charset. Note that the string '' has a length of 0, and thus is not valid if minlen is greater than 0. Returns 0 if no minimum length is required. The minimum length must be smaller or equal to the maxlen.


Return the maximum length of a valid string as defined by it's charset. Returns 0 if no maximum length is required. The maximum length must be greater or equal to the minlen.


Return the number of characters in the resulting string (aka it's length). The zero string '' has a length of 0.

This is faster than doing length("$string"); because it doesn't need to do the costly creation of the string version from the internal number representation.

Note: The length() will be always in characters. If your characters in the charset are longer than one byte/character, you need to multiply the length by the character length to find out how many bytes the string would have.

This is nearly impossible if your character set has characters with different lengths (aka if it has a separator character). In this case you need to construct the string to find out the actual length in bytes.


Return a string representing the internal number with the given charset. Since this omitts the sign, you can not distinguish between negative and positiv values. Use as_number() or sign() if you need the sign.

This returns undef for 'NaN', since with a charset of [ 'a', 'N' ] you would not be able to tell 'NaN' from true 'NaN'! '+inf' or '-inf' return undef for the same reason.


Return a reference to the charset of the Math::String object.


Just like new, but you can import it to save typing.


For the actual math, the same limits as in Math::BigInt apply. Negative Math::Strings are possible, but produce no different output than positive. You can use as_number() or sign() to get the sign, or do math with them, of course.

Also, the limits detailed in Math::String::Charset apply, like:

No doubles

The sets must not contain doubles. With a set of "eerr" you would not be able to tell the output "er" from "er", er, if you get my drift...

Charset items

All charset items must have the same length, unless you specify a separator string:

        use Math::String;

        $b = Math::String->new( '',
           { start => [ qw/ the green car a/ ], sep => ' ', }

        while ($b ne 'the green car')
          print ++$b,"\n";      # print "a green car" etc

Writing things like

        $a = Math::String::bsub('hal', 'aaa');

does not work, unlike with Math::BigInt (which just knows how to treat the arguments to become BigInts). The first argument must be a reference to a Math::String object.

The following two lines do what you want and are more or less (except output) equivalent:

        $a = new Math::String 'vms'; $a -= 'aaa';
        $a = new Math::String 'ibm'; $a->badd('aaa');

Also, things like

        $a = Math::String::bsub('hal', 5);

does not work, since Math::String can not decide whether 5 is the number 5, or the string '5'. It could, if the charset does not contain '0'..'9', but this would lead to confusion if you change the charset. So, the second paramter must always be a Math::String object, or a string that is valid with the charset of the first parameter. You can use Math::String::from_number():

        $a = Math::String::bsub('hal', Math::String::from_number(5) );


Fun with Math::String:

        use Math::String;

        $ibm = new Math::String ('ibm');
        $vms = new Math::String ('vms');
        $ibm -= 'aaa';
        $vms += 'aaa';
        print "ibm is now $ibm\n";
        print "vms is now $vms\n";

Some more serious examples:

        use Math::String;
        use Math::BigFloat;

        $a = new Math::String 'henry';                  # default a-z
        $b = new Math::String 'foobar';                 # a-z

        # Get's you the amount of passwords between 'henry' and 'foobar'.
        print "a  : ",$a->as_numbert(),"\n";
        print "b  : ",$b->as_bigint(),"\n";
        $c = $b - $a; print $c->as_bigint(),"\n";

        # You want to know what is the first or last password of a certain
        # length (without multiple charsets this looks a bit silly):
        print $a->first(5),"\n";                        # aaaaa
        print Math::String::first(5,['a'..'z']),"\n";   # aaaaa
        print $a->last(5),"\n";                         # zzzzz
        print Math::String::last(5,['A'..'Z']),"\n";    # ZZZZZ

        # Lets assume you had a password of length 4, which contained a
        # Capital, some lowercase letters, somewhere either a number, or
        # one of '.,:;', but you forgot it. How many passwords do you need
        # to brute force in the worst case, testing every combination?
        $a = new Math::String '', ['a'..'z','A'..'Z','0'..'9','.',',',':',';'];
        # produce last possibility ';;;;;' and first 'aaaaa'
        $b = $a->last(4);   # last possibility of length 4
        $c = $a->first(4);  # whats the first password of length 4

        print $c->as_bigint(),"\n";             # all of length 4
        print $b->as_bigint(),"\n";             # testing length 1..3 too

        # Let's say your computer can test 100.000 passwords per second, how
        # long would it take?
        $d = $c->bdiv(100000);
        print $d->as_bigint()," seconds\n";     #

        # or:
        $d = new Math::BigFloat($c->as_bigint()) / '100000';
        print "$d seconds\n";                   #

        # You want your computer to run for one hour and see if the password
        # is to be found. What would be the last password to be tested?
        $c = $b + (Math::BigInt->new('100000') * 3600);
        print "Last tested would be: $c\n";

        # You want to know what the 10.000th try would be
        $c = Math::String->from_number(10000,
        print "Try #10000 would be: $c\n";


For simple things, like generating all passwords from 'a' to 'zzz', this is expensive and slow. A custom, table-driven generator or the build-in automagic of ++ (if it would work correctly for all cases, that is ;) would beat it anytime. But if you want to do more than just counting, then this code is what you want to use.




  • Charsets with bi-grams do not work fully yet.

  • Adding/subtracting etc Math::Strings with different charsets treats the second argument as it had the charset of the first. This is thought as a feature, not a bug.

    Only if the first charset contains all the characters of second string, you could convert the second string to the first charset, but whether this is usefull is questionable:

            use Math::String;
            $a = new Math::String ( 'a',['a'..'z']);        # is 1
            $z = new Math::String ( 'z',['z'..'a']);        # is 1, too
            $b = $a + $z;                                   # is 2, with set a..z
            $y = $z + $a;                                   # is 2, with set z..a

    If you convert $z to $a's charset, you would get either an 1 ('a'), or a 26 ('z'), and which one is the right one is unclear.

  • Please report any bugs or feature requests to bug-math-string at, or through the web interface at (requires login). We will be notified, and then you'll automatically be notified of progress on your bug as I make changes.


You can find documentation for this module with the perldoc command.

    perldoc Math::String

You can also look for information at:


This program is free software; you may redistribute it and/or modify it under the same terms as Perl itself.


If you use this module in one of your projects, then please email me. I want to hear about how my code helps you ;)

Tels 2000 - 2005.

Maintained by Peter John Acklam, 2017-