These are considerably more concise :

use constant  BASE62 => join '', 'a'..'z', 0 .. 9, 'A'..'Z';

sub toBase62 {
    use integer;
    my $n = shift;
    my @out;
    push( @out, substr( BASE62, $n % 62, 1 ) ), $n /= 62 while $n;
    join '', @out;
}
sub fromBase62 {
    my $t = reverse shift;
    my $n = 0;
    $n *= 62, $n += index( BASE62, substr( $t, 0, 1, '') ) while lengt
+h $t;
    return $n
}
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and quite a bit more efficient:

#! perl -slw
use 5.010;
use strict;
use Benchmark qw[ cmpthese ];
use Math::Random::MT qw[ rand ];

...

our @data = map int( rand 2**32 ), 1 .. 1000;

cmpthese -1, {
    yourn => q[
        $_ == alphaID( alphaID( $_, 0 ), 1 )
            or die "Yourn $_" for @data;
    ],
    mine  => q[
        $_ == fromBase62( toBase62( $_ ) )
            or die "mine $_" for @data;
    ],
};

__END__
C:\test>junk25
        Rate yourn  mine
yourn 49.2/s    --  -48%
mine  93.9/s   91%    --
[download]