This is not a full solution for you, but rather some interesting fact.

You can use bit operations, like & and | directly against bit strings, but you have to make sure those strings have the same length (if not, add 0's to the beginning of the shorter one):

use strict;
my $a = "0" . "1011";
my $b = "10011";
print $a & $b, "\n";#00011
print $a | $b;#11011
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However, ^ and ~ do not work in this way.

The reason why it works for & and |:

use strict;
print ord("0") & ord("0"), "\n";#48, which is "0"
print ord("0") & ord("1"), "\n";#48, which is "0"
print ord("1") & ord("0"), "\n";#48, which is "0"
print ord("1") & ord("1"), "\n";#49, which is "1"
print ord("0") | ord("0"), "\n";#48, which is "0"
print ord("0") | ord("1"), "\n";#49, which is "1"
print ord("1") | ord("0"), "\n";#49, which is "1"
print ord("1") | ord("1"), "\n";#49, which is "1"
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The reason why it does not work for ^:

use strict;
print ord("0") ^ ord("0"), "\n";#0, which is not "0"
print ord("0") ^ ord("1"), "\n";#1, which is not "1"
print ord("1") ^ ord("0"), "\n";#1, which is not "1"
print ord("1") ^ ord("1"), "\n";#0, which is not "0"
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