Here’s the surprisingly simple solution to a fairly challenging problem. I do not understand why PHPs GMP extension does not include a gmp_not() function. function gmp_not($n) { # convert to binary string $n = gmp_strval($n, 2); # invert each bit, one at a time for($i = 0; $i < strlen($n); $i++) { $n[$i] = ~$n[$i]; } # convert back to decimal return gmp_strval(gmp_init($n, 2), [...]

XOR (exclusive OR) is a boolean operator, like && and ||, but with the following logic: It is successful if the expression on either side is true (like ||), but not if both sides are true (like !( x && y )). Sometimes, in your JavaScript, you might want to do the following: if( foo XOR bar [...]