One of the most widely used and useful operations in cryptographic algorithms is XOR. It is worth understanding just why XOR is such a helpful operation. XOR, as it is used in cryptography, is a bitwise numeric function with a domain of a bit pair, and the range of a result bit. (It has a slightly different, but isomorphic, use in formal logic.) Most readers are probably already familiar with XOR's result table, but let us take a look at it as a reminder:

		    XOR(1, 1) --> 0
		    XOR(1, 0) --> 1
		    XOR(0, 1) --> 1
		    XOR(0, 0) --> 0
		    

We write the XOR function in the above table in a prefix notation, but most programming languages use an infix form. Don't worry about the notation -- the above just helps illustrate the functional nature of XOR. Also, in most programming languages, the operation called XOR (or more accurately "bitwise XOR") does more than the above table shows, but only as a generalization. That is, in an operation like C, Perl, or Python, "^" is actually the Boolean XOR of each corresponding bit in two bit-fields (or ASCII characters, integers, etc., considered as bit-fields). In principle, a language with only a single-bit XOR could simulate the bit-field XOR behavior by looping through each bit position (but computational efficiency benefits greatly from the compound bit-field XOR).