ple that the same ciphertext character may correspond to different plaintext characters, namely the ;S corresponds to I and u and the 1s corresponds to i and y, respectively. We may, therefore, rapidly conclude that this XOR operation with a key as long as the plaintext (or sufficiently long and repeated) and as many as 128 possible characters would produce quite a strong cipher. But this is not the case. To provide here a simple explanation, consider indeed the first codeword of the ciphertext, 0111010. Because of the XOR operation, it is certain that (1) For bit numbers 7, 3, and 1: the operands must be of the same parity, namely coming from the operations 0 XOR 0 or 1 XOR 1; (2) For bit numbers 6, 5, 4, and 2: the operands must be of opposite parities, namely, coming from the operations 0 XOR 1 or 1 XOR 0. Since there are two operand choices for each bit, the number of possibilities is 27 = 128. But we can reduce this number by correlating the two different conditions for each bit. To establish such a correlation, we can tabulate all correspondence possibilities between each seven-bit ciphertext block and key block in the form of a 128 X 128 = 16 384 element array, where the cases that violate the two conditions are left empty. A straightforward frequency analysis of both plaintext and key, searching for the most common letters, digrams, trigrams, and other most commonly used English words will allow a small computer to finish the job in seconds! This description suggests that in the binary system, encryption with a secret key on a one-to-one character basis is unsafe, unless complex random keys are used. As we saw earlier, the binary system offers a tremendous advantage, which is the possibility of mingling the characters together in the coding process, on a bit-by-bit or block-by-block basis (a block being a string of bits of any prescribed length). This is the principle used by the first attempt at a standard cipher, developed in the 1960s by IBM and called Lucifer (as a pun on cipher). Figure 25.1 shows th