xt provides a means for both encryption and decryption. Such mixing is performed by simple Boolean operations. In the binary system, the four operations, called Boolean, are NO (logical inversion), AND (multi- plication), OR (addition or subtraction), and XOR (exclusive OR). The rules of these operations on one (NO) or two (AND, OR, XOR) bits or "operands" have been intro- duced in Chapter 15. We note that with binary numbers, addition and subtraction are the same (1 + 1 = 1 - 1 = 0, 1 + 0 = 1 - 0 = 0 + 1 = 0 1). We also note that XOR is the same as OR, except for the rule 1 + 1 = 0, which is similar to 9 + 1 = 0 in the ordinary digital system. The operators (NO, AND, OR, and XOR) are also noted by mathematicians -, ^, V, and +, respectively. Thus NO a, a AND b, and a XOR b are noted -a, a ^ b, and a + b, respectively. The other basic Boolean operators are < (noted and = (noted Thus, a b is zero if a > b, and one otherwise. Similarly, b is zero if a # b, and one if a = b. It is easily checked that the binary result of the operation (b c) is unity if and only if the condition a b = C is fulfilled. In this chapter, we will not be concerned by Boolean logic other than just using XOR. Classical and quantum cryptography Table 25.3 Ciphering with the (case-sensitive) key sAXOPHON. Plaintext PT-ASCII 1001001 1101101 1101001 1110011 1110011 1111001 1101111 1110101 K-ASCII 1110011 1000001 1011000 1001111 1010000 1001000 1001111 1001110 CT-ASCII 0111010 0101100 0110001 0111100 0100011 0110001 0100000 0111011 Ciphertext ; space An interesting property of XOR is that its double application to a with the same operand b restores operand a, i.e., a + b b = a. We can, thus, use this property for the purpose of encryption with a key. Indeed, if we perform XOR between each bit of the plaintext and the key, we obtain a ciphertext. If we repeat the operation, this time between the ciphertext (CT) and the key (K), we obtain the plaintext (PT). Table 25.3 shows an example, with sAXOPHON as a (case-sensitive) key. We see from this exam