ecting complex permutations thereof through an "S-box" look-up table. Cryptanalysts have shown that with a 256-bit key, AES has a complexity of 2100 (and not 2256), which comes close, SO to speak, to that of 3DES, which is about three orders of magnitude greater (2112). As a concluding statement, it should be stressed herewith that cryptography is hardly limited to encryption, decryption and key exchange protocols. Indeed, to communicate messages safely through any of the above-described cryptosystems is not all that mat- ters in real cryptospace. There exists a broad catalog of other severe issues and more immediate threats, which academics focusing on quantum alternatives generally tend to overlook, while these represent far more serious exposures to attacks, even at basic network layers. For instance, how can Bob be confident that Alice is the real Alice, and the reverse? How are they confident that their messages, as received, are the ones they intended to share? This is the realm of digital signatures and message authentication. As the name indicates, a digital signature is a way of certifying, within reasonable confidence, that the message originator Alice is really the person claimed, or that her signature is technically unforgeable. Authentication is different. It ensures that nothing in the message was altered, even to the level of a single punctuation mark. This protec- tion is important in all matters pertaining to official records, such as titles, patents, or 13 E. Desurvire, Wiley Survival Guide in Global Telecommunications, Broadband Access, Optical Components and Networks, and Cryptography (New York: John Wiley and Sons, 2004), Ch. 3, pp. 345-477. 25.7 Quantum cryptography tax declarations (contents and time stamping), contracts, authorizations, and confidential orders, for instance. The key to these processes is the use of hash functions, which can be viewed as one-way encryption. For short, and to convey a flavor, a 10000-page plaintext book can be "hashed" into a 512-bit signature (the "hash"). 