t be relatively prime with . And as it nicely turns out, the numbers 3, 17, and 64 537 are all primes, therefore, more likely to be relatively prime with . To complicate Eve's task, the recommendation is also that the private key be sufficiently large and that the choice of modulus be broad ("sufficiently" and "broad" being here left to professional appreciation). It is not recommended to use the same private key for encrypted signature and authentication. It can be shown that malicious Eve can retrieve Alice's private key by sending her a specially prepared document and asking her candidly to sign or certify it with her private key! Several variants and extensions of PKC and asymmetric-key cryptography can be also found in the footnote reference. See B. Schneir, Applied Cryptography, 3rd edn. (New York: John Wiley and Sons, 1996). 25.6 Data encryption standard (DES) and advanced encryption standard (AES) Data encryption standard (DES) and advanced encryption standard (AES) The previous sections have introduced the reader to some aspects of cryptography standardization, which concern the encryption and decryption algorithms as well as the formats to be used for keys and ciphers. In this section, I shall briefly describe two leading standard cryptosystems, referred to as DES (data encryption standard) and AES (advanced encryption standard). As we have seen, the story began in the 1960s under IBM's Lucifer initiative, which set the grounds for the adoption, in the 1970s, of DES. The initial DES version had a 56-bit key, a reduction from the proposed 128-bit format, which was meant to facili- tate cracking the code should government authorities need access for national-security reasons. Rumors have circulated about the existence in DES of embedded trapdoor functions, which could enable direct decryption by state-class parties, but such rumors did not prevent its eventual standardization and widespread adoption. At the time, it was considered that cracking DES by brute-force attack, i.e., trying all possible keys at 