t there is no reason not to use greater numbers to give a trillion possibilities. Here is another variant of the Diffie-Hellman-Merk algorithm (referred to as Hughes), where Alice wants the secret key to be some specific number, k. Step 1 Alice chooses a large integer number A and keeps it secret; she computes k = mA mod n. Step 2 Bob chooses a large integer number B and keeps it secret; he computes b = m B mod n and sends the result to Alice. Step 3 Alice computes a=b4 mod n and sends the result to Bob. Step 3 Bob computes c=1/B k = a' mod n. The result of Bob's operation is = = (modulo n). Thus, Alice has successfully communicated her preferred key choice to Bob. The advantage of the Hughes algorithm is that Alice can initiate the communication Classical and quantum cryptography of encrypted messages herself, not only to Bob but to any group of persons. She can send them the key at a later time. These approaches concern secret-key cryptography, which, except for the absence of key exchange, resort to the same classical approach when it comes to encryption and decryption. Furthermore, the two parties share the same (secret) key, therefore, they can both encrypt and decrypt with that key. In the communication channel, their relationship is symmetric, hence, the name symmetric-key cryptography. An interesting alternative would be to have different keys for encryption and decryption: Bob can send Alice encrypted messages using a shared key, but only Alice is able to decrypt them, using her own secret key. Bob's and Alice's communication being, thus, asymmetric, they are using a new form of asymmetric-key cryptography. In the next section, we shall see that modular arithmetic opened the path to asymmetric cryptography, better known as public-key cryptography (PKC). Public-key cryptography and RSA The approach of public-key cryptography (PKC) led to the RSA standard, named after its inventors R. Rivest, A. Shamir, and L. Adleman. A detailed description of PKC and RSA was presented in Chapter 20 concerning Shor's (qua