Now I return to the specifics of the El Gamal cryptosystem, and make some technical remarks. It is important that a new third key be selected every time a new message is sent. (We know from the Venona story the danger of re-using a one-time pad!) Therefore, Alice must select a new number A each time she wants to encrypt a message. What about when someone else (Bob, for example) wants to send a message to Alice? For such occasions, Alice has her own secret key AliceSecret and her own public key AlicePublic. Bob chooses a number B (just to be used for encrypting one message) and so on, using AlicePublic as AlicePart. The point is that the number A that Alice selects when she wants to encrypt a message has nothing to do with the number AliceSecret she uses to decrypt a message sent to her.
Note that because the third key is chosen anew each time by a random process, two encryptions of the same cleartext are very likely to be different. That is, unlike all other cryptosystems we have studied, for this system the cyphertext depends not only on the cleartext and the key but also on some random stuff. This might sound like a disadvantage (``the cyphertext is unpredictable!'') but it is usually an advantage: an eavesdropper who intercepts multiple cyphertexts cannot tell which of them correspond to identical cleartexts. There is a disadvantage: the cyphertext is at least twice as long as the cleartext--it consists of two numbers, each having at least as many digits as the cleartext--because it depends on more than just the cleartext.