Well, even a quantum computer can’t invent information out of thin air. When you have a message that was encrypted using an arbitrary key the same length as the message, there is absolutely no way to recover the original message without knowing the key. As you say, every possible message of that length is a valid plaintext for some key. If you have no information about what the key is, you have no information about what the message is.

It’s like I give you a single integer, 5, and tell you I arrived at it by subtracting one integer (my “key”) from another (my “message”). There are naturally an infinite number of possible messages that could result in ciphertext 5, so without any basis for knowing the key, you have no information about my message. No possible cryptanalysis can discover what isn’t there.

As for your first question, I imagine you’re suggesting that two interested parties could use such a number as a convenient “infinite” source of key data? I suspect it would be a very bad idea, and would undermine most of the security of the one-time pad.

Let’s say you use such a scheme for a while. You’re up to the millionth digit or so of your transcendental number *x*. Then your foes somehow get their dirty paws on the plaintext for some of your old messages, and are able to correlate them with intercepted ciphertext (by context, time, message length, etc.). If you had been using a proper one-time pad, you’d have nothing to fear; those messages were old and the discovery of the plaintext doesn’t worry you. But since all the numbers in your pad are related mathematically, there is the real risk that some mathematical wiseguy in the enemy’s crypto department will look for such a correspondence and find it.

I don’t know how hard that would actually be; obviously if you chose something like *pi* your code would be broken in a jiffy and your opponents would be reading your messages as soon as the intended recipient, if not sooner. My understanding is that known transcendental numbers fall into a small number of classes with strict relationships, so I would not trust my life to a “one-time pad” based on such a scheme.

Basically, though, no matter what the practical difficulties would be, using a transcendental number as your pad would change your code from “theoretically unbreakable” to “theoretically breakable,” which is a pretty big shift.