Public/Private Key Cryptography

This is a system for communicating accross a channel that ensures that only the intended recipient can read the message. We start by generating a public key and a private or secret key. We will consider a communication between two people. It is traditional to call these two people ``Alice'' and ``Bob''. It doesn't take a great deal of thought to realise that this is so we can call their public keys $P_A$ and $P_A$ and their secret keys $S_A$ and $S_B$.

Now assume that all permissable messages form a set (prehaps the set of all finite-length bit sequences). We now define 4 one-one functions on this set (i.e. permutations). The function corresponding to Alice's public key $P_A$ we will call $P_A()$ and the one corresponding to her secret key $S_A$ we will call $S_A()$. These functions are also inverses of each other. So given a message M:

$$M = S_A(P_A(M))$$

(The functions relating to Bob's keys are predictably named $P_B()$ and $S_B()$ and are also inverses).

It is now clear to see how the system can work. Suppose Bob has message $M$ and he wants to pass it securely to Alice. Since he (along with the rest of the world) knows Alice's public key $P_A$ he is able to calculate $C = P_A(M)$. He can now safely pass C to Alice across the insecure channel. Once she has got $C$ Alice can calculate $M = S_A(C)$ without any difficulty since she knows $S_A$ (the inverse function to $P_A$). No-one else (Bob included) can do this calculation since they don't possess $S_A$.

Of course there is still the problem of choosing suitable functions. Remember we must be able to happily reveal $P_A$ and this $P_A()$ while being more than happy that it is sufficiently impractical to calculate it's inverse function $S_A()$ (or equally $S_A$) from it.