Public-key cryptography.

Used to encrypt messages sent between two communicating parties so
that an eavesdropper who overhears the conversation cannot decode
them.  

Also enables a party to append an unforgeable signature to the end of
a message.  This signature cannot be "easily" forged and can be
checked by anyone.

Suppose Alice and Bob are communicating.  And Bob wants to send a
message to Alice.  Alice has two keys, a secret key S_A that only
Alice knows and a public key P_A that Alice advertises to the whole
world.  Both the keys are actually functions that map a message to
another message.

Bob: M -> P_A(M) -------------> Alice: S_A(P_A(M)) -> M

So we want two functions S_A and P_A such that S_A(P_A(M)) = M, for
all permissible messages M.  Furthermore, any eavesdropper, who can
read message P_A(M), cannot extract M from this -- or, in fact, cannot
extract any reasonable information from this.

RSA cryptosystem:

1. Alice selects at random two large primes p and q.

2. Compute n = pq.

3. Select a small odd integer e that is relatively prime to
(p-1)(q-1).

4. Set d so that d*e is 1 modulo (p-1)(q-1).

5. Publish the pair (e,n) as the public key.

6. Keep secret the pair (d,n) as the secret key.

In order to send message M, Bob sends 

P_A(M) = M^e (mod n)

Alice computes

S_A(M') = M'^d (mod n)
        = M^{de} (mod n)
        = M.

Example: Take p = 5, q = 3, n = 15.  (p-1)(q-1) = 8.  Set e = 3.  Set
d = 3.