A Small Example

So the arithmetic is easy to follow, we use small primes $p$ and $q$ and encrypt the single letter ``X''.

1. Choose $p$ and $q$: Let $p=17$, $q=19$, so $n=pq = 323$.
2. Compute $\varphi (n)$:
   $$\varphi (n) = \varphi (p\cdot q)=\varphi (p)\cdot\varphi (q) = (p-1)(q-1) = pq-p-q+1 = 323-17-19+1 = 288.$$

3. Randomly choose an $e\in\mathbb{Z}/323\mathbb{Z}$: We choose $e=95$.
4. Solve
   $$95x \equiv 1\pmod{288}.$$
   Using the GCD algorithm, we find that $d=191$ solves the equation.

The public key is $(323,95)$. So $E:\mathbb{Z}/323\mathbb{Z}\rightarrow \mathbb{Z}/323\mathbb{Z}$ is defined by

$$E(x) = x^{95}.$$

Next, we encrypt the letter ``X''. It is encoded as the number $24$, since X is the $24$th letter of the alphabet. We have

$$E(24) = 24^{95} = 294 \in \mathbb{Z}/323\mathbb{Z}.$$

To decrypt, we compute $E^{-1}$:

$$E^{-1}(294) = 294^{191} = 24 \in \mathbb{Z}/323\mathbb{Z}.$$