How Nikita Makes an RSA Public Key

Here is how Nikita makes a one-way function $E$:

1. Nikita picks two large primes $p$ and $q$, and lets $n=pq$.
2. It is easy for Nikita to then compute
   $$\varphi (n) = \varphi (p)\cdot \varphi (q) = (p-1)\cdot (q-1).$$

3. Nikita next chooses a ``random'' integer $e$ with
   $$1<e<\varphi (n)$$ and $$\gcd(e,\varphi (n))=1.$$

4. Finally, Nikita uses the algorithm from Lecture 7 to find a solution $d$ to the equation
   $$ex \equiv 1\pmod{\varphi (n)}.$$

The Encoding Function:
Nikita defines a function $E:\mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{Z}/n\mathbb{Z}$

$$E(x) = x^e.$$

(Recall that $\mathbb{Z}/n\mathbb{Z}= \{0,1,\ldots,n-1\}$ with addition and multiplication modulo $n$.) Then anybody can compute $E$ fairly quickly using the repeated-squaring algorithm from Lecture 7.

Nikita's public key is the pair of integers $(n,e)$, which is just enough information for people to easily compute $E$. Nikita knows a number $d$ such that $ed\equiv 1\pmod{\varphi (n)}$, so, as we will see below, she can quickly compute $E^{-1}$.

Now Michael or even The Collective can send Nikita a message whenever they want, even if Nikita is asleep. They look up how to compute $E$ and compute $E($their message$)$.