In this series of videos, we will introduce the basics of the RSA cryptosystem.

Alice wants to send a message to Bob securely.

First, Bob generates a public and private key.

He chooses two (very large) primes \(p\) and \(q\) and sets \(N = pq\).

Then he chooses \(1\leq e\leq N\) with \(\gcd(e,\varphi(N)) = 1\).

Bob’s public key is \((N,e)\). He tells everyone his public key.

Next he computes his private key. This is secret.

He computes \(\varphi(N) = (p-1)(q-1)\).

Then he computes \(1\leq d\leq N\) such that \(de\equiv 1\bmod{\varphi(N)}\).

Bob’s private key is \((\varphi(N),d)\).

Now, \(m\) is our message we want to send and \(1\leq m\leq N\). Before we send \(m\) we encrypt it by computing \(c = m^e\bmod{N}\). Then we send \(c\) to Bob.

Bob receives the integer \(c\), called the ciphertext. He computes \(c^d\bmod{N}\) (that is why he has kept \(d\) for himself).

Proposition. Assume \(\gcd(m,N) = 1\). Then \(m\equiv c^d\bmod{N}\).

In the following video, we briefly discuss the security of the private key. It turns out, figuring out the private key given only the private key seems to be very difficult.