RSA encryption in the real world?

[geoffrey159]

Homework Statement

It is about an exercise called "RSA encryption". The problem statement was :
Let p,q be distinct prime numbers so that $n = pq$. If c,d are two integers so that the Euler totient function of n, $\phi(n)$, divides $cd - 1$, show that for any $t\in\mathbb{Z}$, $n$ divides $t^{cd} - t$.

I have found the solution but I don't understand how it works in the real world although it is the most interesting part of the exercise. Can you give a simple, real world example ?

Homework Equations

The Attempt at a Solution

 

[STEMucator]

RSA is also known as public key cryptography. The RSA algorithm allows one to create a private key and a public key. These keys can be used to encrypt and decrypt messages between different parties.

The public and private keys are really just prime numbers produced by the algorithm; Although both of these numbers should be kept a secret by the creator and any associates.

 

[geoffrey159]

hmmm... but how does it work ? Say you want to send me the crypted message "Hello", how do we do ?