Question: I can imagine that finding $p,q,n$ are easy, but my text book also says the $e$ can be found rapidly, so how does modern RSA find this $e$?

![RSA as a Public Key System](https://i.stack.imgur.com/LBR7F.png)

Answer: 
Secret Sharing may be another option to consider. It allows you to take a value, break it up into arbitrarily many pieces, and possession of a subset of these pieces, not the entire set, makes reconstruction of the original value impossible. This is done by multiple means, the simplest of which is with respect to additive secret sharing. Given a secret value $x$, $m$ shares of $x$ in a group $Z\_N$ may be generated by selecting $m-1$ random values and assigning them to shares $$\forall i \in \{1,\dots,m-1\},[x]\_N^{P\_i}\in\_RZ\_N $$ The final share satisfies the equation
$$[x]\_N^{P\_m}=(x-\sum\_{i=1}^{m-1}[x]\_N^{P\_i})\mod N$$ 
This way every share is uniform random and is secure in an information theoretic sense. Additionally, this holds up to an individual possessing any $m-1$ magnitude subset of shares. Only possession of all $m$ shares will allow the original secret to be reconstructed, and this holds without respect to assumptions or limitations on computational power. If an individual does posess all $m$ shares, reconstruction of the secret is very easy since:
$$x=(\sum\_{i=1}^{m}[x]\_N^{P\_i})\mod N$$
The "key" is secret data that are used in encryption with known "algorithm". Of course you can make keyless algorithm - get permutation, shifting stuff around, bit inversion, whatever. Employ any known strong encryption with built-in values. Mix and match all you want! However, this effectively makes algorithm itself the key. Once someone has access to your encryptor/decryptor and/or any built-in values, they don't need to know any additional data - which is ENTIRE REASON for separate key - and can decrypt whatever they want.

Incidentally that's primary and inevitable point of failure of futile attempts for DRM.