Textbook RSA:
Choose two large primes $p$ and $q$. Let $n=p\cdot q$. Choose $e$ such that $gcd(e,\phi(n))=1$ (where $\phi(n)=(p-1)\cdot (q-1)$). Find $d$ such that $e\cdot d\equiv 1\mod\phi(n)$.
To encrypt a message $m$ compute $c\equiv m^e\mod n$. To decrypt a ciphertext $c$, compute $c^d\mod n$.
Signing and verifying messages is also defined (omitted for brevity).
Some (Undesirable) Properties of Textbook RSA:
- It is malleable. I.e., if you give me a ciphertext $c$ which encrypts $m$, I can compute $c'\equiv c\cdot 2^e\mod n$. When the owner of the private key decrypts $c'$, she will get $2m\mod n$. In other words, I can make predictable changes to ciphertexts.
- It is not semantically secure. I.e., I can distinguish between the encryptions of $0$ and $1$.
Differences with Deployed RSA:
- Padding
- Chinese Remainder Theorem is sometimes used in deployed systems
- $e$ is often statically set to $65567$ for speed (since there are only two set bits in that number)
- Side-channel attack mitigations can be put in place for deployed systems too
In no way is my list comprehensive, but hopefully this helps.