The Math Behind RSA #3: Implementing RSA from Scratch in Python

This article has a more math-focused version with formal proofs on Folio . In the first two parts of this series, we built up the number theory toolkit: modular arithmetic, Euler's totient function, the extended Euclidean algorithm, and Fermat's little theorem. Now it's time to put all of that to work and implement RSA from scratch in Python . By the end of this article you will have a working RSA implementation, understand why each mathematical piece is needed, and see a formal proof that decryption actually recovers the original message. RSA in Three Steps The entire RSA cryptosystem reduces to three operations: Key Generation -- produce a public key (n,e)(n, e) ( n , e ) and a private key (n,d)(n, d) ( n , d ) . Encryption -- given plaintext mm m , compute ciphertext c=me mod nc = m^e \bmod n c = m e mod n . Decryption -- given ciphertext cc c , recover m=cd mod nm = c^d \bmod n m = c d mod n . The security rests on one assumption: given n=pqn = pq n = pq , it is computationally inf