RSA

RSA is likely the most popular public key cryptographic set of algorithms.

Step 1:
Choose two secret large random primes p, q.

Step 2:
Compute p*q = n. n is disclosed as part of the public key.

Step 3:
Compute Phi(n) = (p-1)*(q-1). This is the totient function answering the number of co-prime numbers between 1…n.

Step 4:
Choose a number e < n which is co-prime to Phi(n).

(n, e) is then the public key.

Step 5:
Find the modular multiplicative inverse of e modulo Phi(n) e such that e*d ≡ 1 (mod Phi(n)).

(n, d) is then the private key.

Step 6:
Encrypt a message m such that cipher = m^e(mod n).

Step 7:
Decrypt an encrypted message cipher such that m = cipher^d(mod n).

Question 1:

Why is deriving the private d from (n, e) extremely challenging for large secret random p, q prime factors?

Answer:

While there has been progress in factoring at least one large n number up to 193 digits (RSA-640 bits,) cracking 1024-bit numbers or even as large as 2048-bit numbers appear to out of reach for the time being.

Question 2:

Does that mean that RSA is uncrackable for large n (> 2048 bits)?

Answer 2:

Yes, if the chosen random p, q primes are truly random and truly primes which has not always been the case commercially.

Note: there is a known exploit for cracking large keys by pairing same-sized n public RSA keys and calculating hcf on them to find any factors other than 1. If one is computed (poor p, q randomness) it is likely a prime factor of both n and the 2nd factor can then be computed instantly.

Pollard’s Rho is a relatively efficient probabilistic algorithm for factoring small sized n.

The Rho algorithm is employed to reverse an encrypted RSA message in this repo.