RSA Modulus

The whole of RSA rests on a gap between two operations that look symmetrical and are not. Multiplying $61 \times 53$ to get 3233 takes no time at all. Starting from 3233 and recovering 61 and 53 also takes no time — at this size. Scale both primes to 1024 bits each and the multiplication is still instant while the factoring is, as far as anyone has published, beyond every machine on Earth. The modulus n is published as half of the public key; p and q are destroyed or locked away, because anyone who has them can compute the private key in a few microseconds.

61 and 53 are the numbers from Rivest, Shamir and Adleman's original 1977 paper and they turn up in every textbook since, which makes 3233 a useful sanity check on any implementation. Real keys use primes of similar size to each other but not too similar: if p and q are close, n is close to a perfect square and Fermat's factorisation method peels them apart quickly. That is a genuine failure mode found in deployed hardware, not a theoretical one.

What this page does not do is prove your inputs are prime. Give it 9 and 15 and it will return 135 quite happily. It is doing arithmetic, not key generation — checking primality of a 1024-bit candidate is a separate algorithm, and the Miller–Rabin test that does it is not a formula in the sense this site means.