Bruno Prime — Prime numbers as a working topic: how | brunoprime.com
RSA multiplies two secret primes into a public 617-digit modulus and bets that no one can split it back apart.
Key generation for RSA picks two large primes p and q of roughly equal size and multiplies them. The product n = p · q becomes the public modulus: 2048 bits for RSA-2048, about 617 decimal digits in total. Everyone working with the key sees n; only the holder knows p and q.
Multiplication is a fast operation — milliseconds even for thousand-bit inputs — while factoring the result back into its prime factors is a search with no known shortcut. Trial division would have to probe candidates up to about 10^308, and the best general factoring methods tested to date do not recover factors of a 2048-bit modulus in practical time.
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The security therefore rests on an asymmetry, not on secrecy of the algorithm. The scheme is published; the factors are not. Change one prime and the entire key changes, which is why key rotation means regenerating both factors.
Estimating how many candidate primes exist is part of key design. The prime number theorem puts the count of primes below x near x / ln x, so designers can size p and q to make accidental collisions negligible.
Related constructions — signatures, key exchange, and other public-key primitives — inherit the same hardness assumption: multiplying is easy, factoring the product is not.
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