What is a "provably secure" cryptosystem, and how does the Rabin cryptosystem differ from RSA in this regard?

A provably secure cryptosystem is one where breaking the cipher is mathematically proven to be as hard as solving a known hard problem — the Rabin system has this property, but RSA does not.

Rabin cryptosystem:

• Breaking Rabin is provably equivalent to solving the integer factorization problem
• This means: if you can break Rabin, you can factor large numbers (and vice versa)
• This gives a mathematical guarantee — Rabin is exactly as secure as factoring is hard

RSA:

• It's often claimed that breaking RSA is equivalent to factoring, but this is NOT proven
• What IS proven: if you can factor $N$ into its primes, you can compute the $e$-th root $\bmod N$ (and thus break RSA)
• But the reverse is not proven: it's possible (though unlikely) that someone could break RSA without being able to factor $N$
• RSA security could theoretically be weaker than the factoring problem

Why this matters:

• "Provably secure" doesn't mean "unbreakable" — it means "as hard as problem X"
• If someone finds an efficient factoring algorithm, both Rabin and RSA would be broken
• The distinction is about the quality of the security guarantee, not the absolute security level