# Zero knowledge proofs that one has broken a crypto system

Suppose someone has figured out how to factor integers in polynomial time, thus breaking the RSA system. Also suppose this person does not want to reveal the secret of how to factor integers in polynomial time, only to prove he or she has broken it (or at least give evidence that he has broken it). One way to do this would be to present factors of the RSA numbers https://en.m.wikipedia.org/wiki/RSA_numbers

My question is are there other ways of giving zero knowledge proofs that one has broken a cryptosystem?

Sure! For example, you could prove in zero-knowledge that you know the prime factors of $$n$$, where $$n$$ is an online RSA challenge (i.e. something we know for sure you did not create yourself).
Such a zero-knowledge proof would look as follow: you commit to the two prime factors $$p, q$$. You prove that the commitments contain primes, and you prove that their product is $$n$$.
This is actually a relatively common zero-knowledge proof in cryptography, but in a different context: it is used to prove, when you generate an RSA key $$n$$, that you generated it correctly and that it has the right structure. The seminal paper describing a zero-knowledge proof for this task is this one.