What conditions does a provable secure cryptosystem satisfy?
Provably secure doesn't actually mean secure. Plenty of provably secure cryptosystems were broken and not because the proof had a flaw. When something is provably secure, it just meant some statement regarding it's security has been proven. There are any number of such statements.

A proof always starts off with some assumptions, such as the hardness of some underlying problem, or security property of some building block. A statement about something else can then be build on top of those assumptions. But when that statement is formalized, it is never as broad as: "$X$ is secure".  Instead the statement would be a more specific statement such as: "If you can invert $F$ you can also invert $G$, but $G$ is believed to be hard to invert. In this protocol under such and such assumption an attacker learns nothing more then he knew initially."



WPA is provably secure. They went all out on this proof. It was not just proven that the specification was secure in that no key material could leak. The implementation was proven to match the specification and the compiler was proven to work correctly.

And yet WPA was horribly broken. The proof didn't say anything about an attacker being able to coerce an all zero key. The attacker learned nothing, the security proof was not violated.



Provably secure does not imply Secure. It implies only whichever claims were actually proven.