I was talking with a friend last night, and she raised the topic of the Clay Millennium Prize problems. I mentioned that my "favorite" problem is the Riemann Hypothesis; I explained what it posits and mentioned that, if proven, it would have great impact on cryptography. Her immediate response was, "Why do we have to wait to prove it? We're pretty sure it's true, so why can't cryptographers assume so until shown otherwise?" As weird as it sounds, I was unable to give a good response.
In "popular mathematics" culture (i.e. not necessarily accurate but well-known), I've heard over and over that "proving the Riemann Hypothesis would cause a lot of problems in Cryptography." (See, for instance, the Numb3rs episode where a mathematician supposedly solves it and puts all of encryption at risk.) However, if a positive answer to the Riemann Hypothesis would result in us being able to break much of modern encryption, why couldn't we just assume that it has a positive answer and use that "maybe-valid" result to break codes now?
My best guess is that the idea that R.H. puts cryptography at risk is actually false; instead, either the process to reach the solution is the risky part, or it just produces a theoretical attack strategy that is still infeasible. That is, it would prove modern encryption insecure to some attack, but that attack technique is not possible at this time (much like creating a collision with SHA-1; theoretically, but not practically, possible).
tl;dr: I've been told that a positive answer to the Riemann Hypothesis is bad for cryptography. Why can't we assume a positive answer now to break codes, but just realize that what we're doing isn't guaranteed correct?