TITLE: Proving the Riemann Hypothesis and Impact on Cryptography
QUESTION [29 upvotes]: 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?

REPLY [3 votes]: As with all these very difficult conjectures which we are sure to be true, but are unproven, the truth of the statement itself holds some profound insight which we can bring to bear on various matters - however the means of the proof also usually reveals some deep insight into numbers.
So taking Fermat's last theorem as an example, the proof arrived at by Andrew Wiles demonstrated without ambiguity a powerful relationship between modular elliptic curves and number theory.  As a result of the proof we now know that further research in this area has the power to yield very difficult results.
Bringing this back to the last part of your question... the reason we need the proof in order to attack cryptography, is that the means by which it is proven will reveal to cryptographers the means by which they can successfully attack very difficult encryption algorithms.  This will, of course, show us how to create more resilient encryption systems in due course but it will also, in the interim, make current systems less resilient.