Would a proof to the Riemann Hypothesis affect security?

Question

If a solution was found to the Riemann Hypothesis, would it have any effect on the security of things such as RSA protection? Would it make cracking large numbers easier?

Answer 1

If by "solution" you mean confirmation or counterexample, then no. Clearly, one could just in good faith assume the result, and go on to produce some algorithmic whose validity requires the Riemann hypothesis, and use it to break RSA codes. Simply knowing the Riemann hypothesis is true doesn't help you construct any factorization method (although it does tell you a theoretical bound on how good certain algorithms can run).

There is one other possibility however. The idea of a solution to the Riemann Hypothesis really gets a mathematicians tongue salivating, not necessarily because it's so important itself that all the non-trivial zeros lie on the critical strip, but because chances are that a proof of RH requires genuinely new, creative techniques/ways of thinking. Hopefully we can expand on the proof and use these new techniques for other problems. For example, the methods introduced by Andrew Wiles in proving Fermat's Last theorem have been extended to make some (limited) progress into special cases of the Birch-Swinnerton Dyer conjecture. It could well be the case that the proof of the Riemann Hypothesis introduces extremely strong tools and new methods which give insight into the distribution of the primes, in even more detail than RH yields, which could potentially increase the efficiency of factorization algorithms to a point where a supercomputer could realistically crack an RSA code.

Answer 2

No. Practical applications can simply assume the truth of the Riemann hypothesis; proving it would increase knowledge but not itself affect security.

Answer 3

No, I don't believe so. The Riemann hypothesis controls (in some statistical sense) the distribution of primes, and one can prove stronger results about the running time of various number-theoretic algorithms if one knows that RH (or some its generalizations) are true.

However, in practice (e.g. if you are an intelligence agency trying to crack encrypted data) I think you can assume that all running time estimates whose truth hinges on RH are in fact true, since there is every reason to believe that RH is true. Thus these algorithms will in fact (with near certainty) behave as RH predicts that they would.

Answer 4

If RH is shown to be false, it would mean that the distribution of primes has more structure than anticipated (or a different type of structure, such as unusual correlations where independence was expected) and conceivably that could be leveraged for cryptanalysis of RSA or other number theoretic codes.

If RH is shown to be true, it is likely to be as a result of proving that some symmetries or algebraic structures connected to prime numbers, structures that are presently conjectural and not precisely understood, do exist. There are such structures that exist for function fields and have applications to cryptography (Frobenius map, Weil pairings, and etale cohomology come to mind) and it's possible that the expected but currently undiscovered constructions in the number field case would also be useful for making or breaking codes.

Of course any big leap in understanding of number theory would lead to a reconsideration of number theoretic algorithms in general, especially the ones in cryptography.