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I agree that studying pure mathematics is meaningful by intellectual curiosity itself.

However, after AKS algorithm is found, I have a question "Is still Riemann hypothesis practically important after discovery AKS algorithm?"

I read two non-formal textbooks "Prime Obsession" and "The music of the primes".

Those book are published before discovery of AKS algorithm.

Summarizing importance of proving Riemann hypothesis in those books is "If Riemann hypothesis is true, Miller-Rabin algorithm became deterministic polytime O(bit$^4$) algorithm. And it is very helpful to fortifying RSA by increasing bit.

But I have two doubts:

AKS algorithm is also a poly prime determining algorithm with O(bit$^{12}$). It slower than Miller-Rabin but it is poly-time without any assumption and probably time complexity can be reduced. Miller-Rabin is still strong without Riemann-hypothesis because it can answer "it's a composite" with probability 3/4 at each iteration if it is a composite number. I agree it has crucial vulnerability: Miller-Rabin can be false negative.

RSA is practically strong(safeness unproved but unbroken) and can use digital signature. However, there are combinatorial cryptosystem (and still many things are suggesting) without using number theory such as tool prime number. (But some crypto such as Merkle–Hellman are broken)

So, is proving Riemann-hypothesis still practically important?

Maddy
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1 Answers1

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It is as practically important as it was before, as AKS is impractical for the large numbers required for security.

The AKS algorithm is a "galactic algorithm", and therefore not used in practice...not practical for large numbers. It is an enormous theoretical result, but not "practically important" in terms of encryption. Therefore, if proof of the RH were practically important prior, then it remains practically important today. Prime testing algorithms used in practice now are the same as before, still relying on the RH being true.

That said, mathematical proofs are only practically important for other math problems, not for matters of trust, which is security. Encryption is a matter of trust, that depends not on the possibility of it being broken (proof), but the practicality of it being broken (likelihood).

How one may define how practically important it was prior, however, is a matter of philosophical thought. But factually...its importance remains unchanged.

Jeff
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  • The $\pi$ example is a bit strange, as $\pi$ is defined as that ratio. Perhaps something more like “you don’t have to prove $\pi$ is irrational for engineers to avoid using any particular rational approximation”? – Kevin Carlson Feb 21 '20 at 17:27
  • @Kevin, yes, and think the example is unnecessary anyway. Thanks – Jeff Feb 21 '20 at 19:54