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What does Riemann's hypothesis have to do with encryption?
In the TV show "Numb3rs" , episode Prime Suspect , they describe that Riemann's hypothesis, if solved, would give great insight to internet encryption. They say that R's hypothesis predicts where prime numbers are, and encryption depends on the fact people do not know where they are (in other words, once solved, we can know what and where prime numbers are, rendering encryption's foundation worthless)
Is that theoretically true? If so, does that mean there is no such thing as encryption & protection if Riemann's hypothesis (or whatever encryption depends on) is fully understood?
Or, are there measures against such events? Is asymmetrical encryption designed for this purpose (maintaining the integrity of security even if the foundation was solved)?
Does the show accurately explain these concepts?
"Riemann's hypothesis does not facilitate the factorization of composite numbers (which is required for some types of encryption). "
Is the method and means of factorization the key secret to disabling or decoding encryption?
3 Answers
- Anonymous10 years ago
The above answer is right, although I am not an expert but have read a lot about it. In principle, if RH is solved to be true, then it only confirms that the distributions of zero's of the complex function is indeed as stated by Riemann. These zeros are connected to the distrubutions of the 'real' primes, but finding the zero's and the primes one by one will still mean using a slow algorithm probably not far from the Sieve of Erathosthenes, so in other words, there still won't be a quick way to find prime number 2910 for instance, nor will it be possible to find quickly that the number 21 can be factorized by 3 and 7 (for big numbers I mean).
But, as I asked the same question to a guru in this field, Terence Tao, he said that the transformative power of a proof itself, that could indeed show RH to be true or not, might just do amazing things, so who knows.
- 10 years ago
Riemann's hypothesis does not facilitate the factorization of composite numbers (which is required for some types of encryption). The hypothesis "corrects" the error term for the current estimation of distribution of primes. Further, it is not a matter of "solving" RH which suggests it is some sort of high school algebra problem. Rather, it is proving (or disproving) the conjecture that the RH proposes; namely the location of the zeros of the Zeta function. Besides, if the RH could be used in the manner you are worried about you would not need a "proof" of it. Many constructs or theorems in mathematics are created assuming the RH is true. Some have suggested that the RH is, in fact, undecidable which is an even deeper and more squiggly can of worms.
