On Wed, May 27, 2015 at 02:55:39AM +0000, mancha wrote: > On Tue, May 26, 2015 at 03:10:01PM -0400, Daniel Kahn Gillmor wrote: > > Does anyone have a pointer to any decent free software for > > generating and verifying primality proofs? > > > > --dkg > > OpenSSH generates strong pseudoprimes to k random bases (that if prime > would be "safe primes"). I think Darren uses k=100 (confirmation?) so > the way the math works out, each number he generates has at most a > 1/(4^100) probability of being composite. > > In comparison, it's estimated the odds of getting crushed to death by > a vending machine are 1 in 112 million. > > Death-by-chocolate-bar is about > 14,347,661,109,455,270,317,338,947,253,046,094,665,376,812,444,489,221 > more likely to happen to you than having a given "Darren-prime" turn > out to be composite. > > The take-away, of course, is to always snack responsibly. > > Nonetheless, the most we can currently say is the numbers are almost > surely prime. Certainty would be nice. > > ECPP is the fastest prime proving algorithm that also provides > certificates (I think) and PRIMO is the king-of-the-hill > implementation (others exist but are significantly slower with bigger > numbers). > > Though as you mentioned PRIMO's not libre (just free), the math in the > certificates is open-source and that's really the critical part here. > > The proofs themselves are sequences of steps that at each step (except > the last) prove a number N is prime if a smaller number R is prime. > Every subsequent step takes its N from the previous step's R and this > continues until we arrive at an R < 34*10^13 because such an R can be > proven prime if it is a strong pseudoprime to the base of the 7 primes > smaller than 19 (i.e. 2, 3, 5, 7, 11, 13, 17) > > TL;DR > > With PRIMO's help, I've proved the first 200 strong pseudoprimes in > the latest v1.12 moduli file are indeed prime. Darren, so far you're > batting a thousand! > > I've uploaded my proofs to factordb.com for independent verification > and complementary confirmation proofs. > > For example, check out the 4th prime in Darren's new moduli file: > > https://tinyurl.com/pg66aq5 > > As I write this I've checked through #209. Those with spare CPU cycles > on 64-bit Linux can help with the 65 strong pseudoprimes remaining > (#210 to #274). > > Those who wish to help should get PRIMO [1] and grab the full set of > input files I made: > > http://sf.net/projects/mancha/files/misc/openssh-moduli-20150522_primo.tar.bz2 > > The 7680-bit moduli (#225 - #248) and 8192-bit moduli (#249 - #274) > are up for grabs. Probably best if you claim your range to avoid > effort duplication. > > --mancha > > PS In addition to factordb.com, ecpp-dj [2] can verify PRIMO certs. > ecpp-dj also proves primality but it is slower than PRIMO and more > importantly its cert format can't be independently verified. > > ------ > > [1] http://www.ellipsa.eu/public/primo/files/primo-411-lx64.7z > [2] http://sti15.com/nt/ecpp-dj-1.04.tar.gz I was not as clear in my above postscript as I should have been. What I meant to say is there aren't independent software implementations that can verify ecpp-dj generated proofs. That said, the certificate format is documented so one can be manually check ecpp-dj certs by verifying requisite conditions are satisfied for the math theorems used by its tests. Time permitting, I plan to dig deeper into this promising program. --mancha
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