Weak DH primes and openssh

[mancha]

On Wed, May 27, 2015 at 02:06:12PM +0200, Aris Adamantiadis wrote:
> If the primality test is such a problem, one could implement a variant
> to the AKS polynomial time primality test:
> https://en.wikipedia.org/wiki/AKS_primality_test
> https://math.dartmouth.edu/~carlp/aks041411.pdf
> 
> This test (and variants) are not statistical. I have no idea if they
> work in reasonable time on 1024-4096bits prime candidates.
> 
> Aris
> 
> Le 27/05/15 04:55, mancha a écrit :
> > 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

Though PRIMES IN P has been heavily optimized since it's first version,
it's still O(log^7 n) or so. Compare that to Fast-ECPP which is around
O(log^4 n). Also, for our purposes we're going up to 8192 bit numbers
not just 4096.

Unfortunately, neither PRIMES IN P nor tests such as the near poly-time
APR generate proofs. So, there's no quick way to verify someone else's
result.  

--mancha
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