Weak DH primes and openssh

[Aris Adamantiadis]

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
>
>
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