Re: [TLS] DH generator 2 problem?

[Dan Brown <[e-mail]>]

Dear Michael,

Your concern that special primes in Diffie--Hellman might be weaker has some
truth: the famous and ingenious special number field sieve (SNFS) works
against primes with special structure.  See 
https://eprint.iacr.org/2016/961 
for a recent application (which, as an extra wrinkle, is also able to hide
the special structure). 

Therefore, applied cryptography like TLS already uses vetted primes in
Diffie--Hellman, with careful guidance from cryptographers such as those in
CFRG, which specialize in such fundamentals.  (Full disclosure: I proposed
some primes for DH to the CFRG list (for reductionist security), but never
followed through with an I-D.)

Scott explains below very well, as one'd expect, why the generator choice
should generally not matter (random self-reducibility).

Best regards,

Dan

> -----Original Message-----
> From: TLS <[e-mail]> On Behalf Of Scott Fluhrer (sfluhrer)
> Sent: Thursday, October 8, 2020 3:09 PM
> To: Michael D'Errico <[e-mail]>; TLS List <[e-mail]>
> Subject: Re: [TLS] DH generator 2 problem?
> 
> > -----Original Message-----
> > From: TLS <[e-mail]> On Behalf Of Michael D'Errico
> > Sent: Thursday, October 08, 2020 1:54 PM
> > To: TLS List <[e-mail]>
> > Subject: [TLS] DH generator 2 problem?
> >
> > Using finite-field Diffie-Hellman with a generator of 2 is probably
> > not the best choice.  Unfortunately all of the published primes (RFCs
> > 2409, 3526, and
> > 7919) use 2 for the generator.  Any other generator would likely be
> > (not sure how much?) more secure.
> 
> No, that is known to be not true.
> 
> In particular, if you can compute discrete logs to the base 2, you can
compute
> discrete logs to any base (except in the cases where 2 generates an
> anomalously small subgroup, which is not the case in the above groups).
> 
> Here's how it works; suppose you were given the problem of solving the
> discrete log problem g^x = h, for some g, h.  Then, if you can solve
discrete logs
> to base 2, you would solve these two problems:
> 
> 2^y = g
> 2^z = h
> 
> Once you have solved those two problems, then you have x = y z^-1 mod p-1.
> 
> It's a little more complex if g, h is not in the subgroup that 2
generates, but not
> that much more (unless, as above, the size of that subgroup is far smaller
than
> p-1).
> 
> >
> > The problem is that 2^X consists of a single bit of value 1 followed
> > by a huge string of zeros.  When you then reduce this modulo a large
> > prime number, there will be a pattern in the bits which may help an
> > attacker discern the value of X.  This is further helped by the fact
> > that all of the published primes have 64 bits of 1 in the topmost and
bottom-
> most bits.
> > In addition, the larger published primes are very similar to the
> > shorter ones, the shorter ones closely matching truncated versions of
the
> larger primes.
> >
> > If you were to manually perform the modulo-P operation yourself, you
> > would add enough zeros to the end of P until the topmost bit is just
> > to the right of the 1 bit from 2^X, and then you'd subtract.  This bit
> > pattern will always be the same, no matter the value of X.  In
> > particular, the top 64 bits disappear since they're all one.
> > Continuing the mod-P operation, you adjust the number of zeros after
> > the prime P and then subtract again, reducing the size of the operand.
> > The pattern of bits again will be the same, regardless of the value of
X, the
> only difference being the number of trailing zeros.
> 
> Actually, for these group, the value of 2^x mod p can take on (p-1)/2
different
> values; there is no chance that the bit pattern will be trapped in some
cul-de-
> sac, as you appear to be suggesting...
> 
> >
> > I have not looked at the cyclic patterns which happen as you do this,
> > but I wouldn't be surprised to find that the "new" primes based on e
> > (RFC 7919) have easier-to-spot bit patterns than those based on pi.
> 
> I would be surprised; do you have some reason that would suggest why bits
> derived from the binary expansion of 'e' would be somehow qualitatively
> different from bits derived from the binary expansion of 'pi'?
> 
> >
> > This is speculation of course.
> 
> Might I suggest you learn a bit of number theory to go along with your
> speculation?
> 
> >
> > Should we define some new DH parameters which use a different
> > generator?  Maybe the primes are fine....
> 
> If the prime is fine, so is the generator...

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