On 09/09/13 12:53, Alexander Klimov wrote:
On Sun, 8 Sep 2013, Peter Fairbrother wrote:
You can use any one of trillions of different elliptic curves,which should be
chosen partly at random and partly so they are the right size and so on; but
you can also start with some randomly-chosen numbers then work out a curve
from those numbers. and you can use those random numbers to break the session
key setup.
Can you elaborate on how knowing the seed for curve generation can be
used to break the encryption? (BTW, the seeds for randomly generated
curves are actually published.)
Move along please, there is nothing to see here.
This is just a wild and disturbing story. It may upset you to read it,
so please stop reading now.
You may have read a bit about the story in the papers or internet or
elsewhere, but isn't actually true. Government Agencies do not try to
break the internet's encryption, as used by Banks and Doctors and
Commerce and Government Departments and even Government Agencies
themselves - that wouldn't be sensible.
Besides which, there is no such agency as the NSA.
But ..
Take FIPS P-256 as an example. The only seed which has been published is
s= c49d3608 86e70493 6a6678e1 139d26b7 819f7e90 (the string they hashed
and mashed in the process of deriving c).
I don't think they could reverse the perhaps rather overly-complicated
hashing/mashing process, but they could certainly cherry-pick the s
until they found one which gave a c which they could use.
c not being one of the usual parameters for an elliptic curve, I should
explain that it was then used as c = a^3/b^2 mod p.
However the choice of p, r, a and G was not seeded, and the methods by
which those were chosen are opaque.
I don't really know enough about ECC to say whether a perhaps
cherry-picked c = a^3/b^2 mod p is enough that the resulting curve is
secure against chosen curve attacks - but it does seem to me that there
is a whole lot of legroom between a cherry-picked c and the final curve.
And as I said, it's only a story. We don't know much about what the NSA
knows about chosen curve attacks, although we do know that they are
possible. Don't go believing it, it will just upset you.
They wouldn't do that.
-- Peter Fairbrother
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