On Aug 10, 2009, at 4:42 AM, Alexander Klimov wrote:
When the first results about exponential speedup of factoring came
out, people assumed that this was true in general. But it isn't. In
particular, simple search, where you have only an equality test so you
can't build a hash table or some kind of ordered structure - is O(N)
on a traditional computer - and O(sqrt(N)) on a quantum computer. I'm
not sure what the current knowledge about what a quantum machine can
do for NP computations, but there's no "probably" here.
On Sun, 9 Aug 2009, Jerry Leichter wrote:
Since people do keep bringing up Moore's Law in an attempt to justify
larger keys our systems "stronger than cryptography," it's worth
keeping in mind that we are approaching fairly deep physical limits.
I wrote about this on this list quite a while back. If current
physical theories are even approximately correct, there are limits to
how many "bit flips" (which would encompass all possible binary
operations) can occur in a fixed volume of space-time.
A problem with this reasoning is that the physical world and the usual
digital computers have exponential simulation gap (it is known at
least in one direction: to simulate N entangled particles on a digital
computer one needs computations exponential in N). This can be
considered as a reason to suspect that physical world is
non-polynomially faster than the usual computers (probably even to an
extent that all NP computations can be simulated).
The physical arguments to which I was referring say *nothing* about
how the computation is done. It can be a QM computation as well.
While it is possible to use physical world to simulate usual computers
in the straightforward way (namely by using voltage levels of a
circuit to represent separate bits), it is not clear that doing
computations in this way is the best way to do computations, for
example, if the meaning of our computations are simulation of the
physical world, then it can be better to use direct
physical-to-physical mapping instead of physical-to-usual followed by
usual-to-physical: analog computers, such as wind tunnels, are still
I am not aware of any plausible argument why a brute force search in
general (a quintessence of NP class, by the way) or a key search
against any particular algorithm cannot be implemented in a direct way
significantly faster than in the indirect way, that is NP-to-physical
instead of NP-to-usual followed by usual-to-physical. All the fuss
about quantum computing is exactly because people believe that a
different mapping (not thru usual computers) can be more efficient (if
I understand correctly, right now neither the class of algorithms that
can be sped up this way is understood, nor the quantum computers of
practical capacity exist).
In any case, the simple search result above applies directly to brute
force: For that problem, you only get a polynomial speedup anyway.
That's a ... bizarre point of view. :-) Should freedom from related-
key attacks be part of the definition of a "secure" encryption
algorithm? We should decide that on some rational basis, not on
whether, with care, we can avoid such attacks. Clearly, a system that
*is* secure against such attacks is more robust. Do we know how to
build such a thing? What's the cost of doing so? But to say it's an
*advantage* to have a weakness seems like some kind of odd moral
argument: If you're hurt by this it's because you *deserve* to be.
All the protocols and standards out there calling for AES-256 - it's
obviously "better" than AES-128 because after all 256 is *twice as
large* as 128! - were just a bunch of nonsense. And, perhaps,
I see the situation in the positive way: the recent AES attacks
stress the fact that the key management should be done
correctly, in particular, keys should be derived thru KDF (not
a simple xor) and must be authenticated. With this attack in
hand it is much easier for us now to say why one should not use
K to encrypt messages of one type and K+1 for another type, or
why it is a bad idea to encrypt a key in CTR mode and store the
result without a MAC. I doubt it is possible to find any
professionally designed protocol or standard that becomes weak
due to the recent discovery.
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