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Date: Mon, 20 Jan 2014 16:18:22 +0400
From: Solar Designer <>
Subject: multiplies (Re: [PHC] escrypt memory access speed)

On Sun, Jan 12, 2014 at 11:58:24AM -0500, Bill Cox wrote:
> On Sun, Jan 12, 2014 at 11:43 AM, Solar Designer <> wrote:
> > And yes, I was already thinking of throwing in some multiplies.  We can
> > easily do several of them one after another while waiting for data to
> > arrive.  If each is 3 cycles latency not only on our CPU, but also on
> > custom ASIC, then we can achieve a delay of roughly the same number of
> > cycles for the attacker that we normally incur because of our memory
> > latency.  The attacker's memory might have lower latency, but with the
> > multiplies the attacker will be forced to slow down and match our speed.
> I like that idea.  Maybe make the number of serial multiplies in the
> hashing loop a parameter?  It would be simple to auto-detect a good
> setting for such a parameter.

Maybe, but for good efficiency this needs to be compile-time, or we'd
need several code versions or JIT.  I'd rather avoid JIT.

> BTW, I'm sure you know this, but for a
> 32x32 multiply -> 32 bit result to be reversible, one of the
> parameters needs to be odd.  I'm only using reversible operations in
> the theory that it helps avoid leaking away entropy.

This isn't exactly reversible even with one of the factors being odd,
but yes, there is this aspect.  I've been considering the following
three options (or a mix of them):

1. Use a suitable constant factor, e.g. like it is done during Mersenne
twister initialization (in fact, I'd even reuse their 32-bit constant,
if I use this approach, use 32-bit multiplies, and need only one constant).
Drawback: this probably makes the corresponding specialized ASIC circuit
roughly twice simpler than a generic two variable inputs multiplier
would be.  Correct?

2. Keep the variable factors as they are, coming from memory or directly
from other computations.  Yes, when a factor happens to be even, it's
like a left shift for the other factor, shifting out some of its most
significant bits.  However, left shifts are present in many crypto
primitives.  As long as there's enough other processing on the same
inputs, with all of their bits in use, this might or might not be fine.

3. Forcibly set the least significant bit, like you chose to do.
Drawbacks: extra operation for defender (but none for ASIC attacker),
doesn't use the original value of the replaced bit.  Is your effective
memory size 31/32 of what's allocated?  Or are you using the same values
without the OR as well?  If you are, then possibly you could use
approach #2 above instead.

> Is that all I
> have to do?  It seems that if I can show that toggling any input bit
> in the inputs to the hash function toggles ~half of the output bits
> regardless of the other values, then when these things are cascaded,
> toggling any input but should toggle ~half of all memory.  Is this the
> right approach for proving the value of a hash function?

Disclaimer: I am not an expert at this.

This is a good and relevant test, yes.  Another test is to see whether
the number of different hash values is as expected for uniform
distribution given a certain number of hashes computed (for different
non-random yet unique inputs - e.g., sequential numbers).  For example,
for 32-bit hash values and 100M hashes computed, you should get around
98844829.5 different hash values.  If you consistently (or on average)
get significantly fewer (across many different 100M samples, for
different sets of inputs), then your hash is significantly worse than
perfect.  If you usually get significantly more, this would suggest that
your hash preserves properties of the input (seeds) distribution, which
is also bad.

And yes, I agree that for our use having a nearly perfect hash function
is not critical.


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