On Tue, 28 Oct 2008, John Denker wrote:

| Date: Tue, 28 Oct 2008 12:09:04 -0700
| From: John Denker <[EMAIL PROTECTED]>
| To: "Leichter, Jerry" <[EMAIL PROTECTED]>,
|     Cryptography <cryptography@metzdowd.com>
| Cc: IanG <[EMAIL PROTECTED]>
| Subject: Re: combining entropy
| 
| On 10/28/2008 09:43 AM, Leichter, Jerry wrote:
| 
| > | We start with a group comprising N members (machines or 
| > | persons).  Each of them, on demand, puts out a 160 bit 
| > | word, called a "member" word.  We wish to combine these 
| > | to form a single word, the "group" word, also 160 bits 
| > | in length.
| > This isn't enough.  Somehow, you have to state that the values emitted
| > on demand in any given round i (where a round consists of exactly one
| > demand on all N member and produces a single output result) cannot
| > receive any input from any other members.  Otherwise, if N=2 and member
| > 0 produces true random values that member 1 can see before it responds
| > to the demand it received, then member 1 can cause the final result to
| > be anything it likes.
| 
| 
| Perhaps an example will make it clear where I am coming
| from.  Suppose I start with a deck of cards that has been 
| randomly shuffled.  It can provide log2(52!) bits of 
| entropy.  That's a little more than 225 bits.  Now suppose 
| I have ten decks of cards all arranged alike.  You could 
| set this up by shuffling one of them and then stacking 
| the others to match ... or by some more symmetric process. 
| In any case the result is symmetric w.r.t interchange of 
| decks.  In this situation, I can choose any one of the 
| decks and obtain 225 bits of entropy.  The funny thing 
| is that if I choose N of the decks, I still get only 225 
| bits of entropy, not N*225....
| The original question spoke of "trusted" sources of
| entropy, and I answered accordingly.  To the extent
| that the sources are correlated, they were never eligible 
| to be considered trusted sources of entropy.  To say 
| the same thing the other way around, to the extent
| that each source can be trusted to provide a certain
| amount of entropy, it must be to that extent independent 
| of the others.
Rest of example omitted.  I'm not sure of the point.  Yes, there are
plenty of ways for correlation to sneak in.

As far as I can see, only the second piece I quoted is relevant, and it
essentially gets to the point:  The original problem isn't well posed.
It makes no sense *both* to say the sources and trusted *and* to say
that they may not deliver the expected entropy.  If I know the entropy of
all the sources, that inherently includes some notion of trust - call
it source trust:  I can trust them to have at least that much entropy.
I have to have that trust, because there is no way to measure the
(cryptographic) entropy.  (And don't say I can analyze how the source
is constructed, because then I'm left with the need to trust that what
I analyzed is actually still physically there - maybe an attacker has
replaced it!)

Given such sources it's easy to *state* what it would mean for them to
be independent:  Just that if I consider the source produced by
concatenating all the individual sources, its entropy is the sum of the
entropies of the constituents.  Of course, that's an entropy I can again
measure - at least in the limit - in the information theoretical sense,
but not in the cryptographic sense; another aspect of trust - call it
independence trust - has to enter here.

All that's fine, but how then are we supposed to construe a question
about what happens if some of the sources fail to deliver their rated
entropy?  That means that source trust must be discarded.  (Worse, as
the original problem is posed, I must discard source trust for *some
unknown subset of the sources*.)  But given that, why should I assume
that independence trust remains?

Sure, I can make additional assumptions.  If I'm concerned only about,
say, physical failures of sources implemented as well-isolated modules,
it might well be a reasonable thing to do.  In fact, this is essentially
the independent- failure model we use all the time in building reliable
physical systems.  Of course, as we know well, that model is completely
untenable when the concern is hostile attack, not random failure.  What
do you replace it with?

Consider the analogy with reliable distributed systems.  People have
basically only dealt with two models:

        1.  The fail-stop model.  A failed module stops interacting.
        2.  The Byzantine model.  Failed modules can do anything
                including cooperating by exchanging arbitrary
                information and doing infinite computation.

The Byzantine model is bizarre sounding, but it's just a way of expressing
a worst-case situation:  Maybe the failed modules act randomly but just by
bad luck they do the worst possible thing.

We're trying to define something different here.  Twenty-odd years ago,
Mike Fischer at Yale proposed some ideas in this direction (where
modules have access to true random numbers and the only thing a failed
module *cannot* do is determine what random values another module drew
unless that module chooses to make them available), but I don't recall
much that came out of this work (not that I specifically tried to keep
track).  That seems related to the underlying problem here.  (If we want
to ignore intelligent attacks, we get something close to the fail-stop
model, where a failed module can deliver anything that depends only on
its internal state and its private random number source.)

| > If the issue is how to make sure you get out at least all the randomness
| > that was there,
| 
| I'm going to ignore the "At least".  It is very hard to 
| get out more than you put in.
Yes.  I'm not sure what the sentence was originally supposed to mean,
but what I ended up saying didn't make a whole load of sense....

                                                        -- Jerry

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