statistical inferences and PRNG characterization

[leichter_jerrold at emc.com]

| Hi,
| 
| I've been wondering about the proper application of statistics with
| regard to comparing PRNGs and encrypted text to truly random sources.
| 
| As I understand it, when looking at output, one can take a
| hypothetical source model (e.g. "P(0) = 0.3, P(1) = 0.7, all bits
| independent") and come up with a probability that the source may have
| generated that output.  One cannot, however, say what probability such
| a source had generated the output, because there is an infinite number
| of sources (e.g. "P(0) = 0.29999.., P(1) = 7.000...").  Can one say
| that, if the source must be A or B, what probability it actually was A
| (and if so, how)?
That's not the way it's done.  Ignore for a moment that we have a sequence
(which is probably irrelevant for this purpose, but might not be).  Instead,
just imagine we have a large collection of values generated by the PRNG -
or, looked at another way, a large collection of values alleged to have been
drawn from a population with P(0) = 0.3 and P(1) = 0.7.  Now take a truely
random sample from that collection and ask the question:  What is the
probability that I would have seen this result, given that the collection
I'm drawing from is really taken from the alleged distribution?  You don't
need any information about *other* possible distributions.  (Not that there
aren't other questions you can ask.  Thus, if the collection could have
been drawn from either of two possible distributions, you can ask which
is more probable to have resulted in the random sample you saw.)

The randomness in the sampling is essential.  When you have it, you wipe out
any underlying bias in the way the collection was created.

| Also, it strikes me that it may not be possible to prove something
| cannot be distinguished from random, but that proofs must be of the
| opposite form, i.e. that some source is distinguishable from random.
Actually, what one tends to prove are things like:  If X is uniformally
randomly distributed over (0,1), then 2X is uniformally randomly distributed
over (0,2).  (On the other hand, X + X, while still random, is *not*
uniformally distributed.)  That's about as close as you are going to get
to a "proof of randomness".

| Am I correct?  Are there any other subtleties in the application of
| statistics to crypto that anyone wishes to describe?  I have yet to
| find a good book on statistics in these kinds of situations, or for
| that matter in any.
Statistics in general require subtle reasoning.

| As an aside, it's amusing to see the abuse of statistics and
| probability in the media.  For example, when people ask "what's the
| probability of <some non-repeating event or condition>?"
That may or may not be a meaningful concept.  If I toss a coin, and
depending
on the result, blow up a building - there is no way to repeat the blowing up
of the building, but still it's meaningful to say that the probability that
the building gets blown up is 50%.

							-- Jerry


| -- 
| "Curiousity killed the cat, but for a while I was a suspect" -- Steven
Wright
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