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From: mjohnson@netcom11.netcom.com (Mark Johnson)

Subject: Re: Maurer's Universal Test
In-Reply-To: stevalln@dorsai.org's message of Mon, 31 Oct 1994 15:04:02 GMT
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Date: Tue, 1 Nov 1994 14:45:45 GMT
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In article  stevalln@dorsai.org (Steve Allen) writes:
>
>   How universal is Ueli Maurer's Universal Statistical Test For
> Random Bit Generators?
>

Regrettably, Maurer's test does not reject the infamous RANDU
algorithm.  This algorithm is singled out for special scorn
and abuse in Knuth's Volume 2 (section 3.3.4, page 104 of the
2nd edition):

" ... its very name RANDU is enough to bring dismay into
the eyes and stomachs of many computer scientists.
... the generator fails most three dimensional criteria
for randomness and it should never have been used."

The RANDU algorithm is a linear congruential, defined thus

X[n+1] = (65539 * X[n]) mod 2^31

Maurer's test doesn't reject RANDU because of a regrettable
deficiency in modern computers: they're just not big enough
and fast enough to run his testing program.  His problem can
be found at the top of page 416 ("A Universal Statistical
Test for Random Bit Generators", proceedings of Crypto 90,
pp. 409-420).

"The test can be implemented by using a table
(denoted below as TAB) of size 2^L that stores for
each L-bit block the time index of its most recent
occurrence."

Unfortunately for Maurer, RANDU has L=31, so he's building
and using an array of 2^31 elements. The third for-loop
of his test algorithm, ("for n:= ...") performs a total
of 10000 * (2^L) iterations.  This takes a long time :-).

Equally unfortunately, other generators (e.g. the
Marsaglia generator posted here less than a week
ago) use L=250 or more.

Maurer's test seems best adapted for inspecting the
output of _physical_ random number generators ... things
that contain nuclear scintillation counters and