True hardware random number generators
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The statistical conspicuities can have drastically
consequences. Quote from a textbook:
"If all scientific paper whose results are in doubt because of
bad rand()s were to disappear from library shelves, there would
be a gap on each shelf about as big as your fist."
Numerical Recipes (in C), Chapter 7.1
Another problem is the estimation of the degree of
non-computability or true randomness / really randomness, also
called unpredictability or degree of nondeterminateness.
Because a hardware random number generator is a stochastic
automat with a finite memory, it's true randomness can be
calculated by the theory of markov
chains.
The true randomness, more precisely the average randomness of a
random number of a stochastic automat, is known as the entropy
cover in this theory. In contrast of the entropy, the entropy
cover incorporates conditional probabilities and it's value is
invariant against basis transformations.
In the theory the memory of length m means that a random number
depends on the m previous produced random numbers.
The number of interior states is n^m (n to the power of m) with
n=number of possible random numbers, e. g. (2^(16))^(1000) =
3,02E4816 with a 16-bit random number generator and a memory
length of m=1000. Because of this huge number of states and a
greater number of transitions the entropy cover can generally not
be calculated. Another problem is that before this calculation
the parameters, the ergodic probabilities of the states and
transitions probabilities, have to be estimated. Additionally the
independence from outside influences, at all allowed parameter
values, has to be validated. Because of Bell's
theorem it is not necessary to care about hidden
variables.
The due to the huge number of interior states and conditional
probabilities practically impossible effort is the only reason
why hardware random number generators have to be tested; in
theory they can be calculated completely.
It is also the reason why true random numbers often are
after-treated deterministic. A complete deterministic
after-treatment like exclusive oring (so called XOR mode) with
pseudorandom numbers increases the entropy in most cases but
never the entropy cover. For cryptographically application such
after-treatments are senseless and should be avoided, because
they can only mask the weaknesses of the hardware but they can
not cancel the weaknesses. Through this, such after-treatments do
fool the user and cozenage a randomness which is not really
there.
Because the entropy cover can generally not be estimated, in
practice only tests like an entropy test or the diehard tests and
estimating of simplifying details like an entropy be done.
In special cases an entropy and the entropy cover can be
calculated. An example are pseudorandom number generators because
they do have N interior states and all transition probabilities
are 0 or 1. Therefore the term p*ld(p) in the definition formula
of the entropy cover is always zero and also the whole entropy
cover.
As can bee seen at the simple definition formula, the entropy is
also zero when, and only when, the random numbers where so
designed (the number width so choosen) that they meet an integer
multiple of the length M>0 of the period length P of the
generator. For this the period P has to be finite, because
otherwise the entropy would always be zero, even at
nondeterministic random number generators. Pseudorandom number
generators (PRNGs) with an infinite period, for example an
algorithmically random sequence like the
decimal places of a
normal number, e. g. the
Champernowne constant,
the entropies are (close to) one because every newly calculated
decimal place is a surprise (because it was unknown before)
while the entropy cover is zero because the decimal places
are constant and do exist, indendently from a calculation.
In practice often there is no distinction between
pseudorandomness and true randomness (nondeterminism) and named
entropy what is accurately named entropy cover. With this there
is no distinction between appearances and reality of
randomness.
Generally the physical states do not meet the markov-states,
but the physical states can be approximated by markov-states.
With a m growing to infinity this approximation is arbitrary
accurate. That's why every random number generator is a markov
process and can be described by the theory of markov
chains.
The nondeterminism is the criteria for information theoretical
safety of the random numbers and is important e. g. for
preventing that someone can calculate or guess with a high
probability transaction numbers (TANs), even if that someone
knows all previous TANs (e. g. from the recovered paper
container). For this another quote:
"There is a DEFINITE NEED FOR DATA WITH ENTROPY NOT BASED ON
ALGORITHMS. As to whether we need *true* random numbers, that's a
subject for debate. Typically we get data that's as good as we
need it to be from a hardware generator by running the initial
data (which may have some kind of pattern or bias) through an
algorithm which sort of mixes it. But it is very useful to have
this entropy come from a hardware device, because it affords an
EXTRA LEVEL OF SECURITY; pseudorandom numbers can be compromised
if the algorithm is known and the seed is discovered, but THERE
IS NO SUCH EQUIVALENT FOR RANDOM NUMBERS GENERATED BY A HARDWARE
DEVICE."
Moses Liskov (faq-editor@rsa.com), http://www.rsa.com/ , year
2000
For solving these two problems, producing random numbers without
statistical conspicuities and with high entropy and entropy cover,
the below-mentioned hardware random number generators have been
developed.
There are virtual for free hardware random number generators like
/dev/random (the standard hardware random number generator on
Linux and other Unix derivatives) which also do produce high
quality random numbers but only with a speed of approximately 2
Bytes/s, which is million times slower than the below-mentioned
generators.
As tests the FIPS, AIS, NIST, diehard and special tests, e. g.
with a hardware spectrum analyser, were used. Because the
diehard tests
are most often stronger and more versatile than the other tests,
they were used most often.
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The main advantage is that while production of a new random
number takes only one clock cycle, these random numbers do pass
all tests, i. e. the 15 diehard tests,
without after-treatment and that this can be shown both
experimental and in theory. Furthermore no shielding is necessary
and the slowest generators here are at minimum eight million
times faster than /dev/random, the standard hardware random
number generator on Linux and other Unix derivatives.
This mathematical proven technique is patented (german patent
number 199 26 640)
and makes these random number generators unique.
For showing that the random numbers do need no after-treatment
the software is completely open source. Because no
after-treatment is necessary, these generators can be used
without software. That's the reason why some of them (rw2) do
have one mono and two stereo outputs with the random bits, which
can be used e. g. as perfect white noise for audio measurements
or for switching a Laser for crosscorrelation measurements.
It could be shown that these generators do pass all statistical
tests even in climate chambers at -10...+60°C, as certified e. g.
from Kryptografics.
These generators are optimised for a) autonomous operation, b)
hard real time and c) high quality at high speed. Therefore they
can also be used for
noise
radar, stealth radar, as interfering
transmitter and as random clock generator (with discrete
exponential distributed cycle duration).
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Links
An associate article from uni ulm intern, November 1999, here.
An article about my first generation of random number generators here.
If the random number generator
/dev/random
is not fast enough for you and if the generators above are too fast (or expensive, or ...), you can use the
free ALSA sound card related entropy gathering daemon (abbrev. EGD) randomsound to speed up /dev/random by a factor of about 100:
http://packages.debian.org/en/sid/admin/randomsound
http://v2.www.rpmseek.com/rpm-pl/randomsound.html?hl=de&cx=857:R:0.
For randomsound the onboard sound or a small USB sound card for
about 4 Euro is sufficient.
A similar alternative is the EGD
RandD.
A further alternative are the rng-tools (
http://www.kernel.org/doc/Documentation/hw_random.txt).
c't-RandCam & Co - for generating and testing random numbers.
Dieharder: A (new) Random Number Test Suite.
TestU01 - Empirical Testing of Random Number Generators.
NIST test suite
FIPS 140 standard
AIS 20 and AIS 31 for evaluation of deterministic (AIS 20) and nondeterministic (AIS 31) random number generators:
L'Ecuyer's test suite (also a number of pseudo rngs)
A method for recycling physical random numbers, from Art Owen
Public Domain random bit files, Freeware (e. g. a program for creation of true random and exact equally distributed lottery numbers) and further information here.
Information Theory with Entropy, Entropy Cover and more: Book
"Grundlagen der Information", Akademie Verlag, Berlin,
1991.
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