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Re: SRFI-27




Hi Sven,

Thanks for pointing out the typos. I will check with the editors if how
the corrections can be published.

It is possible to have a Scheme-only implementation with only 30 bit
integers---the only thing to do is to implement one's own 56-bit arithmetics
on pairs of 30 bit integers. Presumably, this implementation will be
very slow, probably to an extent which is annoying in practice.
Moreover, there is not much from it to be learned for implementors.
Therefore I have chosen not to provide any such implementation
and hope that a Bigloo expert eventually implements the SRFI in
a meaningful way.

Cheers,

Sebastian.

----
Dr. Sebastian Egner
Senior Scientist
Philips Research Laboratories
Prof. Holstlaan 4 (WY2)
5656 AA Eindhoven
The Netherlands
tel:       +31 40 27-43309
fax:      +31 40 27-44918
email: sebastian.egner@xxxxxxxxxxx



Sven Hartrumpf <Sven.Hartrumpf@xxxxxxxxxxxxxxxx>

08/21/02 18:01

       
        To:        Sebastian Egner/EHV/RESEARCH/PHILIPS@EMEA3
        cc:        
        Subject:        SRFI-27

        Classification:        




Hi Sebastian.
Thanks for this nice SRFI!
Unfortunately, I missed the discussion phase. So, here are just some minor
comments.

reference implementation mentions mrg32k3a-b.scm (2 times). Should be mrg32k3a-b.c.

attached: some typos in srfi-27.html

finally, is it possible to get a useful Scheme-only implementation of mrg32k3a with
only 30-bit integers? (bigloo)

Greetings
Sven

-----

<!-- X-URL: http://srfi.schemers.org/srfi-27/srfi-27.html -->
<!-- Date: Tue, 20 Aug 2002 13:53:06 GMT -->
<BASE HREF="">
<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN">
<html>
 <head>
   <title>SRFI 27: Sources of Random Bits</title>
 </head>

 <body>

<H1>Title</H1>

Sources of Random Bits

<H1>Author</H1>

Sebastian Egner

<H1>Status</H1>

This SRFI is currently in ``final'' status. To see an explanation of
each status that a SRFI can hold, see <a
href=""  
You can access
previous messages via
<a href=""> the archive of the mailing list</a>.

<UL>
<LI>Draft: 2002/02/12-2002/04/12</LI>
<LI>Revised: 2002/04/04
<LI>Revised: 2002/04/10
<LI>Revised: 2002/04/10
<LI>Final: 2002/06/03
</UL>

<H1>Abstract</H1>

This document specifies an interface to sources of random bits,
or "random sources" for brevity.
In particular, there are three different ways to use the interface,
with varying demands on the quality of the source and the
amount of control over the production process:

<UL>
<LI>
The "no fuss" interface specifies that
<code>(random-integer </code><I>n</I><code>)</code>
produces the next random integer in {0, ..., <I>n</I>-1} and
<code>(random-real)</code> produces the next random
real number between zero and one.
The details of how these random values are produced may not be
very relevant, as long as they appear to be sufficiently random.
<LI>
For simulation purposes, on the contrary, it is usually necessary
to know that the numbers are produced deterministically by a pseudo
random number generator of high quality and to have explicit access
to its state.
In addition, one might want to use several independent sources of
random numbers at the same time and it can be useful to have some
simple form of randomization.
<LI>
For security applications a serious form of true randomization
is essential, in the sense that it is difficult for an adversary to
exploit or introduce imperfections into the distribution of random bits.
Moreover, the linear complexity of the stream of random bits is more
important than its statistical properties.
In these applications, an entropy source (producing truly random
bits at a low rate) is used to randomize a pseudo random number
generator to increase the rate of available bits.
</UL>
<p>

Once random sources provide the infrastructure to obtain
random bits, these can be used to construct other random deviates.
Most important are floating point numbers of various distributions
and random discrete structures, such as permutations or graphs.
As there is an essentially unlimited number of such objects (with
limited use elsewhere), we do not include them in this SRFI.
In other words, this SRFI is <em>not</em> about making
all sorts of random objects---it is about obtaining random
bits in a portable, flexible, reliable, and efficient way.

<H1>Rationale</H1>

This SRFI defines an interface for sources of random bits
computed by a pseudo random number generator.
The interface provides range-limited integer and real numbers.
It allows accessing the state of the underlying generator.

Moreover, it is possible to obtain a large number of
independent generators and to invoke a mild form of true
randomization.<p>

The design aims at sufficient flexibility to cover the
usage patterns of many applications as diverse as
discrete structures, numerical simulations, and cryptographic protocols.
At the same time, the interface aims at simplicity,
which is important for occasional use.
As there is no "one size fits all" random number generator,
the design necessarily represents some form of compromise
between the needs of the various applications.<p>

Although strictly speaking not part of the specification,
the emphasis of this proposal is on <em>high quality</em>
random numbers and on <em>high performance</em>.
As the state of the art in pseudo random number generators
is still advancing considerably, the choice of method for
the reference implementation should essentially be
considered preliminary.

<H1>Specification</H1>

<DL>

<DT>
<code>(random-integer </code><I>n</I><code>) -> </code><I>x</I><code></code>
</DT>

<DD>
The next integer <I>x</I> in {0, ..., <I>n</I>-1}
obtained from <code>default-random-source</code>.
Subsequent results of this procedure appear to be independent
uniformly distributed over the range {0, ..., <I>n</I>-1}.
The argument <I>n</I> must be a positive integer,
otherwise an error is signalled.
</DD>

</DL>

<DL>

<DT>
<code>(random-real) -> </code><I>x</I><code></code>
</DT>

<DD>
The next number 0 < <I>x</I> < 1 obtained from
<code>default-random-source</code>.
Subsequent results of this procedure appear to be
independent uniformly distributed.
The numerical type of the results and the
quantization of the output range depend on the implementation;
refer to <code>random-source-make-reals</code> for details.
</DD>

</DL>

<DL>

<DT>
<code>default-random-source</code>
</DT>

<DD>
A random source from which <code>random-integer</code> and
<code>random-real</code> have been derived using
<code>random-source-make-integers</code> and
<code>random-source-make-reals</code>.
Note that an assignment to <code>default-random-source</code>
does not change <code>random</code> or <code>random-real</code>;
it is also strongly recommended not to assign a new value.
</DD>

</DL>
<HR ALIGN=left WIDTH="10%">
<DL>

<DT>
<code>(make-random-source) -> </code><I>s</I><code></code>
</DT>

<DD>
Creates a new random source <I>s</I>.
Implementations may accept additional, optional arguments in
order to create different types of random sources.
A random source created with <code>make-random-source</code>
represents a deterministic stream of random bits generated
by some form of pseudo random number generator.
Each random source obtained as <code>(make-random-source)</code>
generates the same stream of values, unless the state is modified
with one of the procedures below.
</DD>

</DL>

<DL>

<DT>
<code>(random-source? </code><I>obj</I><code>)  ->  </code><I>bool</I><code></code>
</DT>

<DD>
Tests if <I>obj</I> is a random source.
Objects of type random source are distinct from all
other types of objects.
</DD>

</DL>

<DL>

<DT>
<code>(random-source-state-ref </code><I>s</I><code>)  ->  </code><I>state</I><code></code><br>
<code>(random-source-state-set! </code><I>s</I><code> </code><I>state</I><code>)</code>
</DT>

<DD>
Get and set the current state of a random source <I>s</I>.  The
structure of the object <I>state</I> depends on the implementation;
the only portable use of it is as argument to
<code>random-source-state-set!</code>.
It is, however, required that a state possess an external
representation.
</DD>

</DL>

<DL>
<DT>
<code>(random-source-randomize! </code><I>s</I><code>)</code>
</DT>

<DD>
Makes an effort to set the state of the random
source <I>s</I> to a truly random state.
The actual quality of this randomization depends on the implementation
but it can at least be assumed that the procedure sets <I>s</I> to a
different state for each subsequent run of the Scheme system.
</DD>
</DL>

<DL>
<DT>
<code>(random-source-pseudo-randomize! </code><I>s</I><code> <I>i</I> <I>j</I>)</code>
</DT>

<DD>
Changes the state of the random source <I>s</I> into the initial
state of the (<I>i</I>, <I>j</I>)-th independent random source,
where <I>i</I> and <I>j</I> are non-negative integers.
This procedure provides a mechanism to obtain a large number of
independent random sources (usually all derived from the same backbone
generator), indexed by two integers.
In contrast to <code>random-source-randomize!</code>,
this procedure is entirely deterministic.
</DD>
</DL>


<HR ALIGN=left WIDTH="10%">

<DL>

<DT>
<code>(random-source-make-integers </code><I>s</I><code>)  ->  </code><I>rand</I><code></code>
</DT>

<DD>
Obtains a procedure <I>rand</I> to generate random integers
using the random source <I>s</I>.
<I>Rand</I> takes a single argument <I>n</I>,
which must be a positive integer, and returns the next uniformly
distributed random integer from the interval {0, ..., <I>n</I>-1}
by advancing the state of the source <I>s</I>.<p>

If an application obtains and uses several generators for the same
random source <I>s</I>, a call to any of these generators advances
the state of <I>s</I>. Hence, the generators <em>do not</em> produce
the same sequence of random integers each but rather share a state.
This also holds for all other types of generators derived from
a fixed random sources.
Implementations that support concurrency make sure that
the state of a generator is properly advanced.<p>
</DD>

</DL>

<DL>

<DT>
<code>(random-source-make-reals </code><I>s</I><code>)  ->  </code><I>rand</I><code></code><br>
<code>(random-source-make-reals </code><I>s</I><code> </code><I>unit</I><code>)  ->  </code><I>rand</I><code></code><br>
</DT>

<DD>
Obtains a procedure <I>rand</I> to generate random real numbers
0 < <I>x</I> < 1 using the random source <I>s</I>.
The procedure <I>rand</I> is called without arguments.<p>

The optional parameter <I>unit</I> determines the type of numbers
being produced by <I>rand</I> and the quantization of the output.
<I>Unit</I> must be a number such that 0 < <I>unit</I> < 1.
The numbers created by <I>rand</I> are of the same numerical
type as <I>unit</I> and the potential output values are
spaced by at most <I>unit</I>. One can imagine <I>rand</I>
to create numbers as <I>x</I>*<I>unit</I> where <I>x</I>
is a random integer in {1, ..., floor(1/<I>unit</I>)-1}.
Note, however, that this need not be the way the values
are actually created and that the actual resolution of
<I>rand</I> can be much higher than <I>unit</I>.
In case <I>unit</I> is absent it defaults to a reasonably
small value (related to the width of the mantissa of an
efficient number format).
</DD>

</DL>


<H1>Design Rationale</H1>

<H3>Why not combine <code>random-integer</code> and
<code>random-real</code>?</H3>

The two procedures are not combined into a single variable-arity
procedures to save a little time and space during execution.
Although some Scheme systems can deal with variable arity as
efficiently as with fixed arity this is not always the case
and time efficiency is very important here.

<H3>Why not some object-oriented interface?</H3>

There are many alternatives to the interface as specified in this SRFI.
In particular, every framework for object-orientation can be used to
define a class for random sources and specify the interface for the
methods on random sources.
However, as the object-oriented frameworks differ considerably
in terms of syntax and functionality, this SRFI does not make
use of any particular framework.

<H3>Why is there not just a generator with a fixed range?</H3>

A bare fixed-range generator is of very limited use.
Nearly every application has to add some functionality

to make use of the random numbers.
The most fundamental task in manipulating
random numbers is to change the range and quantization.
This is exactly what is provided by
<code>random-integer</code> and <code>random-real</code>.
In addition, is saves the user from the pitfall of changing
the range with a simple <code>modulo</code>-computation
which may substantially reduce the quality of the
numbers being produced.<p>

The design of the interface is based on three prototype applications:
<OL>
<LI>
Repeatedly choose from relatively small sets:
As the size of the set is likely to vary from call to call,
<code>random-integer</code> accepts a range argument <I>n</I> in every call.
The implementation should try to avoid boxing/unboxing of values
if the ranges fit into immediate integers.
<LI>
Generate a few large integers with a fixed number of bits:
As generating the random number itself is expensive,
passing the range argument in every call does not hurt performance.
Hence, the same interface as in the first application can be used.
<LI>
Generate real numbers:
Unlike the choose-from-set case,
the range and the quantization is constant over a
potentially very large number of calls.
In addition, there are usually just a few distinct instances of
quantization and number type, most likely corresponding to
underlying <code>float</code> and <code>double</code>
representations.
Therefore,
<code>random-real</code> does not accept any parameters but
the procedure <code>random-source-make-reals</code> creates
a properly configured <code>random-real</code> procedure.
</OL>

<H3>Why bother about floating point numbers at all?</H3>

A proper floating point implementation of a random number generator
is potentially much more efficient that an integer implementation
because it can use more powerful arithmetics hardware.
If in addition the application needs floating point random numbers
it would be an intolerable waste to run an integer generator to
produce floating point random numbers.
A secondary reason is to save the user from the 'not as easy as
it seems' task of converting an integer generator into a real
generator.

<H3>Why are zero and one excluded from <code>random-real</code>?</H3>

The procedure <code>random-real</code> does not return
<I>x</I> = 0 or <I>x</I> = 1 in order to allow
<code>(log </code><I>x</I><code>)</code> and
<code>(log (- 1 </code><I>x</I>)<code>)</code>
without the danger of a numerical exception.

<H1>Implementation</H1>

<H3>Choice of generator</H3>

The most important decision about the implementation is
the choice of the random number generator.
The basic principle here is: <em>Let quality prevail!</em>
In the end, a performance penalty of a better generator may be
a cheap price to pay for some avoided catastrophes.
It may be unexpected, but I have also seen many examples
where the better generator was also the faster.
Simple linear congruential generator cannot be recommended
as they tend to be ill-behaved in several ways.<p>

For this reason, my initial proposal was George Marsaglia's
COMBO generator, which is the combination of a 32-bit
multiplicative lagged Fibonacci-generator with a 16-bit
multiply with carry generator.
The COMBO generator passes all tests of Marsaglia's
<a href=""> testsuite for random number generators and has
a period of order 2^60.<p>

As an improvement, Brad Lucier suggested
<a href=""> Pierre L'Ecuyer's
<a href=""> generator which is combination of two recursive generators
of degree three, both of which fit into 54-bit arithmetics.
The MRG32k3a generator also passes DIEHARD and in addition,
has desirable spectral properties and a period in the
order of 2^191.
As a matter of fact, multiple recursive generators (MRGs) are
theoretically much better understood than special constructions
as the COMBO generator.
This is the reason why the implementations provided here
implements the MRG32k3a generator.
When implemented in floating point arithmetics with sufficient
mantissa-width, this generator is also very fast.<p>

<H3>Choice of arithmetics</H3>

The next important decision about the implementation is
the type of arithmetics to be used.
The choice is difficult and depends heavily on the
underlying Scheme system and even on the underlying
hardware platform and architecture.
For the MRG32k3a generator, use 64-bit arithmetics if you
really have it. If not, use a floating point ALU if
it gives you 54 or more bits of mantissa.
And if you do not have floats either, then at least
try to make sure you work with immediate integers
(instead of allocated objects).
Unfortunately, there is no portable way in Scheme to
find out about native and emulated arithmetics.<p>

As performance is critical to many applications,
one might want to implement the actual
generator itself in native code.
For this reason, I provide three different
implementations of the backbone generator
as a source of inspiration.
See the code below.

<H3>Data Type for Random Sources</H3>

An important aspect of the specification in this SRFI
is that random sources are objects of a distinct type.
Although this is straight-forward and available in nearly
every Scheme implementation, there is no portable way
to do this at present.
One way to define the record type is to use
<a href="">
The reference implementations below define a record
type to contain the exported procedures.
The actual state of the generator is stored in the
binding time environment of <code>make-random-source</code>.
This has the advantage that access to the state is fast
even if the record type would be slow (which need not be
the case).

<H3>Entropy Source for Randomization</H3>

Another problematic part of the specification with respect
to portability is <code>random-source-randomize!</code> as
it needs access to a real entropy source.<p>

A reasonable choice for such as source is to use the system
clock in order to obtain a value for randomization, for example
in the way John David Stone recommends (see reference below).
This is good enough for most applications with the
notable exception of security related programs.
One way to obtain the time in Scheme is to use
<a href="">
<H3>Implementation of the specified interface</H3>

Once the portability issues are resolved,
one can provide the remaining functionality as
specified in this SRFI document.<p>

For the reference implementation, a relatively large part
of the code deals with the more advanced features of the
MRG32k3a generator,
in particular <code>random-source-pseudo-randomize!</code>.
This code is inspired by Pierre L'Ecuyer's own implementation
of the MRG32k3a generator.<p>

Another part of this generic code deals with changing
the range and quantization of the random numbers and
with error checking to detect common mistakes and abuses.<p>

<H3>Implementation Examples</H3>

<a href="" are three alternative implementations of the SRFI.
(<a href="" are all files, tar-gzipped, 13020 bytes.)
Keep in mind that a SRFI is a "request for implementation",
which means these implementations are merely examples
to illustrate the specification and inspire people to implement
it better and faster.
The performance figures below are rough indications measured
on a Pentium3, 800 Mhz, Linux; <i>x</i> ints/s, <i>y</i> reals/s
means <code>(random-integer 2)</code> can be computed about <i>x</i>
times a second and <code>(random-real)</code> about <i>y</i> times a second.
The implementations are

<OL type="a">

<LI> for Scheme 48 0.57, using 54-bit <code>integer</code> only.
This implementation aims at portability, not at performance
(30000 ints/s, 3000/s reals/s).

<LI> for Scheme 48 0.57 with the core generator being implemented
in C using <code>(double)</code>-arithmetics.
The generator is made available in Scheme 48 via the
<a href=""
interface</a>.
The performance of this generator is good
(160000 ints/s, 180000 reals/s).

<LI> for Gambit 3.0, using <code>flonum</code> and
54-bit <code>integer</code>.
This code is inspired by a program by Brad Lucier as
<a href=""> to the discussion archive of this SRFI.

The performance of this generator is good when compiled
(5000 ints/s, 25000/s reals/s when interpreted,
200000 ints/s, 400000/s reals/s when compiled;
see acknowledgements).

</OL>

In addition to the implementations there is a small
collection of <a href="" tests</a>
for the interface specified.
The tests merely check a few assertions expressed by the specification.
It is not the intention to provide a complete test of the interface here.
It is even less the intention to provide statistical tests of the
generator itself.
However, there is a function to write random bits from
the generators to a file in a way readable by the <em>DIEHARD</em>
testsuite. This makes it easier for implementors to find out
about their favorite generators and check their implementation.<p>



<H1>Recommended Usage Patterns</H1>

Unless the functionality defined in this SRFI is sufficient,
an application has to implement more procedures to construct
other random deviates.
This section contains some recommendation
on how to do this technically by presenting
examples of increasing difficulty
with respect to the interface.
Note that the code below is not part of the specification,
it is merely meant to illustrate the spirit

<H3>Generating Random Permutations</H3>

The following code defines procedures to generate random
permutations of the set {0, ..., <I>n</I>-1}.
Such a permutation is represented by a <code>vector</code>
of length <I>n</I> for the images of the points.<p>

Observe that the implementation first defines the procedure
<code>random-source-make-permutations</code> to
turn a random source <I>s</I> into a procedure to generate
permutations of given degree <I>n</I>.
In a second step, this is applied to the default source
to define a ready-to-use procedure for permutations:
<code>(random-permutation </code><I>n</I><code>)</code>
constructs a random permutation of degree <I>n</I>.

<code><pre>
(define (random-source-make-permutations s)
 (let ((rand (random-source-make-integers s)))
   (lambda (n)
     (let ((x (make-vector n 0)))
                (do ((i 0 (+ i 1)))
                    ((= i n))
                  (vector-set! x i i))
                (do ((k n (- k 1)))
                    ((= k 1) x)
                  (let* ((i (- k 1))
                                  (j (rand k))
                                  (xi (vector-ref x i))
                                  (xj (vector-ref x j)))
                    (vector-set! x i xj)
                    (vector-set! x j xi)))))))

(define random-permutation
 (random-source-make-permutations default-random-source))
</pre></code>

For the algorithm refer to Knuth's "The Art of Computer Programming",
Vol. II, 2nd ed., Algorithm P of Section 3.4.2.

<H3>Generating Exponentially-Distributed Random Numbers</H3>

The following code defines procedures to generate exponentially
Exp(mu)-distributed random numbers.
The technical difficulty of the interface addressed here is
how to pass optional arguments to <code>random-source-make-reals</code>.

<code><pre>
(define (random-source-make-exponentials s . unit)
 (let ((rand (apply random-source-make-reals s unit)))
   (lambda (mu)
     (- (* mu (log (rand)))))))

(define random-exponential
 (random-source-make-exponentials default-random-source))
</pre></code>

The algorithm is folklore. Refer to Knuth's "The Art of Computer
Programming", Vol. II, 2nd ed., Section 3.4.1.D.

<H3>Generating Normally-Distributed Random Numbers</H3>

The following code defines procedures to generate
normal N(mu, sigma)-distributed real numbers using
the polar method.<p>

The technical difficulty of the interface addressed here
is that the polar method generates two results per computation.
We return one of the result and store the second one to be
returned by the next call to the procedure.
Note that this implies that <code>random-source-state-set!</code>
(and the other procedures modifying the state) does not necessarily
affect the output of <code>random-normal</code> immediately!

<code><pre>
(define (random-source-make-normals s . unit)
 (let ((rand (apply random-source-make-reals s unit))
                (next #f))
   (lambda (mu sigma)
     (if next
                  (let ((result next))
                    (set! next #f)
                    (+ mu (* sigma result)))
                  (let loop ()
                    (let* ((v1 (- (* 2 (rand)) 1))
                                    (v2 (- (* 2 (rand)) 1))
                                    (s (+ (* v1 v1) (* v2 v2))))
                      (if (>= s 1)
                                   (loop)
                                   (let ((scale (sqrt (/ (* -2 (log s)) s))))
                                     (set! next (* scale v2))
                                     (+ mu (* sigma scale v1))))))))))

(define random-normal
 (random-source-make-normals default-random-source))
</pre></code>

For the algorithm refer to Knuth's "The Art of Computer Programming",
Vol. II, 2nd ed., Algorithm P of Section 3.4.1.C.

<H1>Acknowledgements</H1>

I would like to thank all people who have participated in the discussion,
in particular Brad Lucier and Pierre l'Ecuyer.
Their contributions have greatly improved the design of this SRFI.
Moreover, Brad has optimized the Gambit implementation quite substantially.

<H1>References</H1>

<OL>

<LI>
      G. Marsaglia:
      Diehard -- Testsuite for Random Number Generators.
      <a href="">       (Also contains some generators that do pass Diehard.)
</LI>

<LI>
      D. E. Knuth:
      The Art of Computer Programming;
      Volume II Seminumerical Algorithms.
      2nd ed. Addison-Wesley, 1981.
      (The famous chapter on random number generators.)
</LI>

<LI>
      P. L'Ecuyer:
      "Software for Uniform Random Number Generation:
      Distinguishing the Good and the Bad",
      Proceedings of the 2001 Winter Simulation Conference,
      IEEE Press, Dec. 2001, 95--105.
      <a href="">       (Profound discussion of random number generators.)
</LI>

<LI>
      P. L'Ecuyer:
      "Good Parameter Sets for Combined Multiple Recursive
      Random Number Generators",
      Shorter version in Operations Research, 47, 1 (1999), 159--164.
      <a href="">       (Actual numbers for good generators.)
</LI>

<LI>
      P. L'Ecuyer:
      "Software for Uniform Random Number Generation:
      Distinguishing the Good and the Bad",
      Proceedings of the 2001 Winter Simulation Conference,
      IEEE Press, Dec. 2001, 95--105.
</LI>

<LI>
      MIT Scheme v7.6:
        <code>random flo:random-unit *random-state* make-random-state
        random-state?</code>
      <a href="">       (A mechanism to run a fixed unspecified generator.)
</LI>

<LI>
      A. Jaffer:
      SLIB 2d2 with (require 'random):
        <code>random *random-state* copy-random-state seed->random-state
        make-random-state random:uniform random:exp random:normal-vector!
        random-hollow-sphere! random:solid-sphere!</code>
      <a href="">       (Based on the MIT Scheme mechanism.)
</LI>

<LI>
      R. Kelsey, J. Rees:
      Scheme 48 v0.57 'random.scm':
        <code>make-random make-random-vector</code>
      (Internal procedures of Scheme48; a fixed 28-bit generator.)
</LI>

<LI>
      M. Flatt:
      PLT MzScheme Version 200alpha1:
        <code>random random-seed current-pseudo-random-generator
        make-pseudo-random-generator pseudo-random-generator?</code>
      <a href="">       (A mechanism to run a generator and to exchange the generator.)

</LI>

<LI>
      H. Abelson, G. J. Sussmann, J. Sussman:
      Structure and Interpretation of Computer Programs.
      <a href="">       (The <code>rand</code>-example shows a textbook way to define a
      random number generator.)
</LI>

<LI>
      John David Stone:
      A portable random-number generator.
      <a href="">       (An implementation of a linear congruential generator in Scheme.)
</LI>

<LI>
      Network Working Group:
      RFC1750: Randomness Recommendations for Security.
      <a href="">       (A serious discussion of serious randomness for serious security.)
</LI>

<LI>
      <a href="">       <a href="">       (Resources on random number generators and randomness.)
</LI>

</OL>


<H1>Copyright</H1>
Copyright (C) Sebastian Egner (2002). All Rights Reserved.
<p>
This document and translations of it may be copied and furnished to
others, and derivative works that comment on or otherwise explain it
or assist in its implementation may be prepared, copied, published and
distributed, in whole or in part, without restriction of any kind,
provided that the above copyright notice and this paragraph are
included on all such copies and derivative works. However, this
document itself may not be modified in any way, such as by removing
the copyright notice or references to the Scheme Request For
Implementation process or editors, except as needed for the purpose of
developing SRFIs in which case the procedures for copyrights defined
in the SRFI process must be followed, or as required to translate it
into languages other than English.
<p>
The limited permissions granted above are perpetual and will not be
revoked by the authors or their successors or assigns.
<p>
This document and the information contained herein is provided on an
"AS IS" basis and THE AUTHOR AND THE SRFI EDITORS DISCLAIM ALL
WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY
WARRANTY THAT THE USE OF THE INFORMATION HEREIN WILL NOT INFRINGE ANY
RIGHTS OR ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A
PARTICULAR PURPOSE.


   <hr>
   <address>Editor: <a href="" Sperber</a></address>
   <address>Author: <a href="" Egner</a></address>
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