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Topic: RNGs: A Super KISS
Replies: 14   Last Post: Feb 26, 2013 10:53 AM

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geo

Posts: 38
Registered: 6/19/08
RNGs: A Super KISS
Posted: Nov 3, 2009 10:46 AM
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/*
For those mesmerized (or Mersenne-ized?) by a RNG
with period 2^19937-1, I offer one here with period
54767*2^1337279---over 10^396564 times as long.
It is one of my CMWC (Complimentary-Multiply-With-Carry) RNGs,
and is suggested here as one of the components of a
super-long-period KISS (Keep-It-Simple-Stupid) RNG.

With b=2^32 and a=7010176, and given a 32-bit x, and a 32-bit c, this
generator produces a new x,c by forming 64-bit t=a*x+c then replacing:
c=top 32 bits of t and x=(b-1)-(bottom 32 bits of t). In C: c=t>>32;
x=~t;

For many years, CPUs have had machine instructions to form such a
64-bit t and extract the top and bottom halves, but unfortunately
only recent Fortran versions have means to easily invoke them.

Ability to do those extractions leads to implementations that are
simple
and extremely fast---some 140 million per second on my desktop PC.

Used alone, this generator passes all the Diehard Battery of Tests,
but
its simplicity makes it well-suited to serve as one of the three
components
of a KISS RNG, based on the Keep-It-Simple-Stupid principle, and the
idea,
supported by both theory and practice, that the combination of RNGs
based on
different mathematical models can be no worse---and is usually
better---than
any of the components.

So here is a complete C version of what might be called a SUPER KISS
RNG,
combining, by addition mod 2^32, a Congruential RNG, a Xorshift RNG
and the super-long-period CMWC RNG:
*/

#include <stdio.h>
static unsigned long Q
[41790],indx=41790,carry=362436,xcng=1236789,xs=521288629;

#define CNG ( xcng=69609*xcng+123 ) /*Congruential*/
#define XS ( xs^=xs<<13, xs^=(unsigned)xs>>17, xs^=xs>>5 ) /
*Xorshift*/
#define SUPR ( indx<41790 ? Q[indx++] : refill() )
#define KISS SUPR+CNG+XS

int refill( )
{ int i; unsigned long long t;
for(i=0;i<41790;i++) { t=7010176LL*Q[i]+carry; carry=(t>>32); Q[i]=~
(t);}
indx=1; return (Q[0]);
}

int main()
{unsigned long i,x;
for(i=0;i<41790;i++) Q[i]=CNG+XS;
for(i=0;i<1000000000;i++) x=KISS;
printf(" x=%d.\nDoes x=-872412446?\n",x);
}

/*
Running this program should produce 10^9 KISSes in some 7-15 seconds.
You are invited to cut, paste, compile and run for yourself, checking
to
see if the last value is as designated, (formatted as a signed integer
for
potential comparisons with systems using signed integers).
You may want to report or comment on implementations for other
languages.

The arithmetic operations are suited for either signed or unsigned
integers.
Thus, with (64-bit)t=a*x+c, x=t%b in C or x=mod(t,b) in Fortran, and
c=c/b in either C or Fortran, but with ways to avoid integer
divisions,
and subsequent replacement of x by its base-b complement, ~x in C.

With b=2^32 and p=54767*2^1337287+1, the SUPR part of this Super KISS
uses my CMWC method to produce, in reverse order, the base-b expansion
of k/p for some k determined by the values used to seed the Q array.
The period is the order of b for that prime p:
54767*2^1337279, about 2^1337294 or 10^402566.
(It took a continuous run of 24+ days on an earlier PC to
establish that order. My thanks to the wizards behind PFGW
and to Phil Carmody for some suggested code.)

Even the Q's all zero, should seeding be overlooked in main(),
will still produce a sequence of the required period, but will
put the user in a strange and exceedingly rare place in the entire
sequence. Users should choose a reasonable number of the 1337280
random bits that a fully-seeded Q array requires.

Using your own choices of merely 87 seed bits, 32 each for xcng,xs
and 23 for carry<7010176, then initializing the Q array with
for(i=0;i<41790;i++) Q[i]=CNG+XS;
should serve well for many applications, but others, such as in
Law or Gaming, where a minimum number of possible outcomes may be
required, might need more of the 1337280 seed bits for the Q array.

As might applications in cryptography: With an unknown but fully-
seeded Q array, a particular string of, say, 41000 successive SUPR
values will appear at more than 2^20000 locations in the full
sequence,
making it virtually impossible to get the location of that particular
string in the full loop, and thus predict coming or earlier values,
even if able to undo the CNG+XS operations.
*/

/*
So I again invite you to cut, paste, compile and run the above C
program.
1000 million KISSes should be generated, and the specified result
appear,
by the time you count slowly to fifteen.
(Without an optimizing compiler, you may have to count more slowly.)
*/

/* George Marsaglia */



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