geo
Posts:
38
Registered:
6/19/08


RNGs: A Super KISS
Posted:
Nov 3, 2009 10:46 AM


/* For those mesmerized (or Mersenneized?) by a RNG with period 2^199371, I offer one here with period 54767*2^1337279over 10^396564 times as long. It is one of my CMWC (ComplimentaryMultiplyWithCarry) RNGs, and is suggested here as one of the components of a superlongperiod KISS (KeepItSimpleStupid) RNG.
With b=2^32 and a=7010176, and given a 32bit x, and a 32bit c, this generator produces a new x,c by forming 64bit t=a*x+c then replacing: c=top 32 bits of t and x=(b1)(bottom 32 bits of t). In C: c=t>>32; x=~t;
For many years, CPUs have had machine instructions to form such a 64bit 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 fastsome 140 million per second on my desktop PC.
Used alone, this generator passes all the Diehard Battery of Tests, but its simplicity makes it wellsuited to serve as one of the three components of a KISS RNG, based on the KeepItSimpleStupid principle, and the idea, supported by both theory and practice, that the combination of RNGs based on different mathematical models can be no worseand is usually betterthan 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 superlongperiod 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 715 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 (64bit)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 baseb 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 baseb 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 fullyseeded 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 */

