Despite the fact that Marsaglia's MWC PRNG (multiply-with-carry random number generator) is considered to be "the mother of all RNGs", it does not seem to be considered to be a CSPRNG (cryptographically secure pseudo-random number generator) even though it passes several statistical randomness tests, including ENT and Die-Hard.
Considering it's possible to define a seed etc., what is the MWC RNG missing that a CSPRNG has? Simpler asked: "What stops the MWC PRNG from being a CSPRNG?"
CSPRNGs have to pass statistical randomness tests, which MWC RNG seems to satisfy. Also, MWC PRNG allows you to choose random initial as long as c < 809430660, which seems big enough to make it hard for people to analyse which of the 809430659 options you chose to seed the MWC PRNG. And last but not least people call it the "mother of all RNGs" — see George Marsaglia's "Mother of all RNGs" (C code with comments) (via archive.org) — which implies it's about the best (P)RNG out there.
Comparing MWC PRNG with cryptographically secure PRNGs, it shows both are pseudo-random, both produce comparable randomness, and both pass the same statistical tests. Some even claim it has cryptographic quality. Yet, I can not find a single indication that it is cryptographically secure.
Therefore I am wondering and asking: "What stops the MWC PRNG from being a CSPRNG?"
Or, asking in other words:
- What is the MWC PRNG missing a CSPRNG should have to be cryptographically secure, and/or
- What does the MWC PRNG have that a CSPRNG should not have to be cryptographically secure?
Currently, I am suspecting I am missing some security definition related to cryptographically secure pseudo-random number generators, or maybe I'm not seeing the obvious security issue which makes the MWC PRNG cryptographically insecure. Therefore, every explanation which could help me understand why the MWC PRNG fails to be classified as a cryptographically secure PRNG is highly appreciated.
In case someone needs some references related to MWC PRNG…
- General Wikipedia Description of MWC
- George Marsaglia's "Mother of all RNGs" (C code with comments) (via archive.org)
Since Berkeley obviously removed all of George Marsaglia's work during a server cleanup, which is why I’m posting a copy of stat.berkeley.edu/classes/s243/mother.c here for reference purposes:
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From: Bob Wheeler <[email protected]>
Newsgroups: sci.stat.consult,sci.math.num-analysis
Subject: Marsaglia's Mother of all RNG's (Long?)
Date: Fri, 28 Oct 94 19:32:08 EDT
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Several people have asked me to post this:
First the C program, then George Marsaliga's post with details
about the RNG. He claims a period of about 2^250 for this and
that it passes all of the usual tests. I've tried it enough
to be sure his claim is reasonable.
The program:
*/
#include <string.h>
static short mother1[10];
static short mother2[10];
static short mStart=1;
#define m16Long 65536L /* 2^16 */
#define m16Mask 0xFFFF /* mask for lower 16 bits */
#define m15Mask 0x7FFF /* mask for lower 15 bits */
#define m31Mask 0x7FFFFFFF /* mask for 31 bits */
#define m32Double 4294967295.0 /* 2^32-1 */
/* Mother **************************************************************
| George Marsaglia's The mother of all random number generators
| producing uniformly distributed pseudo random 32 bit values
with
| period about 2^250.
| The text of Marsaglia's posting is appended at the end of the function.
|
| The arrays mother1 and mother2 store carry values in their
| first element, and random 16 bit numbers in elements 1 to 8.
| These random numbers are moved to elements 2 to 9 and a new
| carry and number are generated and placed in elements 0 and 1.
| The arrays mother1 and mother2 are filled with random 16 bit values
| on first call of Mother by another generator. mStart is the
switch.
|
| Returns:
| A 32 bit random number is obtained by combining the output of the
| two generators and returned in *pSeed. It is also scaled by
| 2^32-1 and returned as a double between 0 and 1
|
| SEED:
| The inital value of *pSeed may be any long value
|
| Bob Wheeler 8/8/94
*/
double Mother(unsigned long *pSeed)
{
unsigned long number,
number1,
number2;
short n,
*p;
unsigned short sNumber;
/* Initialize motheri with 9 random values the first time */
if (mStart) {
sNumber= *pSeed&m16Mask; /* The low 16 bits */
number= *pSeed&m31Mask; /* Only want 31 bits */
p=mother1;
for (n=18;n--;) {
number=30903*sNumber+(number>>16); /* One line
multiply-with-cary */
*p++=sNumber=number&m16Mask;
if (n==9)
p=mother2;
}
/* make cary 15 bits */
mother1[0]&=m15Mask;
mother2[0]&=m15Mask;
mStart=0;
}
/* Move elements 1 to 8 to 2 to 9 */
memmove(mother1+2,mother1+1,8*sizeof(short));
memmove(mother2+2,mother2+1,8*sizeof(short));
/* Put the carry values in numberi */
number1=mother1[0];
number2=mother2[0];
/* Form the linear combinations */
number1+=1941*mother1[2]+1860*mother1[3]+1812*mother1[4]+1776*mother1[5]+
1492*mother1[6]+1215*mother1[7]+1066*mother1[8]+12013*mother1[9];
number2+=1111*mother2[2]+2222*mother2[3]+3333*mother2[4]+4444*mother2[5]+
5555*mother2[6]+6666*mother2[7]+7777*mother2[8]+9272*mother2[9];
/* Save the high bits of numberi as the new carry */
mother1[0]=number1/m16Long;
mother2[0]=number2/m16Long;
/* Put the low bits of numberi into motheri[1] */
mother1[1]=m16Mask&number1;
mother2[1]=m16Mask&number2;
/* Combine the two 16 bit random numbers into one 32 bit */
*pSeed=(((long)mother1[1])<<16)+(long)mother2[1];
/* Return a double value between 0 and 1 */
return ((double)*pSeed)/m32Double;
}
/*
*********************
Marsaglia's comments
Yet another RNG
Random number generators are frequently posted on
the network; my colleagues and I posted ULTRA in
1992 and, from the number of requests for releases
to use it in software packages, it seems to be
widely used.
I have long been interested in RNG's and several
of my early ones are used as system generators or
in statistical packages.
So why another one? And why here?
Because I want to describe a generator, or
rather, a class of generators, so promising
I am inclined to call it
The Mother of All Random Number Generators
and because the generator seems promising enough
to justify shortcutting the many months, even
years, before new developments are widely
known through publication in a journal.
This new class leads to simple, fast programs that
produce sequences with very long periods. They
use multiplication, which experience has shown
does a better job of mixing bits than do +,- or
exclusive-or, and they do it with easily-
implemented arithmetic modulo a power of 2, unlike
arithmetic modulo a prime. The latter, while
satisfactory, is difficult to implement. But the
arithmetic here modulo 2^16 or 2^32 does not suffer
the flaws of ordinary congruential generators for
those moduli: trailing bits too regular. On the
contrary, all bits of the integers produced by
this new method, whether leading or trailing, have
passed extensive tests of randomness.
Here is an idea of how it works, using, say, integers
of six decimal digits from which we return random 3-
digit integers. Start with n=123456, the seed.
Then form a new n=672*456+123=306555 and return 555.
Then form a new n=672*555+306=373266 and return 266.
Then form a new n=672*266+373=179125 and return 125,
and so on. Got it? This is a multiply-with-carry
sequence x(n)=672*x(n-1)+ carry mod b=1000, where
the carry is the number of b's dropped in the
modular reduction. The resulting sequence of 3-
digit x's has period 335,999. Try it.
No big deal, but that's just an example to give
the idea. Now consider the sequence of 16-bit
integers produced by the two C statements:
k=30903*(k&65535)+(k>>16); return(k&65535);
Notice that it is doing just what we did in the
example: multiply the bottom half (by 30903,
carefully chosen), add the top half and return the
new bottom.
That will produce a sequence of 16-bit integers
with period > 2^29, and if we concatenate two
such:
k=30903*(k&65535)+(k>>16);
j=18000*(j&65535)+(j>>16);
return((k<<16)+j);
we get a sequence of more than 2^59 32-bit integers
before cycling.
The following segment in a (properly initialized)
C procedure will generate more than 2^118
32-bit random integers from six random seed values
i,j,k,l,m,n:
k=30903*(k&65535)+(k>>16);
j=18000*(j&65535)+(j>>16);
i=29013*(i&65535)+(i>>16);
l=30345*(l&65535)+(l>>16);
m=30903*(m&65535)+(m>>16);
n=31083*(n&65535)+(n>>16);
return((k+i+m)>>16)+j+l+n);
And it will do it much faster than any of several
widely used generators designed to use 16-bit
integer arithmetic, such as that of Wichman-Hill
that combines congruential sequences for three
15-bit primes (Applied Statistics, v31, p188-190,
1982), period about 2^42.
I call these multiply-with-carry generators. Here
is an extravagant 16-bit example that is easily
implemented in C or Fortran. It does such a
thorough job of mixing the bits of the previous
eight values that it is difficult to imagine a
test of randomness it could not pass:
x[n]=12013x[n-8]+1066x[n-7]+1215x[n-6]+1492x[n-5]+1776x[n-4]
+1812x[n-3]+1860x[n-2]+1941x[n-1]+carry mod 2^16.
The linear combination occupies at most 31 bits of
a 32-bit integer. The bottom 16 is the output, the
top 15 the next carry. It is probably best to
implement with 8 case segments. It takes 8
microseconds on my PC. Of course it just provides
16-bit random integers, but awfully good ones. For
32 bits you would have to combine it with another,
such as
x[n]=9272x[n-8]+7777x[n-7]+6666x[n-6]+5555x[n-5]+4444x[n-4]
+3333x[n-3]+2222x[n-2]+1111x[n-1]+carry mod 2^16.
Concatenating those two gives a sequence of 32-bit
random integers (from 16 random 16-bit seeds),
period about 2^250. It is so awesome it may merit
the Mother of All RNG's title.
The coefficients in those two linear combinations
suggest that it is easy to get long-period
sequences, and that is true. The result is due to
Cemal Kac, who extended the theory we gave for
add-with-carry sequences: Choose a base b and give
r seed values x[1],...,x[r] and an initial 'carry'
c. Then the multiply-with-carry sequence
x[n]=a1*x[n-1]+a2*x[n-2]+...+ar*x[n-r]+carry mod b,
where the new carry is the number of b's dropped
in the modular reduction, will have period the
order of b in the group of residues relatively
prime to m=ar*b^r+...+a1b^1-1. Furthermore, the
x's are, in reverse order, the digits in the
expansion of k/m to the base b, for some 0<k<m.
In practice b=2^16 or b=2^32 allows the new
integer and the new carry to be the bottom and top
half of a 32- or 64-bit linear combination of 16-
or 32-bit integers. And it is easy to find
suitable m's if you have a primality test: just
search through candidate coefficients until you
get an m that is a safeprime---both m and (m-1)/2
are prime. Then the period of the multiply-with-
carry sequence will be the prime (m-1)/2. (It
can't be m-1 because b=2^16 or 2^32 is a square.)
Here is an interesting simple MWC generator with
period> 2^92, for 32-bit arithmetic:
x[n]=1111111464*(x[n-1]+x[n-2]) + carry mod 2^32.
Suppose you have functions, say top() and bot(),
that give the top and bottom halves of a 64-bit
result. Then, with initial 32-bit x, y and carry
c, simple statements such as
y=bot(1111111464*(x+y)+c)
x=y
c=top(y)
will, repeated, give over 2^92 random 32-bit y's.
Not many machines have 64 bit integers yet. But
most assemblers for modern CPU's permit access to
the top and bottom halves of a 64-bit product.
I don't know how to readily access the top half of
a 64-bit product in C. Can anyone suggest how it
might be done? (in integer arithmetic)
George Marsaglia [email protected]
*/
And here's a copy of Marsaglia's original publication, which was first published at sci.stat.math…
good C random number generator
From: George Marsaglia ([email protected])
Subject: Re: good C random number generator
Newsgroups: comp.lang.c
Date: 2003-05-13 08:55:05 PST
Organization: Florida State University
Lines: 89Most RNGs work by keeping a certain number, say k, of the most recently generated integers, then return the next integer as a function of those k. The initial k integers, the seeds, are assumed to be randomly chosen, usually 32-bits. The period of the RNG is related to the number of choices for seeds, usually 2^(32k), so to get longer periods you need to increase k.
Probably the most common type has k=1, and needs a single seed, with each new integer a function of the previous one. An example is this congruential RNG, a form of which was the system RNG in VAXs for many years:
/* a random initial x to be assigned by the calling program */ static unsigned long x=123456789; unsigned long cong(void ) { return (x=69069*x+362437); }Simple, k=1, RNGs can perform fairly well in tests of randomness such as those in the new version of Diehard,
csis.hku.hk/~diehardbut experience has shown that better performances come from RNGs with k's ranging from 4 or 5 to as much as 4097.
Here is an example with k=5, period about 2^160, one of the fastest long period RNGs, returns more than 120 million random 32-bit integers/second (1.8MHz CPU), seems to pass all tests:
/* replace defaults with five random seed values in calling program */ static unsigned long x=123456789,y=362436069,z=521288629,w=88675123,v=886756453; unsigned long xorshift(void) { unsigned long t; t=(x^(x>>7)); x=y; y=z; z=w; w=v; v=(v^(v<<6))^(t^(t<<13)); return (y+y+1)*v; }Another example has k=257, period about 2^8222. Uses a static array Q[256] and an initial carry 'c', the Q array filled with 256 random 32-bit integers in the calling program and an initial carry c<809430660 for the multiply-with-carry operation. It is very fast and seems to pass all tests.
/* choose random initial c<809430660 and 256 random 32-bit integers for Q[] */ static unsigned long Q[256],c=362436; unsigned long MWC256(void) { unsigned long long t,a=809430660LL; static unsigned char i=255; t=a*Q[++i]+c; c=(t>>32); return(Q[i]=t); }The Mersenne Twister (check Google) is an excellent RNG, with k=624. But it requires an elaborate C program and is slower than many RNGs that do as well in tests, have comparable or longer periods and require only a few lines of code.
Here is a complimentary-multiply-with-carry RNG with k=4097 and a near-record period, more than 10^33000 times as long as that of the Twister. (2^131104 vs. 2^19937)
/* choose random initial c<809430660 and 4096 random 32-bit integers for Q[] */ static unsigned long Q[4096],c=362436; unsigned long CMWC4096(void) { unsigned long long t, a=18782LL; static unsigned long i=4095; unsigned long x,r=0xfffffffe; i=(i+1)&4095; t=a*Q[i]+c; c=(t>>32); x=t+c; if(x<c) { x++; c++; } return(Q[i]=r-x); }You will find several more CMWC RNGs and comments on choice of seeds in the May 2003 Communications of the ACM.
George Marsaglia
