This is version 3.0 of the PRNG-manual,
last updated 12 March 2001
Table of Contents
*****************
PRNG - Pseudo-Random Number Generator
Features
1 Installing PRNG
Documentation
Profiling and Verification
2 Usage of PRNG
2.1 Interface Description
2.2 PRNG Functions
2.3 Examples
3 The Theory behind PRNG
3.1 General Remarks
3.2 Generator Definitions and Parameters
3.2.1 EICG (explicit inversive congruential generator)
3.2.2 ICG (inversive congruential generator)
3.2.3 LCG (linear congruential generator)
3.2.4 QCG (quadratic congruential generator)
3.2.5 MT19937 (Mersenne Twister)
3.2.6 MEICG (modified explicit inversive congruential generator)
3.2.7 DICG (digital inversive congruential generator)
3.2.8 EXTERNAL (Interface to fixed-parameter generators)
3.2.9 COMPOUND
3.2.10 SUB
3.2.11 ANTI
3.2.12 CON
3.2.13 AFILE (ASCII file)
3.2.14 BFILE (Binary file)
3.3 Recommended Reading
Appendix A Tables of Parameters
A.1 Parameters for LCG (linear congruential generators)
A.2 Parameters for ICG (inversive congruential generator)
PRNG - Pseudo-Random Number Generator
*************************************
PRNG is a collection of algorithms for generating pseudorandom numbers
as a library of C functions, released under the GPL
(http://www.gnu.org/copyleft/gpl.html). It has been written by Otmar
Lendl (<lendl@cosy.sbg.ac.at>) and is now maintained by Josef Leydold
(<leydold@statistik.wu-wien.ac.at>).
The current version of this package can always be found on the ARVAG
(Automatic Random VAriate Generation) project group
(http://statistik.wu-wien.ac.at/arvag/) in Vienna, or the pLab server
(http://random.mat.sbg.ac.at/) in Salzburg.
In the case of any troubles, bug reports or need of assistance please
contact the maintainer via <prng@statistik.wu-wien.ac.at>. Please let
us also know about your experiencies with the library.
Features
========
* Portability. This library should compile on any computer with an
ANSI C compiler. A verification program is included.
* General Implementations. This library does not implement certain
fixed generators like RANDU or `rand', but implements the general
PRNG algorithms to which all parameters can be supplied by the
user.
* Consistent and object-oriented interface. This interface
simplifies the PRNG handling inside the main application.
* Possibility of independent copies of the same generator.
* Extensibility. New generators are easily integrated into the
framework of this library.
* Fully supported Pseudorandom number generating methods: (free
parametrization)
- LCG (linear congruential generator)
- ICG (inversive congruential generator)
- EICG (explicit inversive congruential generator)
- mEICG (modified explicit inversive congruential generator)
- DICG (digital inversive congruential generator)
- QCG (quadratic congruential generator)
Fixed parameter PRNG (external generators):
- MT19937 (Mersenne Twister by M. Matsumoto)
- TT800 (a large TSFR by M. Matsumoto)
- CTG (Combined Tausworthe Generator by P. L'Ecuyer)
- MRG (Multiple Recursive Generator by P. L'Ecuyer)
- CMRG (Combined (Multiple Recursive Generator by P.
L'Ecuyer)
plus the following methods (meta-generators):
- C (Compound generator)
- SUB (Subsequences)
- ANTI (antithetic random variables)
- CON (Consecutive blocks)
- AFILE (Ascii file)
- BFILE (Binary file)
1 Installing PRNG
*****************
While the code is plain ANSI C and thus quite portable, the following
adaptions might be neccessary for compile this library.
All configurations are done in the file `src/prng.h'. Each option is
extensively commented there. Here is a quick rundown on what to expect
there:
* Definition of the basic numeric data-type `prng_num'. It is not
recommended to change this. For 32 and 64 bit computers all
neccessary auxiliary definitions will be made automatically. For
other architectures, please edit `prng.h' according to the
comments.
* Various constants. See comments on the exact meanings.
* Definition of `prng_inverse'. In previous versions, there was no
algorithm which was fastest on all architectures, thus is was
necessary configure the library for the each platform. Now
`prng_inverse_own', which combines the speedups of all old
algorithms is the fastest one on all tested architectures and thus
_no_ configuration is necessary any more.
The code is optimized for GNU CC (gcc). If your compiler supports the
type (`long long int'), too, you can use this feature by defining
`HAVE_LONGLONG' in `prng.h'.
Then do:
./configure --prefix=<prefix_path>
make
This should compile the library (`libprng.a') and example programs.
To install the library (see also GNU generic installation instructions
in file `INSTALL') type:
make install
which installs `<prefix_path>/lib/libprng.a',
`<prefix_path>/include/prng.h', and ` <prefix_path>/info/prng.info'.
If `--prefix' is omitted, then `/usr/local' is used as default.
It is possible to remove these files by
make uninstall
I could not test this code in many environments, thus it might be
necessary to tweak the code to compile it. Please mail me any changes
you made, so that I can include them in the next official release.
Documentation
=============
A manual can be found in directory `doc' in various formats, including
PS, PDF, HTML, Info and plain text.
Profiling and Verification
==========================
Do
make check
to make and run the following executables:
* `iter'
This program counts the number of iterations in the `euclid_table'
algorithm. It's NOT kept up to date. Use at own risk.
* `validate'
Using the supplied file tests.dat, this program tests the
generator library for correct operation. On 32-bit computers it
will fail on generators requiring 64-bit arithmetic.
2 Usage of PRNG
***************
2.1 Interface Description
=========================
The interface has changed dramatically in version 2.0. As more and more
generator types were added to this package, a new generic interface was
needed. While still plain Ansi C, the architecture is now
object-oriented.
All generators are identified by a textual description. This description
is either of the form `"type(parameter1,parameter2, ...)"' or is a
shortcut name for a common PRNG as defined in `src/prng_def.h'.
Calling `prng_new' with such a description as the only argument will
allocate a new generator object, initialize it, and return its handle
(`struct prng *').
All further calls need this handle as the first argument. They are best
explained by example:
#include <prng.h> /* make sure that the compiler can find this file. */
main()
{
struct prng *g;
prng_num seed, n, M;
double next, *array;
int count;
g = prng_new("eicg(2147483647,111,1,0)");
if (g == NULL) /* always check whether prng_new has been successful */
{
fprintf(stderr,"Initialisation of generator failed.\n");
exit (-1);
}
printf("Short name: %s\n",prng_short_name(g));
/* definition as in call to prng_new */
printf("Expanded name: %s\n",prng_long_name(g));
/* Shortcuts expanded */
next = prng_get_next(g); /* get next number 0 <= next < 1 */
prng_get_array(g,array,count); /* fill array with count numbers */
prng_reset(g); /* reset the generator */
prng_free(g); /* deallocate the generator object */
}
These functions work with all generators. For certain generators, the
following functions are available, too:
if (prng_is_congruential(g))
{
n = prng_get_next_int(g); /* return next *unscaled* number */
M = prng_get_modulus(g); /* return the modulus of the prng */
}
if (prng_can_seed(g))
prng_seed(g,seed); /* reseed the generator */
if (prng_can_fast_sub(g))
puts(prng_get_sub_def(g,20,0)); /* Get subsequence definition */
if (prng_can_fast_con(g))
puts(prng_get_con_def(g,20,1)); /* Get block definition */
*NOTE:*
`prng_new' performs only a rudimentary check on the parameters. The
user is responsible for enforcing all restrictions on the parameters,
such as checking that the modulus of an [E]ICG is prime, or that LCG
and ICG are maximum period generators.
Most of these functions are implemented as macros, so be careful with
autoincrements (`++') in parameters.
2.2 PRNG Functions
==================
- Library Function: struct prng prng_new (char *STR)
Create a new generator object. If initialisation of the generator
object fails then `NULL' is returned. Thus the pointer returned by
this routine *must* be checked against `NULL' *before* using it.
Otherwise the program aborts with a segmentation fault.
- Library Function: void prng_reset(struct prng *G)
Reset random number generator.
- Library Function: double prng_get_next (struct prng *G)
Sample from generator (get next pseudo-random number from stream).
- Library Function: void prng_get_array (struct prng *G, double
*ARRAY, int COUNT)
Sample array of length COUNT.
- Library Function: prng_num prng_get_next_int (struct prng *G)
Sample integer random number from generator.
- Library Function: void prng_free (struct prng *G)
Destroy generator object.
- Library Function: char* prng_short_name (struct prng *G)
Get name of generator as in call to `prng_new'.
- Library Function: char* prng_long_name (struct prng *G)
Get name of generator with shortcuts expanded.
- Library Function: int prng_is_congruential (struct prng *G)
`TRUE' if G is a congruential generator.
- Library Function: prng_num prng_get_modulus (struct prng *G)
Return modulus of generator.
- Library Function: int prng_can_seed (struct prng *G)
`TRUE' if generator G can be reseeded.
- Library Function: void prng_seed (struct prng *G, prng_num NEXT)
Reseed generator.
- Library Function: int prng_can_fast_sub (struct prng *G)
`TRUE' if subsequences of the random stream can computed directly.
- Library Function: char* prng_get_sub_def (struct prng *G, int S, int
I)
Get definition for the generator of the subsequence stream of G
with starting point I and stepwidth S. It returns a character
string that can be used a argument for `prng_new'. For generators
where `prng_can_fast_sub' is `TRUE'. (see also *Note SUB::).
- Library Function: int prng_can_fast_con (struct prng *G)
`TRUE' if blocks of the random stream can computed directly.
- Library Function: int prng_get_con_def (struct prng *G, int L, int I)
Get definition for the generator of the blocked stream of G with
position I and block length L. It returns a character string that
can be used a argument for `prng_new'. For generators where
`prng_can_fast_con' is `TRUE'. (see also *Note CON::).
2.3 Examples
============
`examples/pairs.c' is an example how to generate overlapping pairs of
PRN using this package.
`examples/tuples.c' is a more general version of pairs.
3 The Theory behind PRNG
************************
This chapter lists the implemented generators plus a few recommendations
on the parameters.
3.1 General Remarks
===================
* On a b-bit computer, the size of the modulus is limited by 2^(b-1),
that is 2147483648 on a 32 bit machine or 9223372036854775808 on a
64 bit architecture. As of version 1.3 the library will reject
larger moduli.
* The library relies on controlled overflow. If you feel
uncomfortable with that, restrict your choice of moduli to numbers
< 2^(b-2)
and disable the check for power of two moduli in `mult_mod_setup'
(`support.c'). Run the supplied validate program if you have
doubts about this.
* The library does *NOT* test if the parameters are valid for the
chosen generator. The user is responsible for ensuring that the
modulus of an inversive generator is a prime, or that the choice
of parameters will lead to an optimal period length.
*IT IS THUS NOT A GOOD IDEA TO JUST USE ARBITRARY NUMBERS.*
This chapter contains recommended values for all implemented
generator types.
* Do not base your simulation on a single generator. Even if you
picked a good one you should verify the results using a completely
different generator. There is no generator whose output does not
exhibit an intrinsic structure, so it is in theory possible that
this structure correlates to the simulation problem and thus leads
to a skewed result. Do not use just other parameters for the
verification but use a different generator type.
* Small ( < 32767 ) factors will be faster than larger ones.
3.2 Generator Definitions and Parameters
========================================
TeX notation is used.
Most generators operate in the group (or field) Z_p and generate a
sequence y_n, n >= 0 of numbers in Z_p. p is called modulus. In order
to generate U([0,1[) distributed numbers, the y_n are scaled: x_n =
y_n / p.
_Notice:_ If p is prime, one can define the inversion inv() so that
inv(a)*a mod p = 1 (a != 0)
inv(0) = 0
Generator types
...............
3.2.1 EICG (explicit inversive congruential generator)
------------------------------------------------------
* Definition:
y_n = inv(a*(n_0 + n) + b) (mod p) n >= 0
* Name (as given to `prng_new'): `"eicg(p,a,b,n_0)"'
* Properties:
- Period length = p.
- Strong non-linear properties. (e.g. no lattice)
- Parameter selection not sensitive.
- `prng_is_congruential' is `TRUE'
- `prng_can_seed' is `TRUE'.
The parameter of `prng_seed' will be used as "n" in the next
call to `get_next'.
- `prng_can_fast_sub' and `prng_can_fast_con' are `TRUE'.
* Parameter selection: Besides a != 0, no restrictions or even
suggestions are known.
* Introduced in: Eichenauer-Hermann, J. "Statistical independence of
a new class of inversive congruential pseudorandom numbers", Math.
Comp. 60:375-384, 1993
3.2.2 ICG (inversive congruential generator)
--------------------------------------------
* Definition:
y_n = a * inv(y_{n-1}) + b (mod p) n > 0
* Name (as given to `prng_new'): `"icg(p,a,b,y_0)"'
* Properties:
- Period length = p. (for suitable parameters)
- Strong non-linear properties. (e.g. no lattice)
- Parameter selection not sensitive.
- `prng_is_congruential' is `TRUE'.
- `prng_can_seed' is `TRUE'.
The parameter of `prng_seed' will be used as `y_{n-1}' in the
next call to `get_next'.
- `prng_can_fast_sub' and `prng_can_fast_con' are `FALSE'.
* Parameter selection: To ensure that the period length is p, a and
b must be chosen in a way that x^2 - bx -a (\in F_p[x]) is a
primitive polynomial over F_p.
If ICG(p,a,1) has period length p, then ICG(p,a*c^2,c) will have
period length p, too. For recommended parameters *note
Table_ICG::.
* Introduced in: Eichenauer, J. and J. Lehn. "A non-linear
congruential pseudo random number generator", Stat. Papers
27:315-326, 1986
3.2.3 LCG (linear congruential generator)
-----------------------------------------
* Definition:
y_n = a * y_{n-1} + b (mod p) n > 0
* Name (as given to `prng_new'): `"lcg(p,a,b,y_0)"'.
* Properties:
- Period lengths up to p are possible.
- Strong linear properties.
- The quality of the PRN depends very strongly on the choice of
the parameters.
- `prng_is_congruential' is `TRUE'.
- `prng_can_seed' is `TRUE'. The parameter of `prng_seed' will
be used as `y_{n-1}' in the next call to `get_next'.
- `prng_can_fast_sub' and `prng_can_fast_con' are `TRUE'.
Requesting these subsequence may be slow if large skips are
involved and b is not 0.
* Parameter selection: If p is a power of 2, then a mod 4 = 1 and b
odd will guarantee period length = p.
If p is prime and b = 0 then any prime-root modulo p as a will
guarantee period length p-1. (y_0 != 0 )
For recommended parameters *note Table_LCG::.
See also the file `src/prng_def.h' for a list of frequently used
LCGs.
_Hint:_ A rule of thumb suggests not to use more than sqrt(p)
random numbers from an LCG.
References:
Fishman, G.S. "Multiplicative congruential random number
generators ..." Math. Comp. 54:331-344 (1990);
L'Ecuyer, P., "Efficient and portable combined random number
generators" Comm. ACM 31:742-749, 774 (1988)
L'Ecuyer, P., Blouin, F. and Couture R. "A search for good
multiple recursive random number generators" ACM Trans. Modelling
and Computer Simulation 3:87-98 (1993)
* Introduced by D. H. Lehmer in 1948.
The LCG is _the_ classical method. I refer to: Knuth, D. E. "The
Art of Computer Programming, Vol. 2 Seminumerical Algorithms",
Addison-Wesley, second edition, 1981
3.2.4 QCG (quadratic congruential generator)
--------------------------------------------
* Definition:
y_n = a * y_{n-1}^2 + b *y_{n-1} + c (mod p) n > 0
* Name (as given to `prng_new'): `"qcg(p,a,b,c,y0)"'.
* Properties:
- Period lengths up to p are possible.
- Weaker linear properties (tuples fall into union of lattices)
- Reasonable distribution in dimension 2, but not that good in
dimension 3.
- `prng_is_congruential' is `TRUE'.
- `prng_can_seed' is `TRUE'.
The parameter of `prng_seed' will be used as `y_{n-1}' in the
next call to `get_next'.
- `prng_can_fast_sub' and `prng_can_fast_con' are `FALSE'.
* Parameter selection:
If p is a power of 2, then a even, b == a + 1 mod 4, and c odd
will guarantee period length = p.
No table of good parameters has been published.
* Introduced in:
Knuth, D. E. "The Art of Computer Programming, Vol. 2
Seminumerical Algorithms", Addison-Wesley, second edition, 1981
3.2.5 MT19937 (Mersenne Twister)
--------------------------------
* Name (as given to `prng_new'): `"mt19937(seed)"'.
* Properties:
- Period lengths is 2^19937-1.
- `prng_is_congruential' is `TRUE'.
- `prng_can_seed' is `TRUE'.
The parameter of `prng_seed' will be used to seed the array
of coefficients.
- `prng_can_fast_sub' and `prng_can_fast_con' are `FALSE'.
* Introduced in:
Matsumoto, M. and Nishimura, T., "Mersenne Twister: A
623-Dimensionally Equidistributed Uniform Pseudo-Random Number
Generator", ACM Transactions on Modeling and Computer Simulation,
Vol. 8, No. 1, January 1998, pp 3-30.
3.2.6 MEICG (modified explicit inversive congruential generator)
----------------------------------------------------------------
* Definition:
y_n = n * inv(a*(n_0 + n) + b) (mod p) n >= 0
* Name (as given to `prng_new'): `"meicg(p,a,b,n_0)"'.
* Properties:
- Period length = p.
- `prng_is_congruential' is `TRUE'.
- `prng_can_seed' is `TRUE'. The parameter of `prng_seed' will
be used as "n" in the next call to `get_next'.
- `prng_can_fast_sub' and `prng_can_fast_con' are `FALSE'.
Experimental generator: *USE AT OWN RISK*
* Parameter selection:
- For prime moduli, a != 0 suffices.
- It's possible to use a powe of 2 as modulus, which requires a
= 2 (mod 4) and b = 1 (mod 2).
* Introduced in:
Eichenauer-Hermann, J. "Modified explicit inverive congruential
pseudorandom numbers with power of 2 modulus" Statistics and
Computing 6:31-36 (1996)
3.2.7 DICG (digital inversive congruential generator)
-----------------------------------------------------
* Definition:
y_n = a * inv(y_{n-1}) + b (mod p) n > 0
*All operations are in the field F_{2^k} !!*
* Name (as given to `prng_new'): `"dicg(k,a,b,y_0)"'.
* Properties:
- Period length = 2^k.
- Strong non-linear properties.
- Parameters seem not to be sensitive
- `prng_is_congruential' is `TRUE'.
- `prng_can_seed' is `TRUE'. The parameter of `prng_seed' will
be used as `y_{n-1}' in the next call to `get_next'
* `prng_can_fast_sub' and `prng_can_fast_con' are `FALSE'.
* Parameter selection:
Tricky.
* Introduced in:
Eichenauer-Herrmann and Niederreiter, "Digital inversive
pseudorandom numbers", ACM Transactions on Modeling and Computer
Simulation, 4:339-349 (1994)
3.2.8 EXTERNAL (Interface to fixed-parameter generators)
--------------------------------------------------------
These generators are included to provide a uniform interface to a wider
range of PRNG. The only enhancements from the published code is the
support for multiple streams of these generators, as the original code
used global variables.
See the file `src/external.c' for the references. Included are
- TT800 (a large TSFR by M. Matsumoto)
- CTG (Combined Tausworthe Generator by P. L'Ecuyer)
- MRG (Multiple Recursive Generator by P. L'Ecuyer)
- CMRG (Combined (Multiple Recursive Generator by P. L'Ecuyer)
3.2.9 COMPOUND
--------------
* Definition:
This is a "meta"-generator which combines a number of PRNG into
one single generator by adding the respective numbers modulo 1.
* Name (as given to `prng_new'): `"c(generator1,generator2, ...)"'.
Up to `PRNG_MAX_COMPOUNDS' generators are permitted. GENERATORX
may be any valid generator definition, including a compound
generator.
* Properties:
- Period length: Least common multiple of the period length's
of the component generators.
- Generally speaking, most properties of PRNG are preserved if
combining generators of the same type.
- `prng_is_congruential' is `FALSE'.
- `prng_can_seed' is `TRUE'.
- The parameters of `prng_seed' is used to seed all seedable
component generators.
- `prng_can_fast_sub' and `prng_can_fast_con' depend on the
underlying generators.
3.2.10 SUB
----------
* Definition: This is a "meta"-generator which takes a subsequence
out of another generator.
* Name (as given to `prng_new'): `"sub(gen,s,i)"'.
The output of "gen" is spliced into s streams, and the i-th is
used. (0 <= i < s)
* Properties:
- Period length: Typically the period of "gen".
- Generally speaking, most properties of PRNG are preserved
when taking subsequence.
- `prng_is_congruential' and `prng_can_seed' depend on "gen".
- `prng_can_fast_sub' and `prng_can_fast_con' are `FALSE'.
3.2.11 ANTI
-----------
* Definition:
This is a "meta"-generator wich returns 1-U instead of U as random
number.
* Name (as given to `prng_new'): `"anti(gen)"'.
The output of gen (U) is changed to 1-U.
3.2.12 CON
----------
* Definition:
This is a "meta"-generator which takes a block of numbers out of
the output of another generator.
* Name (as given to `prng_new'): `"con(gen,l,i)"'.
The output of "gen" is divided into blocks of length l, and the
i-th is used. (0 <= i < l)
* Properties:
- Period length: The period of "gen".
- `prng_is_congruential' and `prng_can_seed' depend on "gen".
- `prng_can_fast_sub ' and `prng_can_fast_con' are `FALSE'.
3.2.13 AFILE (ASCII file)
-------------------------
* Definition :
This generator takes its numbers from the named file. It expects
each number in plain ascii (atof must be able to parse it) and on
its own line. If EOF is reached, a warning is printed to stderr
and reading continues at the beginning of the file.
* Name (as given to `prng_new'): `"afile(some_file_name)"'.
* Properties:
- `prng_is_congruential' is `FALSE'.
- `prng_can_seed' is `FALSE'.
- `prng_can_fast_sub' and `prng_can_fast_con' are `FALSE'.
3.2.14 BFILE (Binary file)
--------------------------
* Definition:
This generator takes its numbers from the named file. In order to
get good numbers, the file should contain random bytes. If EOF is
reached, a warning is printed to stderr and reading continues at
the beginning of the file.
* Name (as given to `prng_new'): `"bfile(some_file_name)"'.
*WARNING*: The conversion between bytes and numbers in [0,1) is
NOT guaranteed to yield the same results on different computers.
* Properties:
- `prng_is_congruential' is `FALSE'.
- `prng_can_seed' is `FALSE'.
- `prng_can_fast_sub' and `prng_can_fast_con' are `FALSE'.
3.3 Recommended Reading
=======================
Niederreiter, H. "New developments in uniform pseudorandom number and
vector generation" in "Monte Carlo and Quasi-Monte Carlo Methods in
Scientific Computing", Lecture Notes in Statistics, Springer.
Hellekalek, P. "Inversive pseudorandom number generators: Concepts,
Results and Links"
Eichenauer-Herrmann, J. "Pseudorandom Number Generation by Nonlinear
Methods" Int. Statistical Review 63:247-255 (1995)
L'Ecuyer, P. "Uniform random number generation" Ann. Oper. Res.
53:77-120 (1994)
Wegenkittl, S. "Empirical testing of pseudorandom number generators"
Master's thesis, Universitaet Salzburg, 1995
Appendix A Tables of Parameters
*******************************
This chapter lists the implemented generators plus a few recommendations
on the parameters.
A.1 Parameters for LCG (linear congruential generators)
=======================================================
y_n = a * y_{n-1} + b (mod p) n > 0
_Hint:_ A rule of thumb suggests not to use more than sqrt(p) random
numbers from an LCG.
Notice that moduli larger than 2^32 require a computer with
`sizeof(long)>32'.
Generators recommended by Park and Miller (1988), "Random number
generators: good ones are hard to find", Comm. ACM 31, pp. 1192-1201
(Minimal standard).
modul p multiplicator a
----------- --------------------------
2^31 - 1 = 2147483647 16807 (b = 0)
Generators recommended by Fishman (1990), "Multiplicative congruential
random number generators with modulus 2^beta: An exhaustive analysis for
beta=32 and a partial analysis for beta=48", Math. Comp. 54, pp.
331-344.
modul p multiplicator a
----------- --------------------------
2^31 - 1 = 2147483647 950706376 (b = 0)
Generators recommended by L'Ecuyer (1999), "Tables of linear
congruential generators of different sizes and good lattice structure",
Math.Comp. 68, pp. 249-260. (constant b = 0.)
Generators with short periods can be used for generating _quasi-random
numbers_ (_Quasi-Monte Carlo methods_). In this case the _whole_ period
should be used.
(These figures are listed without warranty. Please see also the
original paper.)
modul p multiplicator a
----------- --------------------------
2^8 - 5 = 251 33
55
2^9 - 3 = 509 25
110
273
349
2^10 - 3 = 1021 65
331
2^11 - 9 = 2039 995
328
393
2^12 - 3 = 4093 209
235
219
3551
2^13 - 1 = 8191 884
1716
2685
2^14 - 3 = 16381 572
3007
665
12957
2^15 - 19 = 32749 219
1944
9515
22661
2^16 - 15 = 65521 17364
33285
2469
2^17 - 1 = 131071 43165
29223
29803
2^18 - 5 = 262139 92717
21876
2^19 - 1 = 524287 283741
37698
155411
2^20 - 3 = 1048573 380985
604211
100768
947805
22202
1026371
2^21 - 9 2097143 360889
=
1043187
1939807
2^22 - 3 = 4194301 914334
2788150
1731287
2463014
2^23 - 15 = 8388593 653276
3219358
1706325
6682268
422527
7966066
2^24 - 3 = 16777213 6423135
7050296
4408741
12368472
931724
15845489
2^25 - 39 = 33554393 25907312
12836191
28133808
25612572
31693768
2^26 - 5 = 67108859 26590841
19552116
66117721
2^27 - 39 = 134217689 45576512
63826429
3162696
2^28 - 57 = 268435399 246049789
140853223
29908911
104122896
2^29 - 3 536870909 520332806
=
530877178
2^30 - 35 = 1073741789 771645345
295397169
921746065
2^31 - 1 = 2147483647 1583458089
784588716
2^32 - 5 = 4294967291 1588635695
1223106847
279470273
2^33 - 9 = 8589934583 7425194315
2278442619
7312638624
2^34 - 17179869143 5295517759
41 =
473186378
2^35 - 31 = 34359738337 3124199165
22277574834
8094871968
2^36 - 5 = 68719476731 49865143810
45453986995
2^37 - 25 = 137438953447 76886758244
2996735870
85876534675
2^38 - 274877906899 17838542566
45 =
101262352583
24271817484
2^39 - 549755813881 61992693052
7 =
486583348513
541240737696
2^40 - 1099511627689 1038914804222
87 =
88718554611
937333352873
2^41 - 2199023255531 140245111714
21 =
416480024109
1319743354064
2^42 - 4398046511093 2214813540776
11 =
2928603677866
92644101553
2^43 - 8796093022151 4928052325348
57 =
4204926164974
3663455557440
2^44 - 17592186044399 6307617245999
17 =
11394954323348
949305806524
2^45 - 35184372088777 25933916233908
55 =
18586042069168
20827157855185
2^46 - 70368744177643 63975993200055
21 =
15721062042478
31895852118078
2^47 - 115 140737488355213 72624924005429
=
47912952719020
106090059835221
2^48 - 281474976710597 49235258628958
59 =
51699608632694
59279420901007
2^49 - 562949953421231 265609885904224
81 =
480567615612976
305898857643681
2^50 - 1125899906842597 1087141320185010
27 =
157252724901243
791038363307311
2^51 - 129 2251799813685119 349044191547257
=
277678575478219
486848186921772
2^52 - 47 = 4503599627370449 4359287924442956
3622689089018661
711667642880185
2^53 - 111 9007199254740881 2082839274626558
=
4179081713689027
5667072534355537
2^54 - 33 = 18014398509481951 9131148267933071
3819217137918427
11676603717543485
2^55 - 55 = 36028797018963913 33266544676670489
19708881949174686
32075972421209701
2^56 - 5 = 72057594037927931 4595551687825993
26093644409268278
4595551687828611
2^57 - 13 = 144115188075855859 75953708294752990
95424006161758065
133686472073660397
2^58 - 27 = 288230376151711717 101565695086122187
163847936876980536
206638310974457555
2^59 - 55 = 576460752303423433 346764851511064641
124795884580648576
573223409952553925
2^60 - 93 = 1152921504606846883 561860773102413563
439138238526007932
734022639675925522
2^61 - 1 = 2305843009213693951 1351750484049952003
1070922063159934167
1267205010812451270
2^62 - 57 = 4611686018427387847 2774243619903564593
431334713195186118
2192641879660214934
2^63 - 25 = 9223372036854775783 4645906587823291368
2551091334535185398
4373305567859904186
2^64 - 59 = 18446744073709551557 13891176665706064842
2227057010910366687
18263440312458789471
A.2 Parameters for ICG (inversive congruential generator)
=========================================================
y_n = a * inv(y_{n-1}) + b (mod p) n > 0
Notice that moduli larger than 2^32 require a computer with
`sizeof(long)>32'.
Parameters suggested by P. Hellekalek (1995), "Inversive pseudorandom
number generators: Concepts, Results and Links", in: C. Alexopoulos, K.
Kang, W.R. Lilegdon, and D. Goldsman (eds.), Proceedings of the 1995
Winter Simulation Conference, pp. 255-262:
There are no results that give reason to prefer one set of parameters
over another.
(These figures are listed without warranty. Please see also the
original paper.)
p a b
-------- -------- --------
1031 849 1
345 1
55 1
116 1
441 1
1033 413 1
878 1
595 1
522 1
818 1
1039 173 1
481 1
769 1
1028 1
136 1
2027 579 1
1877 1
390 1
837 1
1048 1
2147483053 858993221 1
22211 11926380
579 24456079
11972 62187060
21714 94901263
4594 44183289
2147483647 1288490188 1
9102 36884165
14288 758634
21916 71499791
28933 59217914
31152 48897674