1. Introduction:
SPRNG is a set of libraries for scalable and transferrable pseudo random number generation, and
has been developed keeping in mind the requirements of users like us involved in parallel Monte
Carlo simulations
We know that Monte Carlo Calculations consume a large fraction of all supercomputing cycles.
The accuracy of these computations is critically influenced by the quality of the random number
generators used. While the issue of random number generation in sequential calculations has
been studied, even though on less powerful computers, there has been comparatively less work
done in the context of parallel Monte Carlo applications. SPRNG seeks to fill this gap by
implementing parallel random number generators that satisfy the requirements given below.
Users like us generally desire the following from any parallel random number generator:
Quality: Provide “high quality” pseudo random number in a computationally inexpensive and
scalable manner.
Reproducibility: Provide totally reproducible streams of parallel pseudo random numbers,
independent of the number of processors used in the computation and of the loading produced by
sharing of the parallel computer.
Locality: Allowing for the creation of unique pseudo random number streams on a parallel
machine with minimal inter-processor communication.
Portability: Should be portable between serial and parallel platforms and must be available on
the most commonly used workstations and supercomputers.
SPRNG authors, intended to provide pseudo random number generators that satisfy the
above requirements.
2. Literature Review:
The ability to generate satisfactory sequences of random numbers is one of the key links between
Computer Science and Statistics. Most computer systems have random number generators
available, and for most purposes they work well. But many of the standard random number
generators (RNGs) are not good enough for increasingly sophisticated Monte Carlo uses, such as
geometric probability, estimating distributed functions.
There are many pseudo random number generators exists. One among them is Parallel pseudo
random number generation. These methods are based on Parameterization of the generators.
2.1 Linear Congruential Generator: It is mostly commonly used pseudo random generator.
These generators use a linear transformation on the ring of residues of some modules m, to
produce a sequence of integers.
Xn = a*Xn-1 + b mod m
There are the most widely used RNGs and they work remarkably well for most general purposes.
But for some purposes they are not satisfactory; points in n-space produced by Congruential
RNG’s fall on a lattice with huge unit cell volume.
Prime Modulus: Parameterization when ‘m’ prime. First we need to determine the family of a’s
then obtain the maximal period (m-1)
If given permittivity, 𝛼 = 𝑎𝑖 (𝑚𝑜𝑑 𝑚)
𝑆𝑖𝑛𝑔𝑙𝑒 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒, 𝑃𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 ′𝑎′ & explicit enumeration of the relative prime to m-1
produces jth primitive element:
PROBLEMS:
Overall efficiency – Where to minimize the cost of computing Lj or to minimize the cost of
modular multiplication modulo ‘m’?
2.2 Lagged-Fibonacci Generators:
Additive lagged-Fibonacci generators can be parameterized through its initial valu It is the most
popular generator for serial as well as scalable machines. These generators with j=5, K=17 and
m=32 was the standard PRNGs.
These are popular for many reasons:
1. Easy to implement.
2. Cheap to compute
3. It does well on statistical tests
Pros and Cons: It is optimal unfortunately there is no proof for this result and improvement of
this analytical is open challenge?
But, this is good for Floating point numbers which avoids the constant conversion from integer
to float. However, care must be taken to maintain the uniqueness of the parallel streams.
3. Analysis of the current techniques:
These authors have developed a library [SPRNG] of parallel random number generators for
Scalable Parallel Random Number Generators; the library collects the top five RNGs into single,
easy-to-use package that runs on almost any computing architecture. Because of extensibility of
library, researchers won’t have to start from the scratch. Also, one can take the advantages of the
high quality random number generators available within this library. Before the existing of
SPRNG the RNGs were not only difficult to find but also many were not performing up to the
standard necessary with more powerful computers. But these authors solved these problems with
SPRNG.
But, one still need to test these generators with their own application. Even in this case SPRNG
made it easier. These authors modified the generators in SPRNG to run on almost any parallel
computing architecture. The program that written is almost compatible in FORTRAN, C and
Java Because of which we can easily change among the generators in the library with a change at
Run time.
Since, SPRNG uses entire families of RNGs; the code written for single processors can run in
parallel i.e., on several processors at a time.
4. Opinion:
The SPRNG library has many generators available so far today and in the future I wish this
library to add generators like WELL and Mersenne Twister generator since these are currently
considered the best random number generator today. Also, the SPRNG library currently
implemented in C++ and uses the FORTRAN 77 interfaces. As we know the growth and
popularity of Java, its good idea to replace there interfaces from low level language like
FORTRAN to Java.
We know that, quasirandom numbers are uniform in distribution and are effective at error
reduction on Monte Carlo application like Numerical integration. The SPRNG should be
designed such that library provides model for quasirandom numbers for parallel and distributed
systems.
Moreover, this library is focused on systems like scalable and distributed but nowhere
cryptographic systems. As we know that the generation of random numbers is critical to
cryptographic systems. Symmetric cyphers such as DES [Data Encryption Standards], RC2 and
RC5 all requires a randomly selected encryption key. Also, public-key algorithms like RSA,
DSA and Diffie-Hellman begin with randomly generated values when generating prime
numbers.
At a higher level, SSL and other cryptographic protocols use random challenges in the
authentication process to stop replay attacks. This poses a challenge for software developers like
we implementing cryptography.
A new kind of generator should be implemented such that it modifies the approach of
construction and utilization of pseudo-random number generators. Since, attacks like brute-force
may cause because of its fixed seed size i.e., 192 bits
5. Results:
Usage of Library:
#include <cstdio>
#define SIMPLE_SPRNG
#include "sprng.h"
using namespace std;
int main(){
/* simple interface
/* SPRNG header file
*/
*/
double rn; int i;
printf(" Printing 3 random numbers in [0,1):\n");
for (i=0;i<3;i++) {
rn = sprng();
/* generate double precision random number */
printf("%f\n",rn);
} return 0;}
$> ./simple-simple Spri
Printing 3 random numbers in [0,1):
0.014267
0.749392
0.007316
Statistical Test: The SPRNG test suites consists the entire test described by the Knuth. These
tests are modified such that these test to test for parallel correlation.
Ex: eqi-distance.lcg 4 2 0 03 1 2 1 00
Mprirun –np 2 equidistance.lcg 4 2 0 0 3 1 2 1 00
This performs equidistance test with 48 bit Linear Congruential generator
Result:
KS value: 0.601252
KS value prob: 17.50
Hence, It’s b/w 2.5% and 97.5% It is passed.
6. CONCLUSION:
This paper is all about pseudo random number generation through
parameterization. The described library is based on this parameterization PRNGs. In detail they
have described about test suites – physical as well statistical. The authors has taken care more in
constructing generators and are based on recursions, which are so simple. Also, they designed
the library such that one can implement their own generators by providing rapid prototyping
tools. As well, parallel and serial tests of randomness which make there SPRNG library unique
among tools for Monte Carlo.