Parallel Processing to Determine Prime Factors
Matt Roberts
Hood College CS566
Abstract
Abstract – Prime number factorization can be
computationally intensive. So much so, that prime
number factorization is used in RSA Encryption [6] as
public and private keys. The Prime number itself has had
the attention of many groups who are trying to find the
largest Mersenne Prime number, such as GIMP [7].
This paper will document a type of research and
experiment used to introduce parallel processing into
computation of prime factorization of large numbers. The
introduction of parallel processing is suggested to speed
up processing of prime factorization and relieve the
computational intensiveness.
If proven useful, this
method of parallel processing could be used in many
areas of prime factorization.
1. Introduction
A prime number is a number greater than 1 with no
divisors other than itself and 1, such as 2 and 11. The
first few prime numbers are 2, 3, 5, 7, 11, 13 and 17. [3]
These primes are fundamentally utilized first when
determining a large number’s prime factorization.
Numbers that are not “prime” contain factors.
Factors are prime numbers multiplied to obtain the
original number.
8 = 2 x 2 x 2 “The 2s are factors of 8.”
This break down of a larger number is also
considered Prime factorization. To determine a larger
number’s prime factors it is best to start working from the
smallest prime number.
18 / 2 = 9
9 is not a prime number, so we continue iterating of
the result, by dividing by small primes.
9/3=3
Finally when hit a prime number as a result, so the
prime factors of 18 are 2 x 3 x 3 or 2 x 3^2.
This method can also be used to determine if a
number is a prime number itself. While iterating over a
large number, if there are no known prime numbers that
can be divided into it, then the large number cannot be
factorized and it is considered a prime number.
The largest prime number known is 12978189. It was
discovered on 08/23/2008 by Edson Smith, George
Woltman and Scott Kurowski. The code they used is
called G10 and it accounts for 4.762% of the active
known primes, 21 total. To reach the largest prime
number known this group of programmers used a
distributed computing project on the internet called the
“Great Internet Mersenne Prime Search” (GIMPS).
A Mersenne Prime Number is a number that is prime.
It is know that if 2^p – 1 is prime then p is also prime. [8]
GIMPS utilize distributed computing to find
Mersenne Prime Numbers. In this paper we shall propose
that basic prime factorization of large numbers can also be
done using parallel processing. In addition, this paper
proposes that a significant decrease in the amount of time
taken to process the factorization of large numbers can be
achieved through parallel processing. If this if true, then it
is possible that parallel processing could enhance GIMPS
search for Mersenne Prime Numbers by incorporating
parallel processing in their own programs.
2. Literature Review
There have been and are many experiments on
parallel processing that try to speed up prime
factorization. One such experiment was performed by the
Transactions of the Institute of Electrical Engineers of
Japan. There experiment was title “Experiments on
Distributed Parallel Processing for Speeding up Prime
Factorization in the Elliptic Curve Method”. [5]
Tsurusawa Hidenobu, Nakayama Shigeru, Fuchida
Takayasu and Murashima Sadayuki, all from Kagoshima
University, carried out distributed parallel processing
using 50 computers connected to one network. Similar to
the 8 computer nodes connected to the Pluto network at
Hood College, but on a large scale. These students
developed a distributed parallel calculation of the prime
factorization in the elliptic curve method to adapt to
object shared space and performed the parallel processing
with a worker-master methodology.
The results of their experiment indicated that the
speed of prime factorization of large integers increases
with each added computer to the distributed network of
slave nodes. In addition, their results illustrated that the
calculation time of prime factorization mainly depends on
the size of the prime factors.
3. Methodology
The algorithm or method to compute primes and their
factors is by repeatedly dividing a number with small
prime factors until the resulting number becomes 1.
Suppose we have the number 50, we find the smallest
number that actually divides 50, which is using 2. Next
50 divided by 2 becomes 25, leaving the resulting number
as 25. The program stores 2 in an array and then iterates
over the result. We find the smallest prime number that
actually divides 25, which is 5. 25 is then divided by 5 to
get 5 as a result. The program stores 5 in an array and
then iterates once again over the result. The program tries
to find a prime that divides into 5, but it cannot since 5 is
a prime. 5 is then stored in the resulting array. The
resulting array now contains the prime factorization of 50.
Number / Prime Number = Result
Resulting Array = [2 x 5 x 5]
Since this research program is to determine if parallel
processing can decrease the amount of time needed to
factorize large numbers and it is not to attempt to find new
prime numbers, a data source of all know prime numbers
will be used during factorization processing. The resource
for prime numbers can be found at http://primes.utm.edu/ ,
which contains the “proclaimed” largest know database of
known prime numbers.
The Message Passing Interface (MPI) will be the
primary methodology used to illustrate parallel processing
of prime factorization. MPI is a directory of C programs
that use the MPI library of message passing routines. The
library allows a user to write a program in a familiar
language like C and carry out a computation or process in
parallel on an arbitrary number of cooperating computers
or processors.
The MPI program will use 8 computer nodes on the
PLUTO network at Hood College’s Computer Science
Linux lab.
4. Expected Results
The expected results will be that parallel processing
greatly enhances prime factorization processing of large
numbers.
The computation may even become
embarrassingly parallel. To compare the results of a
parallel program, this research will perform the same
prime number factorization computations without the
addition of MPI to gain the results of one node performing
the task. The results will then been displayed and plotted
on a graph to show the differences.
Time pending, other alternatives may be added, such
as using CUDA as the parallel processing interface to see
if we get better or worse results when the GPU used to
perform prime factorization. In addition, other resources
have been made available such as NSA’s teraGrid, which
contains a larger scale of nodes. Using a large scale of
computers may provide unexpected results or better
solidify the assumption of this paper that parallel
processing decreases the amount of time needed to
factorize prime numbers.
5. References
[1] Mir Shahriar Emami, Mohammed Reza Meybodi, A Post Processing
Method for Quantum Prime Factorization Algorithm based on
Randomized Approach, http://www.waset.org/journals/waset/v34/v3424.pdf
[2] Washington University in St. Louis, Department of
Mathematics, Find the Prime Factors of a Number,
http://www.math.wustl.edu/cgi-bin/primes
[3] ICRA, Math is Fun, http://www.mathsisfun.com/primefactorization.html
[4] The Largest Known Primes!, http://primes.utm.edu/primes/
[5] TSURUSAWA HIDENOBU(Kagoshima Univ.) NAKAYAMA
SHIGERU(Kagoshima Univ.) FUCHIDA TAKAYASU(Kagoshima
Univ.) MURASHIMA SADAYUKI(Kagoshima Univ.), Experiments
on Distributed Parallel Processing for Speeding up Prime Factorization
in the Elliptic Curve Method, http://sciencelinks.jp/jeast/article/200215/000020021502A0501489.php
[6] Tom Davis, RSA Encryption, October 10, 2003,
http://www.geometer.org/mathcircles/RSA.pdf
[7] GIMP, http://www.mersenne.org/freesoft/#source
[8] Mersenne Prim, Wikipedia,
http://en.wikipedia.org/wiki/Mersenne_prime