CMPT585 Computer

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CMPT-585 Computer Data Security ByAyesha
Mohiuddin Ramazan Burus Advisor Stefan A.
Robila
Generating Large Prime Numbers for Cryptographic
Algorithms Using Distributed Computing
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Prime Numbers

• Used in cryptographic algorithms e.g. RSA
  algorithm, PKE, PKI, Diffie-Helman key exchange
• GIMPS Great Internet Mersenne Prime Search A
  Mersenne prime is a prime of the form 2P-1.
• On February 18, 2005, Dr. Martin Nowak from
  Germany, discovered the 42nd known Mersenne
  Prime, 225,964,951-1. The number is nearly 7.8
  million digits large. It took more than 50 days
  of calculations on his 2.4 GHz Pentium 4
  computer.

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Problem Setting

• Generate and store the prime numbers within a set
  range of values. (such as 1 to a billion)
• Use distributed computing to speed up the
  generation.
• Use database technologies to store the numbers.

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Approach

• Programming language used JAVA
• Database ORACLE
• Object Build a grid of multiple clients
  calculating prime numbers between unique ranges
  of numbers, to obtain a list of large prime
  numbers.

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Computing Structure
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3 Components

• Clients
• Allotted a unique Id and time limit.
• Gets the range of numbers to calculate within.
• Master
• Keeps monitoring the activity.
• Re-assigns range to another client if original
  client does not complete within its allotted
  time. (1 day for our experiment)
• Database
• Stores the client information and the resulted
  Prime numbers sent by the clients.

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Setting

• Client side program responsibilities
• Connect to the database through internet.
• Take a range of numbers to work on, communicate
  that the range has been taken, and start
  calculating primes within that range.
• Connect to database again for each found prime
  number and put that into its corresponding table
• When done communicate completion of task and take
  another range for new calculations.
• Administration Side responsibilities
• Assign different ranges to different clients and
  receive results in tables.
• Keep track of jobs, if a taken job is not done up
  to a certain time by a node, then consider the
  node dead and re-assign the same range to another
  client node. The new node should somehow start
  from where old one left off.

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Number Ranges

• As Numbers keep getting larger, number of prime
  numbers keep decreasing. For example
• Variable ranges required for each clients.

430 primes between 1 to 3000 and 353 primes
between 3000 to 6000, so on.
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Results

• Executable 1.29 MB
• Memory usage 10 MB
• CPU usage 7 to 10
• Total primes stored 376074
• Largest Prime stored 6583813
• In 12 hours using only 6 nodes, Primes within the
  maximum range of 461 million were found.

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Thoughts Conclusion

• In 12 hours using only 6 nodes, which is a really
  small number, we were able to find primes within
  the maximum range of 461 million.
• This speed can be increased further by using more
  client nodes, more efficient algorithm for
  finding prime numbers.
• It would be better if the client side code is
  wrapped into a screen saver, so that it only
  starts executing when the client users computer
  is idle in order not to obstruct their own work.

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Useful Links Used in the Project

• Crow, Jerry. Prime Numbers in Public Key
  Cryptography, GSEC Practical Assignment. SANS
  Institute 2003. http//www.giac.org/practical/GSEC
  /Gerald_Crow_GSEC.pdf
• GIMPS (The Great Internet Mersenne Prime Search),
  2004, http//www.mersenne.org
• Havil, J., Gamma Exploring Euler's Constant,
  Princeton, NJ Princeton University Press, 2003.
• A. Languasco, and A. Perelli. Prime Numbers and
  Cryptography. 2003 http//www.math.unipd.it/lan
  guasc/lavoripdf/R8eng.pdf
• Lewis, John and Loftus, William. Java Software
  Solutions. 2nd edition, Addison Wesley Longman,
  2001
• Pfleeger, Charles and Pfleeger, Shari. Security
  in Computing. Prentice Hall 2003, 3rd Edition
• Weisstein, Eric W. "Prime Number." From
  MathWorld --A Wolfram Web Resource.
  http//mathworld.wolfram.com/PrimeNumber.html