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

2. 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 2 P -1.  On February 18, 2005, Dr. Martin Nowak from Germany, discovered the 42nd known Mersenne Prime, 2 25,964,951 -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.

3. 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.

4. 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.

5. Oracle Database MasterClient 1 Client 2 Client 3 Architecture Computing Structure

6. 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. 3 Components

7. 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.

8.  As Numbers keep getting larger, number of prime numbers keep decreasing. For example:  Variable ranges required for each clients. Number Ranges Client 1Client 2Client 3 0 2 27 ……… 430 primes between 1 to 3000 and 353 primes between 3000 to 6000, so on.

9. 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.

10.  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 user’s computer is idle in order not to obstruct their own work. Thoughts & Conclusion

11. 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/~languasc/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