Generating RSA Primes

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Generating RSA Primes

• Jim Townsend
• CSE633
• Final Results
• Fall 2010

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Importance

• Encryption is harder to secure than ever
• RSA is an important standard in Public Key
  Encryption
• Developed in 1977, it began with relatively small
  keys 128,256 bit keys
• Current standard 1048 bit keys (310 decimal
  digits)
• Math on these numbers is very CPU intensive

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How Keys are Generated

• Use the Miller-Rabin algorithm 
• Tests against a specific few numbers
• Only a probabilistic method
• Probability a number is prime .75
• Repeated passes used to eliminate false positives
• 16 repetitions (1-.75)16 
• Runtime O(ln(N)4)

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Sieve of Eratosphenes

• Decided to implement a small sieve on the numbers
  before using the Miller-Rabin algorithm
• Using all the prime numbers less than 1000 (168
  numbers), see if any of those evenly divide the
  number first
• Decreased serial runtime by more than half

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Current Program

• The program takes in two strings a starting
  value and a range
• Runs a sieve on the range with the first 168
  primes
• Uses the remaining numbers and tests them with
  the Miller-Rabin algorithm up to 16 times on each.

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Serial Results
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Serial Results

• Finding small numbers was relatively fast
• Found 2263 primes 20 digits long in just .68
  seconds
• Large numbers are a different story
• 310 digits (Current RSA standard) took 27.01
  seconds to find only 118 primes

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Parallel Algorithm

• Divided the range among each processor
• Each node checked its set and reported the number
  of primes it found
• Final reduction to sum up the count

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Gains

• Saw incredible speedup due to the minimal
  communication needed
• Most of the real gains came from tweaking the
  serial algorithm
• Using the sieve and only checking odd numbers
• Would see much more by using load balancing using
  OpenMP

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Future Work

• Could be more improved by load balancing the test
  with OpenMP
• Exit on first failed test
• Much better synchronization would be possible
• Could also use this to divide the test into
  smaller pieces as well
• Implementation in CUDA using GPGPUs

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Any Questions?