Analyzing and Testing justified Prime Numbers

1
Analyzing and Testing
justified Prime Numbers

• Concrete Mathematics
• Final Presentation
• 20032047 Jeong-Kyu YANG
• 20032003 Seok-Kyu Kang

2
OUTLINE

• Introduction
• The Primality Testing Algorithms
• Probabilistic Algorithms
• Deterministic Algorithms
• Analyzing
• Solovay-Strassen Algorithm
• Miller-Rabin Algorithm
• AKS Algorithm
• Implements Experiments
• Conclusion Future Works
• References

3
Introduction

• What is Prime Number Primality Testing?
• Prime Number
• Primality Test
• The importance of testing primality
• Applications in cryptography
• RSA, etc. uses primality testing algorithm in the
  part of key generation.
• How fast and efficient?
• Brief History
• 200 BC Eratosthenes Sieve
• 1976 NP(Nondeterministic Polynomial-time), Pratt
• 1977 coRP(Complementary Randomized
  Polynomial-time), Solovay and Strassen
• 1987 RP(Randomized Polynomial-time), adleman and
  Huang
• 1992 UP(Unambiguous Polynomial-time), Fellows
  and Koblitz
• 2002 PRIMES is in P(Polynomial-time), Agrawal et
  al.

4
The Primality Testing Algorithms

• Probabilistic Algorithms
• Lehamann-Peralta
• Solovay-Strassen
• Miller-Rabin
• Deterministic Algorithms
• Eratosthenes Sieve
• Euclidean algorithm
• Fermats Theorem
• Wilsons Theorem
• AKS

5
Analyzing of Solovay-Strassen

• Probabilistic Algorithms
• Solovay-Strassen Algorithm (Cont.)
• Based on Euler Pseudoprime
• More effective than the simpler Fermats test
• A number N called an Euler Pseudoprime to base b,
• if b(N-1)/2 (b/N) (mod N).
• ((b/N) is the Jacobi symbol)
• Legendre symbol, L(a,P)

6
Analyzing of Solovay-Strassen

• Probabilistic Algorithms
• Solovay-Strassen Algorithm

7
Analyzing of Miller-Rabin

• Probabilistic Algorithms
• Miller-Rabin Algorithm (Cont.)
• More efficient than Solovay-Strassen Algorithm
• Emerged by Miller in 1976, modified by Rabin in
  1980
• Definitely correct if it returns COMPOSITE, input
  a maybe
• a pseudoprime if it returns PRIME
• The probability of Miller-Rabin is not greater
  than (1/4)s
• Strong primality test of pseudoprime

8
Analyzing of Miller-Rabin

• Probabilistic Algorithms
• Miller-Rabin Algorithm

Reducing the probability of misjudgment
9
Analyzing of AKS

• Deterministic Algorithm
• AKS Algorithm
• By Manindra Agrawal, Neeraj Kyal and Nitin Saxena
• August 2002
• Always returns right answer
• Works in polynomial time
• Basic Idea
• (x a)n xn a (mod n)
• a, n relatively prime
• if n is prime true
• if n is composite false
• Compare n coefficients O(n) O(2lg n)

10
Analyzing of AKS

• Deterministic Algorithm
• AKS Algorithm

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Analyzing of AKS

• Deterministic Algorithm
• AKS Algorithm

12
Analyzing of AKS

• Complexity
• Filter 1 O(log n)3
• Filter 2 O(log n)3
• Filter 3
• Computation ai mod n0 for all 0ltiltn.
• Using square and multiply method requires O(log
  n)
• multiplications of polynomials of degree
  smaller than r
• Multiplication of 2 such polynomials, takes
  O(r2)
• operations in Z/nZ, whereas, multiplication in
  Z/nZ is
• O(log n)2 additions.
• Then the for loop requires
• O(s r2log n(log n)2)O(2sqrt r log n r2log
  n(log n)2),
• r is O((log n)6) gt O((log n)19)
• O((log n)12f(log log n)), where f is a polynomial
  function

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Implementations SS, MR and AKS

• Environment
• Hardware
• Pentium III 550mhz, 384 RAM
• Language Java (j2sdk1.4.0_02), Boland Jbuilder
  6.0
• The way to implement
• Solovay-Strassen Miller-Rabin
• Run simultaneously with a same random number
  generator
• Same iterations to check better performance
• Same bit lengths
• Demo Program-1
• AKS
• Testing with far smaller lengths (Long integer
  operation is for further works)
• Testing for polynomial time of AKS
• Demo Program-2, Program-3

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Experiments - Probabilistic

• Comparison of primality between Solovay-Strassen
  and Miller-Rabin

15
Experiments - Deterministic

• Testing for polynomial time of AKS
• Limitations with no memory fluctuation
• n 524287
• powerTest output r23159, s5784
• polyTest each for-loop iteration of the
  for-loop takes about 355sec (about 6mins). So,
  overall runtime is 6mins5784 (value of s in this
  case), which is about 34704mins 578.4hours 24
  days!!!
• Solovay-Strassen Miller-Rabin less than 1 sec.

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Experiments Comparison

• Primality Comparisons among tree algorithms
• Limitations
• The range of Positive Odd Integers 3 499
• Iterations 130 (SS MR also has 50 iterations
  internally)

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Conclusion

• The importance of strong very big prime numbers
  from the experiments of this project
• Miller-Rabin has better performance than
  Solovay-Strassen
• However, two algorithms probably declare lots of
  pseudoprimes
• AKS is a breakthrough result
• Always declares real primes
• Solves a long-standing theoretical problem
• AKS has no practical relevance
• Prohibitively slow runtimes
• Not likely to change any time soon
• Polynomial computations are just too inefficient
• Theoretically correctness V.S. practical
  efficiency?
• Depend on purposes

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Future Works

• More analysis of complexity for each algorithms
• Further Experiments for AKS
• Find useful prime numbers and analyze its
  characteristics
• Further Implementation for AKS
• Try to get over inefficiency of AKS Algorithm
• Improving to handle very long integers
• Continue to compare results of each algorithms

19
References

• 1 M.Agrawal, N.Kayal and N.Saxena, PRIMES is
  in P, August 6, 2002
• 2 William Stallings, Cryptography and Network
  security, second edition. Prentice Hall, 1998
• 3 J.Menezes, C.vaz Oorschot and A.Vanstone,
  Handbook of Applied Cryptography CRC,1977
• 4 Takeshi Aoyama, Polynomial Time Primality
  Testing Algorithm, 2003
• 5 Frontline. Volume19-Issue 17, August
  17-30.2002
• 6 http//www.javastudy.co.kr/docs/techtips/02082
  1.html
• 7 http//www-fs.informatik.uni-uebingen.de/rein
  hard/krypto/primzt.html
• 8 http//www.cse.iitk.ac.in/news/primality.html
• 9 http//random.mat.sbg.ac.at/generators/