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Analyzing and Testing justified Prime Numbers

Analyzing and Testing justified Prime Numbers. Concrete Mathematics Final Presentation 20032047 Jeong-Kyu YANG 20032003 Seok-Kyu Kang. OUTLINE. Introduction The Primality Testing Algorithms Probabilistic Algorithms Deterministic Algorithms

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Analyzing and Testing justified Prime Numbers

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  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 • Fermat’s Theorem • Wilson’s Theorem • AKS

  5. Analyzing of Solovay-Strassen • Probabilistic Algorithms • Solovay-Strassen Algorithm (Cont.) • Based on Euler Pseudoprime • More effective than the simpler Fermat’s 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. Legendre’s symbol, L(a, n) Jacobi’s symbol, J(a,n) is generalized from Legendre’s symbol, L(a, n) 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. Reducing the probability of misjudgment 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. Find Useful Prime Brute force can be used Set of congruence Analyzing of AKS • Deterministic Algorithm • AKS Algorithm

  11. Filter 1 Filter 2 Filter 3 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 n=0 for all 0<i<n. • 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* r2*log n*(log n)2)=O(2sqrt r log n* r2*log n*(log n)2), r is O((log n)6) => O((log n)19) O((log n)12f(log log n)), where f is a polynomial function

  13. 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

  14. 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: r=23159, s=5784 • polyTest: each “for-loop” iteration of the for-loop takes about 355sec (about 6mins). So, overall runtime is 6mins*5784 (value of s in this case), which is about 34704mins = 578.4hours = 24 days!!! • Solovay-Strassen & Miller-Rabin: less than 1 sec.

  16. 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)

  17. 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

  18. 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/020821.html [7] http://www-fs.informatik.uni-uebingen.de/~reinhard/krypto/primzt.html [8] http://www.cse.iitk.ac.in/news/primality.html [9] http://random.mat.sbg.ac.at/generators/

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