1. Analyzing and Testing justified Prime Numbers
Concrete Mathematics
Final Presentation
Jeong-Kyu YANG
Seok-Kyu Kang

2. OUTLINE Introduction The Primality Testing Algorithms Analyzing
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?
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. Analyzing of Solovay-Strassen
Probabilistic Algorithms
Solovay-Strassen Algorithm
Jacobi’s symbol, J(a,n) is generalized from Legendre’s symbol, L(a, n)
Legendre’s symbol, L(a, n)

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
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 Find Useful Prime
AKS Algorithm
Find Useful
Prime
Set of congruence
Brute force can be used

11. Analyzing of AKS Deterministic Algorithm Filter 1 Filter 2 Filter 3
AKS Algorithm
Filter 1
Filter 2
Filter 3

12. Analyzing of AKS Complexity Filter 1: O(log n)3 Filter 2: O(log n)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 =
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
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