1. Analyzing and Testing a justified Prime Number 20032047 Jeong-kyu YANG 20032003 Seok-kyu Kang ( Mid-term Presentation )

2. Information Security Group 2 OUTLINE  Overview  Random number generator Linear Congruential Generator (LCG) Linear Feedback Shift Register (LFSR)  Testing Primality algorithms  Remarks  Further work  Reference

3. Information Security Group 3 OVERVIEW Introduction –In the public key Cryptosystems, the big prime number is required as a key. –Difficult to create a big prime number computationally. –Test primality using some algorithms after creating a pseudo prime numbers. Our Goal –Understand the basic concept of a large prime number –Implement Primality testing algorithms and compare the efficiency of each algorithm.

4. Information Security Group 4 R ANDOM NUMBER GENERATOR Create a session key for generating big prime number. Linear Congruential Generator (LCG)  Linear congruential generators are very efficient, are theoretically quite well understood, and work well for many applications.  The maximum period is m-1  Need to use integers with at least 48 and preferably closer to 64 bits.

5. Information Security Group 5 R ANDOM NUMBER GENERATOR Example X n+1 =(3 X n + 4) mod 5, X 0 = 1 X 1 = (3  1 + 4) mod 5 = 2, X 2 = (3  2 + 4) mod 5 = 0, X 3 = (3  0 + 4) mod 5 = 4, X 4 = (3  4 + 4) mod 5 = 1, X 5 = (3  1 + 4) mod 5 = 2, Period of 4 less than modular 5

6. Information Security Group 6 R ANDOM NUMBER GENERATOR Linear Feedback Shift Register (LFSR)  A mechanism for generating a sequence of binary bits.  Consists of a series of cells that are set  LFSRs are fast and easy to implement in both hardware and software.  The sequences generated by single LFSRs are not secure because a powerful mathematical framework has been developed.However, LFSRs are useful as building blocks in more secure systems

7. Information Security Group 7 R ANDOM NUMBER GENERATOR Linear Feedback Shift Register (LFSR) S0S0 S1S1 S2S2 S3S3    C0C0 C1C1 C2C2 C3C3 Output

8. Information Security Group 8 TimeS0S0 S1S1 S2S2 S3S3 Output 110001 200010 300100 401000 510011 600110 701100 811011 910101 1001010 1110111 1201110 1311111 1411101 1511001 f(S 0, S 1, S 2,S 3 ) = S 0  S 1 Time 1 f(1,0,0,0) = 1  0 = 1 Ouptput = S 0 = 1 Time 2 f(0,0,0,1) = 0  0 = 0 Output = S 0 = 0 Each output is always same to S 0 This case has the sequence with a period of 15. The maximum period is 2 n -1. R ANDOM NUMBER GENERATOR

9. Information Security Group 9 T ESTING PRIMALITY ALGORITHMS Deterministic Algorithm Euclidean algorithm n: not prime Fermat ’ s theorem n: prime if (b,n)=1, then Wilson ’ theorem n: prime (n-1)! AKS(Agrawal-Kayal-Saxena) algorithm

10. Information Security Group 10 T ESTING PRIMALITY ALGORITHMS Deterministic Algorithms AKS Algorithm  Simple and unconditional algorithm in polynomial time  A number is prime if only if it is prime.  No percentage margin of error  Slower than other current methods

11. Information Security Group 11 T ESTING PRIMALITY ALGORITHMS AKS Algorithm

12. Information Security Group 12 T ESTING PRIMALITY ALGORITHMS Probabilistic Algorithms Solovay-Strassen Algorithm  Based on Euler pseudoprime  A more effective than the simpler Fermat test.  A number N is called an Euler pseudoprime to base b if b (N-1)/2 =(b/N) (mod N).  ( (b/N) is the Jacobi symbol.) Miller-Rabin Algorithm  More efficient than Solovay-Strassen Algorithm  Emerged by Miller in 1976, modified by Rabin in 1980  Definately correct if it returns COMPOSTIE, input a may be a pseudoprime if it returns PRIME.  The probability of Miller-Rabin is not greater than (1/2)^s

13. Information Security Group 13 T ESTING PRIMALITY ALGORITHMS Probabilistic Algorithm Solovay-Strassen Algorithm

14. Information Security Group 14 T ESTING PRIMALITY ALGORITHMS Probabilistic Algorithms Miller-Rabin Algorithm

15. Information Security Group 15 Milestone Study on the basic concept – Background of Discrete Mathematics – Prime Number & Prime Number Generation Survey on the techniques of algorithms –The existing algorithms of prime number generation –Programming & it ’ s implementation Study on the prime number decision algorithms –The existing algorithms of prime number generation

16. Information Security Group 16 FURTUER WORK Analysis the AKS Algorithm completely Implement the following –Solovary-Strassen Algorithm –Miller-Rabin Algorithm –AKS Algorithm Compare the efficiency of each algorithm with pseudo prime numbers from random number generator

17. Information Security Group 17 REFERENCE William Stallings. Cryptography and Network security, second edition. Prentice Hall, 1998 J.Menezes, C.vaz Oorschot and A.Vanstone. Handbook of Applied Cryptography. CRC,1977 M.Agrawal, N.Kayal and N.Saxena. PRIMES is in P. 2002 Frontline. Volume19-Issue 17. August 17-30.2002 http://www.cse.iitk.ac.in/news/primality.html