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Analyzing and Testing a justified Prime Number

Analyzing and Testing a justified Prime Number. ( Mid-term Presentation ). 20032047 Jeong-kyu YANG 20032003 Seok-kyu Kang. OUTLINE. Overview Random number generator Linear Congruential Generator (LCG) Linear Feedback Shift Register (LFSR) Testing Primality algorithms

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Analyzing and Testing a justified Prime Number

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

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

  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. RANDOM 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. RANDOM NUMBER GENERATOR Example Xn+1 =(3 Xn + 4) mod 5 , X0= 1 X1= (31 + 4) mod 5 = 2, X2= (32 + 4) mod 5 = 0, X3= (30 + 4) mod 5 = 4, X4= (34 + 4) mod 5 = 1, X5= (31 + 4) mod 5 = 2, Period of 4 less than modular 5

  6. RANDOM 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.   C3 C0 C1 C2 S0 S1 S2 S3 Output RANDOM NUMBER GENERATOR • Linear Feedback Shift Register (LFSR)

  8. RANDOM NUMBER GENERATOR f(S0, S1, S2,S3) = S0  S1 Time 1 f(1,0,0,0) = 10 = 1 Ouptput = S0 = 1 Time 2 f(0,0,0,1) = 00 = 0 Output = S0 = 0 Each output is always same to S0 This case has the sequence with a period of 15. The maximum period is 2n-1.

  9. TESTING 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. TESTING 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. TESTING PRIMALITY ALGORITHMS • AKS Algorithm

  12. TESTING 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. TESTING PRIMALITY ALGORITHMS • Probabilistic Algorithm • Solovay-Strassen Algorithm

  14. TESTING PRIMALITY ALGORITHMS • Probabilistic Algorithms • Miller-Rabin Algorithm

  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. 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. 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

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