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Analyzing & Testing Justified Prime Numbers: Mid-term Presentation

This presentation delves into the creation and testing of large prime numbers, focusing on random number generators like LCG and LFSR, primality testing algorithms (including AKS), and their efficiency comparisons.

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Analyzing & Testing Justified Prime Numbers: Mid-term Presentation

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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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