1. Prime Hunting Gábor Farkas Department of Computer Algebra Faculty of Informatics Eötvös Loránd University Jena, Germany 26. May 2008.

2. What does „prime hunting” mean ? What is the main goal of prime hunting? To find the largest known prime number Develop fastest programs than others and we are and curious prime combinations What can we do to reach new world records? ready! almost 2/25

3. MathematicsInformatics Computational Number Theory to study the classical and new theoretical results to invent new algorithms to implement them taken advantage of qualities of the processors 3/25

4. Motivations – the published prime records win laurels for us – the large primes are marketable e. g. public key cryptographic systems need large primes, or the prize of the first known prime 10.000.000 of digits is 100.000 $ – the achieved prime records prove the efficiency of our programs – the fast routines are utilized in software used in practice life 4/25

5. The Top 10 Largest Known Primes in May, 2008 5/25

6. Example Def. An positive integer p is a Sophie Germain prime if p and 2p + 1 are simultaneously primes. Def. A Cunningham chain of length k (of the first kind) is sequence of k primes, each which is twice the proceeding one plus one. {2, 5, 11, 23, 47} {89, 179, 359, 719, 1439, 2879} SG k = 6 k = 5 6/25

7. Let us consider now an RSA public key cryptographic algorithm, where p and q are odd primes, n = pq, e positive integer relatively prime to  (n) and d is a solution of the following linear congruence: Then (n, e) is the public, d is the secret key. How do we choose the parameters ? Naturally we want the probability of a successful cycling attack on the RSA to be as small as possible. if n is a product of only two factors of the same magnitude that are doubly Sophie Germain pairs and e is a primitive root with respect to p – 1 and q – 1 as moduli. The best choice: p and q are the first member of a Cunningham chain of length 3 7/25

8. Description of the „Hunting” Candidates (H = {0, 1, …, N}) Sieving Methods – Production of „Small” Primes – Sieving Tables – Generalized Sieving Probabilistic Primality Test Exact Primality Test 8/25

9. 0,1, N = 2 R –1 Generalized sieve f 1 (x), f 2 (x), …, f s (x)  Z[x] irreducible polynomials H … p „sieving prime” … … if  i  [0, s] : p | f i (h)  h will be „beaten out” 9/25 1............ 111 1 0 ST h 1 0 1 0 1 0 h + p h + 2p … h + kp

10. 2, 3, 5, 7, 11, 13 will never be a prime factor of f i (x) (i = 1, 2, 3) f 1 (x) = (h 0 + c  x)2 e – 1 Particular case 17  p < 2 T = M f 2 (x) = (h 0 + c  x)2 e + 1 h 0 = 5775 c = 30030 e = 171960 f 3 (x) = (h 0 + c  x)2 e+1 + 1 „triple-sieving” 17, 19, 23, …,  2 T/2 „small primes” ~ 51780 digits 10/25

11. 1............ 111 1 0 ST h 1 0 1 0 1 0 h + p h + 2p … h + kp f i (x)  0 (mod p) Sieving with small primes solution: h i = 1, 2, 3 After sieving the elements of H which are represented by 1 have not any „small” primefactor. 11/25

12. Multiprocessing PST 1 PST 2 PST n … ST … sieve of Eratosthenes with small primes proc 1 proc 2 proc n … ST (1) ST (2) ST (n) … Merge ST (j), j = 1, 2, …, n 12/25

13. The more the sieving primes increase, the more the efficiency of the sieve decrease, e. g. if p > N, then p can beat out at most 1 candidate from H. Sooner or later the sieve will be slower than probabilistic primality test. Probabilistic primality test: Miller – Rabin x  2 y – 1: Lucasian type testx  2 y + 1: Brillhart, Lehmer, Selfridge Exact (deterministic) primality test 13/25

14. Theoretical base 14/25 F(n)F(n)

15. The idea behind the conjecture Gauss conjectured (1792) that de la Vallée Poussin and Hadamard (1896) proved Prime number theorem if these events were independent. The probability that the numbers f 1 (n), …, f s (n) are simultaneously prime would be 15/25

16. chance that none of the integers f 1 (n), …, f s (n) is divisible by p chance that none of the integers of an s-tuple is divisible by p But, the prime combinations (s-tuples) are not random! the probability that f 1 (n), …, f s (n) are simultaneously prime 16/25

17. Let us denote by Q(a, b) the expected number of integers n  [a, b) for which f 1 (n), …, f s (n) are simultaneously prime. Then In our case f 1 (x), …, f s (x) are linear polynomials  it is easy to calculate from the constants C 2 = 0.6601618158468695739278121100145… C 1 = 1 C 3 = 0.63516699356280296543… twin prime constant 17/25

18. If we use the sieve with primes a  p < b, than the density of the prime s-tuples is increased by the factor and the number of the elements of H is decreased by this factor. In our cases this formula can be reduced: for p  1.000.033 we do the multiplications  18/25

19. How does the above mentioned calculations estimate the real values? N = 2 33 – 1 The upper bound of small primes is: 305.020.993 8.589.934.592 candidates triple-sieving with the small primes How many candidates remain? theoretical calculation: 27344542 reality: 27347222 Error < 0.01% 19/25

20. 2 33 = 8.589.934.592 candidates triple-sieving: 17  p < 2 48+ε expected ~ 16.13558453 twin and so many SG primes ~ 5.3 million of candidates Prospective value: ~ 1.37 twin and SG primes 2 GB OM 16869987339975 · 2 171960 ±1 twin primes 51779 digits 20/25

21. „The weapons” SGI Altix 3700 Intel Itanium 2 –3 MB cache –128 db processorregister –2 GB operative memory ~ 0-100 processors 21/25

22. Software Redhat GNU/Linux (ia64), kernel 2.4 Compilers (C): –GNU C Compiler (gcc) –Intel C Compiler (icc) Parallelization softwares: –PVM library –MPI library 22/25

23. „The Hunters” Csajbók, Tímea Farkas, Gábor Járai, Antal Járai, Zoltán Kasza, János 23/25

24. The Top 10 Largest Known Twin Primes in May, 2008 24/25

25. The Top 10 Largest Known Sophie Germain Primes in May, 2008 25/25