Prime H unting

1. Prime Hunting Gábor Farkas Department of Computer Algebra Faculty of Informatics Eötvös Loránd University Jena, Germany26. May 2008.

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

3. Computational NumberTheory Mathematics Informatics 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. k = 5 {2, 5, 11, 23, 47} k = 6 {89, 179, 359, 719, 1439, 2879} SG 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 . Naturally we want the probability of a successful cycling attack on the RSA to be as small as possible. How do we choose the parameters ? The best choice: 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. 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. Generalized sieve f1(x), f2(x), …, fs(x)  Z[x] irreducible polynomials p „sieving prime” H … … … 0, 1, N = 2R–1 0 0 0 1 0 1 1 1 … h h + 2p h + p h + kp ST if  i  [0, s] : p | fi(h)  h will be „beaten out” 9/25

10. Particular case f1(x) = (h0 + cx)2e – 1 h0 = 5775 f2(x) = (h0+ cx)2e + 1 „triple-sieving” c= 30030 f3(x) = (h0+ cx)2e+1+ 1 e= 171960 ~51780 digits 2, 3, 5, 7, 11, 13 will never be a prime factor of fi(x) (i = 1, 2, 3) 17p < 2T= M 17, 19, 23, …,  2T/2 „small primes” 10/25

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

12. Multiprocessing sieve of Eratosthenes with small primes PST1 PST2 … PSTn ST ST … ST proc1 proc2 procn … 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 Exact (deterministic) primality test x2y – 1: Lucasian type test x2y + 1: Brillhart, Lehmer, Selfridge 13/25

14. Theoretical base F(n) 14/25

15. The idea behind the conjecture Prime number theorem Gauss conjectured (1792) that de la Vallée Poussin and Hadamard (1896) proved The probability that the numbers f1(n), …, fs(n) are simultaneously prime would be if these events were independent. 15/25

16. But, the prime combinations (s-tuples) are not random! chance that none of the integers f1(n), …, fs(n) is divisible by p chance that none of the integers of an s-tuple is divisible by p the probability that f1(n), …, fs(n) are simultaneously prime 16/25

17. Let us denote by Q(a, b) the expected number of integers n[a, b) for which f1(n), …, fs(n) are simultaneously prime. Then In our case f1(x), …, fs(x) are linear polynomials it is easy to calculate from the constants twin prime constant C1 = 1 C2 = 0.6601618158468695739278121100145… C3 = 0.63516699356280296543… 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 = 233 – 1 8.589.934.592 candidates The upper bound of small primes is: 305.020.993 triple-sieving with the small primes How many candidates remain? reality: 27347222 theoretical calculation: 27344542 Error < 0.01% 19/25

20. 233 = 8.589.934.592 candidates 2 GB OM expected ~ 16.13558453 twin and so many SG primes triple-sieving: 17  p < 248+ε ~ 5.3 million of candidates Prospective value: ~ 1.37 twin and SG primes 51779 digits 16869987339975 · 2171960 ±1 twin primes 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), kernel2.4 • Compilers (C): • GNU C Compiler (gcc) • Intel C Compiler (icc) • Parallelization softwares: • PVM library • MPI library 22/25

23. „The Hunters” Járai, Antal Csajbók, Tímea Járai, Zoltán Farkas, Gábor 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