Evaluation and comparison of two efficient probabilistic primality testing algorithms

AbstractWe analyse two recent probabilistic primality testing algorithms; the first one is derived from Miller [6] in a formulation given by Rabin [7], and the second one is from Solovay and Strassen [9]. Both decide whether or not an odd number n is prime in time O(m, log n M(n)) with an error probability less than αm, for some 0≤a<12. Our comparison shows that the first algorithm is always more efficient than the second, both in probabilistic and algorithmic terms

Computationally efficient search for large primes

To satisfy the speed of communication and to meet the demand for the continuously larger prime numbers, the primality testing and prime numbers generating algorithms require continuous advancement. To find the most efficient algorithm, a need for a survey of methods arises. Concurrently, an urge for the analysis of algorithms\u27 performances emanates. The critical criteria in the analysis of the prime numbers generation are the number of probes, number of generated primes, and an average time required in producing one prime. Hence, the purpose of this thesis is to indicate the best performing algorithm. The survey the methods, establishment of the comparison criteria, and comparison of approaches are the required steps to find the best performing algorithm. In the first step of this research paper the methods were surveyed and classified using the approach described in Menezes [66]. Wifle chapter 2 sorted, described, compared, and summarized primality testing methods, chapter 3 sorted, described, compared, and summarized prime numbers generating methods. In the next step applying a uniform technique, the computer programs were written to the selected algorithms. The programs were installed on the Unix operating system, running on the Sun 5.8 server to perform the computer experiments. The computer experiments\u27 results pertaining to the selected algorithms, provided required parameters to compare the algorithms\u27 performances. The results from the computer experiments were tabulated to compare the parameters and to indicate the best performing algorithm. Survey of methods indicated that the deterministic and randomized are the main approaches in prime numbers generation. Random number generation found application in the cryptographic keys generation. Contemporaneously, a need for deterministically generated provable primes emerged in the code encryption, decryption, and in the other cryptographic areas. The analysis of algorithms\u27 performances indicated that the prime nurnbers generated through the randomized techniques required smaller number of probes. This is due to the method that eliminates the non-primes in the initial step, that pre-tests randomly generated primes for possible divisibility factors. Analysis indicated that the smaller number of probes increases algorithm\u27s efficiency. Further analysis indicated that a ratio of randomly generated primes to the expected number of primes, generated in the specific interval is smaller than the deterministically generated primes. In this comparison the Miller-Rabin\u27s and the Gordon\u27s algorithms that randomly generate primes were compared versus the SFA and the Sequences Containing Primes. The name Sequences Containing Primes algorithm is abbreviated in this thesis as 6kseq. In the interval [99000,1000001 the Miller Rabin method generated 57 out of 87 expected primes, the SFA algorithm generated 83 out of 87 approximated primes. The expected number of primes was computed using the approximation n/ln(n) presented by Menezes [66]. The average consumed time of originating one prime in the [99000, 100000] interval recorded 0.056 [s] for Miller-Rabin test, 0.0001 [s] for SFA, and 0.0003 [s] for 6kseq. The Gordon\u27s algorithm in the interval [1,100000] required 100578 probes and generated 32 out of 8686 expected number of primes. Algorithm Parametric Representation of Composite Twins and Generation of Prime and Quasi Prime Numbers invented by Doctor Verkhovsky [1081 verifies and generates primes and quasi primes using special mathematical constructs. This algorithm indicated best performance in the interval [1,1000] generating and verifying 3585 variances of provable primes or quasi primes. The Parametric Representation of Composite Twins algorithm consumed an average time per prime, or quasi prime of 0.0022315 [s]. The Parametric Representation of Composite Twins and Generation of Prime and Quasi Prime Numbers algorithm implements very unique method of testing both primes and quasi-primes. Because of the uniqueness of the method that verifies both primes and quasi-primes, this algorithm cannot be compared with the other primality testing or prime numbers generating algorithms. The ((a!)^2)*((-1^b) Function In Generating Primes algorithm [105] developed by Doctor Verkhovsky was compared versus extended Fermat algorithm. In the range of [1,10001 the [105] algorithm exhausted an average 0.00001 [s] per prime, originated 167 primes, while the extended Fermat algorithm also produced 167 primes, but consumed an average 0.00599 [s] per prime. Thus, the computer experiments and comparison of methods proved that the SFA algorithm is deterministic, that originates provable primes. The survey of methods and analysis of selected approaches indicated that the SFA sieve algorithm that sequentially generates primes is computationally efficient, indicated better performance considering the computational speed, the simplicity of method, and the number of generated primes in the specified intervals

Solovay-Strassenův test prvočíselnosti

Táto práca sa venuje algoritmu na testovanie prvočíselnosti celého čísla n, založeného na výpočte Jacobiho symbolu, nazývaného Solovay-Strassenov test. Po sformulovaní samotného algoritmu odhadneme pravdepodobnosť, že testo- vané číslo n je skutočne prvočíslo, ak to o ňom vyhlásil Solovay-Strassenov test. Práca ponúka aj vylepšenie výpočtu využívajúce, že n nie je deliteľné konkrét- nymi malými prvočíslami, čo môžeme veľmi jednoducho overiť. V záverečnej časti ide o konštrukciu vlastného testu, ako obdoby k Solovay-Strassenovmu testu, za- loženého na výpočte kvartického symbolu. 1This thesis studies the Solovay-Strassen test for primality of an integer n, which is based on the Jacobi symbol. After formulating the basic algorithm, we compute the probability that the number n being tested is really a prime number if the Solovay-Strassen test declared it so. We further improve the computation of the probability under the assumption that n is not divisible by specific small primes, which can be easily verified. Finally, we construct a new test, as an analogy of the Solovay-Strassen test, based on the quartic residue symbol. 1Department of AlgebraKatedra algebryFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Euler pseudoprimes for half of the bases

We prove that an odd number is an Euler pseudoprime for exactly one half of the admissible bases if and only if it is a special Carmichael number.Comment: 4 pages. To appear in Integer