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Relevant bibliographies by topics / Primality test

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Academic literature on the topic 'Primality test'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 April 2022

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Journal articles on the topic "Primality test"

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Martino, Gabriele. "Primality Test." American Journal of Computational Mathematics 03, no. 01 (2013): 59–60. http://dx.doi.org/10.4236/ajcm.2013.31009.

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Lee, Sang-Un, and Myeong-Bok Choi. "The Primality Test." Journal of the Korea Society of Computer and Information 16, no. 8 (August 31, 2011): 103–8. http://dx.doi.org/10.9708/jksci.2011.16.8.103.

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Valluri, Maheswara Rao. "Combinatorial primality test." ACM Communications in Computer Algebra 54, no. 4 (December 2020): 129–33. http://dx.doi.org/10.1145/3465002.3465004.

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In 1879, Laisant-Beaujeux gave the following result without proof: If n is a prime, then [EQUATION] This paper provides proofs of the result of Laisant-Beaujeux in two cases explicitly: (1) If an integer of the form n = 4k + 1, k > 0 is prime, then ([EQUATION]) and (2) If an integer of the form n = 4k + 3, k ≥ 0 is prime, then [EQUATION]. In addition, the author proposes important conjectures based on the converse of the above theorems which aim to establish primality of n. These conjectures are scrutinized by the given combinatorial primality test algorithm which can also distinguish patterns of prime n whether it is of the form 4k + 1 or 4k + 3.

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Chau, H. F., and H. K. Lo. "Primality Test Via Quantum Factorization." International Journal of Modern Physics C 08, no. 02 (April 1997): 131–38. http://dx.doi.org/10.1142/s0129183197000138.

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We consider a probabilistic quantum implementation of a variation of the Pocklington–Lehmer N - 1 primality test using Shor's algorithm. O ( log 3 N log log N log log log N) elementary q-bit operations are required to determine the primality of a number N, making it (asymptotically) the fastest known primality test. Thus, the potential power of quantum mechanical computers is once again revealed.

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Zhang, Zhenxiang, Weiping Zhou, and Xianbei Liu. "A generalised Lucasian primality test." Bulletin of the Australian Mathematical Society 74, no. 3 (December 2006): 419–41. http://dx.doi.org/10.1017/s0004972700040478.

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We present a primality test for numbers of the form Mh, n = h·2n ±1 (in particular with h divisible by 15), which generalises Berrizbeitia and Berry's test for such numbers with h ≢ 0 mod 5. With our generalised test, the primality of such a number Mh, n can be proved by means of a Lucas sequence with a seed determined by h and πq — primary irreducible divisor of a prime q ≡ 1 mod 4. We call the prime q a judge of the number Mh, n. We prescribe a sequence S of 48 primes ≡ 1 mod 4 in the interval [13, 2593] such that, for all odd h = 15t < 108 and for all n < 7.3 1011, each number Mh, n has a judge q in . Comparisons with Bosma's explicit primality criteria in “a well-defined finite sense” for the case h = 3t < 105 are given.

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Moshonkin, A. G., and I. M. Khamitov. "A New Probabilistic Primality Test." Journal of Mathematical Sciences 249, no. 1 (July 4, 2020): 79–84. http://dx.doi.org/10.1007/s10958-020-04920-z.

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Couveignes, Jean-Marc, Tony Ezome, and Reynald Lercier. "A faster pseudo-primality test." Rendiconti del Circolo Matematico di Palermo 61, no. 2 (May 16, 2012): 261–78. http://dx.doi.org/10.1007/s12215-012-0088-0.

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Ega Gradini. "COMPARISON STUDY OF FERMAT, SOLOVAY-STRASSEN AND MILLER-RABIN PRIMALITY TEST USING MATHEMATICA 6.0." Visipena Journal 3, no. 1 (June 30, 2012): 1–10. http://dx.doi.org/10.46244/visipena.v3i1.48.

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This paper presents three primality tests; Fermat test, Solovay-Strassen test, and Rabin-Miller test. Mathematica software is used to carry out the primality tests. The application of Fermat’s Litle Theorem as well as Euler’s Theorem on the tests was also discussed and this leads to the concept of pseudoprime. This paper is also discussed some results on pseudoprimes with certain range and do quantitative comparison. Those primality tests need to be evaluated in terms of its ability to compute as well as correctness in determining primality of given numbers. The answer to this is to create a source codes for those tests and evaluate them by using Mathematica 6.0. Those are Miller-Rabin test, Solovay-Strassen test, Fermat test and Lucas-Lehmer test. Each test was coded using an algorithm derived from number theoretic theorems and coded using the Mathematica version 6.0. Miller-Rabin test, SolovayStrassen test, and Fermat test are probabilistic tests since they cannot certainly identify the given number is prime, sometimes they fail. Using Mathematica 6.0, comparison study of primality test has been made and given the Miller- Rabin test as the most powerful test than other.

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Ishmukhametov, Shamil Talgatovich, Bulat Gazinurovich Mubarakov, and Ramilya Gakilevna Rubtsova. "On the Number of Witnesses in the Miller–Rabin Primality Test." Symmetry 12, no. 6 (June 1, 2020): 890. http://dx.doi.org/10.3390/sym12060890.

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In this paper, we investigate the popular Miller–Rabin primality test and study its effectiveness. The ability of the test to determine prime integers is based on the difference of the number of primality witnesses for composite and prime integers. Let W ( n ) denote the set of all primality witnesses for odd n. By Rabin’s theorem, if n is prime, then each positive integer a < n is a primality witness for n. For composite n, the power of W ( n ) is less than or equal to φ ( n ) / 4 where φ ( n ) is Euler’s Totient function. We derive new exact formulas for the power of W ( n ) depending on the number of factors of tested integers. In addition, we study the average probability of errors in the Miller–Rabin test and show that it decreases when the length of tested integers increases. This allows us to reduce estimations for the probability of the Miller–Rabin test errors and increase its efficiency.

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Lee, Sang-Un. "A Step-by-Step Primality Test." Journal of the Institute of Webcasting, Internet and Telecommunication 13, no. 3 (June 30, 2013): 103–9. http://dx.doi.org/10.7236/jiibc.2013.13.3.103.

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Dissertations / Theses on the topic "Primality test"

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Siracusa, Mia. "Primality Testing." Scholarship @ Claremont, 2017. http://scholarship.claremont.edu/scripps_theses/1073.

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Hammad, Yousef Bani. "Novel Methods for Primality Testing and Factoring." Queensland University of Technology, 2005. http://eprints.qut.edu.au/16142/.

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From the time of the Greeks, primality testing and factoring have fascinated mathematicians, and for centuries following the Greeks primality testing and factorization were pursued by enthusiasts and professional mathematicians for their intrisic value. There was little practical application. One example application was to determine whether or not the Fermat numbers, that is, numbers of the form F;, = 2'" + 1 were prime. Fermat conjectured that for all n they were prime. For n = 1,2,3,4, the Fermat numbers are prime, but Euler showed that F; was not prime and to date no F,, n 2 5 has been found to be prime. Thus, for nearly 2000 years primality testing and factorization was largely pure mathematics. This all changed in the mid 1970's with the advent of public key cryptography. Large prime numbers are used in generating keys in many public key cryptosystems and the security of many of these cryptosystems depends on the difficulty of factoring numbers with large prime factors. Thus, the race was on to develop new algorithms to determine the primality or otherwise of a given large integer and to determine the factors of given large integers. The development of such algorithms continues today. This thesis develops both of these themes. The first part of this thesis deals with primality testing and after a brief introduction to primality testing a new probabilistic primality algorithm, ALI, is introduced. It is analysed in detail and compared to Fermat and Miller-Rabin primality tests. It is shown that the ALI algorithm is more efficient than the Miller-Rabin algorithm in some aspects. The second part of the thesis deals with factoring and after looking closely at various types of algorithms a new algorithm, RAK, is presented. It is analysed in detail and compared with Fermat factorization. The RAK algorithm is shown to be significantly more efficient than the Fermat factoring algorithm. A number of enhancements is made to the basic RAK algorithm in order to improve its performance. The RAK algorithm with its enhancements is known as IMPROVEDRAK. In conjunction with this work on factorization an improvement to Shor's factoring algorithm is presented. For many integers Shor's algorithm uses a quantum computer multiple times to factor a composite number into its prime factors. It is shown that Shor's alorithm can be modified in a way such that the use of a quantum computer is required just once. The common thread throughout this thesis is the application of factoring and primality testing techniques to integer types which commonly occur in public key cryptosystems. Thus, this thesis contributes not only in the area of pure mathematics but also in the very contemporary area of cryptology.

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Kasarabada, Yasaswy. "A Verilog Description and Efficient Hardware Implementation of the Baillie-PSW Primality Test." University of Cincinnati / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1471347471.

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Giostra, Sara. "Il test di primalità aks." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/9033/.

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La tesi presenta l'algoritmo AKS, deterministico e polinomiale, scoperto dai matematici Agrawal, Kayal e Saxena nel 2002. Esso si basa su una generalizzazione del Piccolo Teorema di Fermat all'anello dei polinomi a coefficienti in Zp.

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Wedeniwski, Sebastian. "Primality tests on commutator curves." [S.l. : s.n.], 2001. http://deposit.ddb.de/cgi-bin/dokserv?idn=963295438.

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Arnault, François. "Sur quelques tests probabilistes de primalité." Poitiers, 1993. http://www.theses.fr/1993POIT2317.

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Nous étudions dans cette thèse quelques tests probabilistes de primalité, en particulier ceux qui, vraisemblablement à cause de leur simplicité et de leur rapidité d'éxécution, sont implantés dans les systèmes de calcul formel usuels. Nous commençons par présenter dans le chapitre 1 le test probabiliste de primalité le plus connu : le test de Rabin. Nous rappelons entre autres le théorème de Rabin, qui permet de majorer la probabilité d'échec de ce test. Nous donnons dans les chapitres 3 et 6 deux méthodes permettant de construire des nombres composés déclarés premiers par le test de Rabin des systèmes de calcul formel comme Axiom et Maple. Quelques rappels sur les lois de réciprocité, utiles pour la première de ces méthodes sont rassemblés dans le chapitre 2. Nous étudions aussi le test de Lucas (chapitre 4), d'un point de vue aussi algébrique que possible, en mettant en valeur l'analogie avec les pseudo-premiers classiques. Nous démontrons un analogue, pour les pseudo-premiers de Lucas, du théorème de Rabin. Précisément, nous montrons que, mises à part quelques rares exceptions bien cernées, le nombre de couples(P,Q) pour lesquels un nombre composé N est pseudo-premier de Lucas est majoré par 4N/15. Nous montrons aussi dans le chapitre 6 que l'une des méthodes proposées pour construire des pseudo-premiers forts classiques s'adapte pour produire des pseudo-premiers forts de Lucas. Nous étudions dans le chapitre 5, une fonction introduite dans le chapitre précédent et qui décrit le nombre d'éléments de norme 1 dans l'anneau des entiers d'un corps quadratique par un entier rationnel. Enfin, nous terminons par un aperçu de quelques autres tests probabilistes de primalité, et utilisons dans quelques cas particuliers, des variantes de la méthodedu chapitre 6 pour produire des nombres admissibles pour un test dû à Adams, Kurtz, Schanks et Williams, et pour produire des pseudo-premiers elliptiques forts.

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Morain, François. "Courbes elliptiques et tests de primalité." Lyon 1, 1990. http://www.theses.fr/1990LYO10170.

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Nous decrivons dans cette these l'application de la theorie des courbes elliptiques definies sur les corps finis a la construction d'algorithmes efficaces de primalite exacte. Nous faisons le lien entre le probleme de la representation des nombres premiers par des formes quadratiques binaires et la theorie du corps de classe. A ce propos, nous donnons un algorithme rapide de construction du corps de classe d'un corps quadratique imaginaire a l'aide des fonctions de weber. Nous en deduisons le calcul des invariants des courbes elliptiques a multiplication complexe dans un corps fini en resolvant par radicaux l'equation de definition du corps de classe, dans le corps des complexes d'abord, modulo un nombre premier ensuite. Nous montrons comment generaliser les algorithmes de preuve de primalite les plus classiques (reciproques du theoreme de fermat) en utilisant les courbes elliptiques. A l'encontre de son concurrent le plus serieux (sommes de jacobi), l'algorithme qui en resulte produit un certificat de primalite. D'un point de vue pratique, nous detaillons toutes les phases de l'implantation de l'algorithme, d'abord sur une station de travail, puis sur plusieurs stations d'une maniere distribuee. A chaque etape, nous presentons les meilleurs algorithmes connus pour resoudre chaque probleme particulier (calculs sur les courbes elliptiques, recherche de racines de polynomes modulo un nombre premier,. . . ). Nous decrivons egalement l'utilisation d'un multiplicateur hardware pour le calcul du produit de grands entiers, qui permet d'accelerer considerablement les calculs. Enfin, nous utilisons le programme pour la recherche de nombres premiers de cent chiffres (utiles en cryptographie) et pour la certification de nombres de trois cents a trois mille chiffres

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Breitenbacher, Dominik. "Paralelizace faktorizace celých čísel z pohledu lámání RSA." Master's thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2015. http://www.nusl.cz/ntk/nusl-234905.

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This paper follows up the factorization of integers. Factorization is the most popular and used method for RSA cryptoanalysis. The SIQS was chosen as a factorization method that will be used in this paper. Although SIQS is the fastest method (up to 100 digits), it can't be effectively computed at polynomial time, so it's needed to look up for options, how to speed up the method as much as possible. One of the possible ways is paralelization. In this case OpenMP was used. Other possible way is optimalization. The goal of this paper is also to show, how easily is possible to use paralelizion and thanks to detailed analyzation the source codes one can reach relatively large speed up. Used method of iterative optimalization showed itself as a very effective tool. Using this method the implementation of SIQS achieved almost 100 multiplied speed up and at some parts of the code even more.

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Ezome, Mintsa Tony Mack Robert. "Courbes elliptiques, cyclotomie et primalité." Toulouse 3, 2010. http://thesesups.ups-tlse.fr/825/.

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L'information est très précieuse, c'est pourquoi au moment de la stocker ou de la transmettre, il est nécessaire de la protéger. La factorisation des grands entiers est un problème diffcile, elle constitue ainsi une base de sécurité en cryptographie asymétrique. Il est donc très utile de pouvoir déterminer la primalité de grands entiers afin de construire des entiers difficilement factorisables qui pourront servir en cryptographie asymétrique. Pour ce faire on a recours aux tests de primalité. Le test AKS (inventé par Agrawal, Kayal et Saxena) est un algorithme polynômial déterministe de preuve de primalité qui a été publié en Août 2002 ("Primes is in P"). L'algorithme ECPP (Elliptic Curves Primality Proving) est un test de primalité probabiliste. Il a été proposé par A. O. L Atkin en 1988 et c'est l'un des tests de primalité les plus puissants utilisés en pratique. L'objet de cette thèse est de donner un critère de primalité de type AKS qui repose sur un anneau des périodes elliptiques. Un tel anneau est obtenu comme anneau résiduel le long d'une section de torsion d'une courbe elliptique définie sur Z/nZ. Cette section joue le rôle dévolu à la racine de l'unité dans le test AKS d'origine. Après avoir énoncé un critère général de primalité en termes d'extension étale de Z/nZ munie d'un automorphisme, nous montrons comment construire de telles extensions à partir d'isogénies entre courbes elliptiques modulo n
Information is very precious, this is the reason why it must be protected both in databasis and during transmission. Integer factoring is a diffcult problem and a cornerstone for safety in asymmetric cryptography. Thus it is very important to be able to check for the primality of big integers for asymetric cryptography. To do this we use primality tests. The AKS test is a deterministic polynomial time primality proving algorithm proposed by Agrawal, Kayal and Saxena in August 2002 ('Primes is in P'). The Elliptic Curves Primality Proving (ECPP), proposed by A. O. L. Atkin in 1988, is a probabilistic test. It is one of the most powerful primality tests that is used in practice. The purpose of this thesis is to give an elliptic version of the AKS primality criterion involving a ring of elliptic periods. Such a ring is obtained as a residue ring at a torsion section on an elliptic curve defined on Z/nZ. This section plays the role of the root of unity in the original AKS test. We give a general criterion in terms of etale extensions of Z/nZ equipped with an automorphism, and we show how to build such extensions using isogenies between elliptic curves modulo n

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Bronder, Justin S. "The AKS Class of Primality Tests: A Proof of Correctness and Parallel Implementation." Fogler Library, University of Maine, 2006. http://www.library.umaine.edu/theses/pdf/BronderJS2006.pdf.

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Book chapters on the topic "Primality test"

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Stiglic, Anton. "Primality Test." In Encyclopedia of Cryptography and Security, 958–59. Boston, MA: Springer US, 2011. http://dx.doi.org/10.1007/978-1-4419-5906-5_469.

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Liskov, Moses. "Fermat Primality Test." In Encyclopedia of Cryptography and Security, 455–56. Boston, MA: Springer US, 2011. http://dx.doi.org/10.1007/978-1-4419-5906-5_448.

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Stiglic, Anton. "Probabilistic Primality Test." In Encyclopedia of Cryptography and Security, 980. Boston, MA: Springer US, 2011. http://dx.doi.org/10.1007/978-1-4419-5906-5_472.

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Kozen, Dexter C. "Miller’s Primality Test." In The Design and Analysis of Algorithms, 201–5. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4612-4400-4_38.

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Atkin, A. "Intelligent primality test offer." In AMS/IP Studies in Advanced Mathematics, 1–11. Providence, Rhode Island: American Mathematical Society, 1997. http://dx.doi.org/10.1090/amsip/007/01.

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Sethi, Gunjan, and Harsh. "Limiting Check Algorithm: Primality Test." In Advances in Intelligent Systems and Computing, 697–705. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-8289-9_67.

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Liskov, Moses. "Miller–Rabin Probabilistic Primality Test." In Encyclopedia of Cryptography and Security, 784–85. Boston, MA: Springer US, 2011. http://dx.doi.org/10.1007/978-1-4419-5906-5_461.

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Kozen, Dexter C. "Analysis of Miller’s Primality Test." In The Design and Analysis of Algorithms, 206–10. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4612-4400-4_39.

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Bressoud, David M. "Primitive Roots and a Test for Primality." In Factorization and Primality Testing, 123–40. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-4544-5_9.

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Dietzfelbinger, Martin. "5. The Miller-Rabin Test." In Primality Testing in Polynomial Time, 73–84. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-540-25933-6_5.

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Conference papers on the topic "Primality test"

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Kaltofen, E., T. Valente, and N. Yui. "An improved Las Vegas primality test." In the ACM-SIGSAM 1989 international symposium. New York, New York, USA: ACM Press, 1989. http://dx.doi.org/10.1145/74540.74545.

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Asaduzzaman, Abu, Chok M. Yip, and Anindya Maiti. "CUDA-assisted energy-efficient primality test." In SOUTHEASTCON 2014. IEEE, 2014. http://dx.doi.org/10.1109/secon.2014.6950678.

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da Silva, Joao Carlos Leandro. "Carmichael numbers and a new primality test." In Electronics Engineers in Israel (IEEEI 2010). IEEE, 2010. http://dx.doi.org/10.1109/eeei.2010.5662138.

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Daghigh, Hassan, Amir Mehdi Yazdani kashani, and Ruholla Khodakaramian Gilan. "Primality test for mersenne numbers using elliptic curves." In 2014 11th International ISC Conference on Information Security and Cryptology (ISCISC). IEEE, 2014. http://dx.doi.org/10.1109/iscisc.2014.6994029.

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Purdy, Carla, Yasaswy Kasarabada, and George Purdy. "Hardware implementation of the Baillie-PSW primality test." In 2017 IEEE 60th International Midwest Symposium on Circuits and Systems (MWSCAS). IEEE, 2017. http://dx.doi.org/10.1109/mwscas.2017.8053007.

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Galligo, André, and Stephen Watt. "A numerical absolute primality test for bivariate polynomials." In the 1997 international symposium. New York, New York, USA: ACM Press, 1997. http://dx.doi.org/10.1145/258726.258788.

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Le Masle, Adrien, Wayne Luk, and Csaba Andras Moritz. "Parametrized hardware architectures for the Lucas primality test." In 2011 International Conference on Embedded Computer Systems: Architectures, Modeling, and Simulation (SAMOS XI). IEEE, 2011. http://dx.doi.org/10.1109/samos.2011.6045453.

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Li, Huiheng, and Rui Wang. "A primality test based on modular hyperbolic curves." In 2020 2nd International Conference on Machine Learning, Big Data and Business Intelligence (MLBDBI). IEEE, 2020. http://dx.doi.org/10.1109/mlbdbi51377.2020.00013.

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Hosung Jo and Heejin Park. "Performance analysis and improvement of JPV primality test for smart IC cards." In 2014 International Conference on Big Data and Smart Computing (BIGCOMP). IEEE, 2014. http://dx.doi.org/10.1109/bigcomp.2014.6741451.

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Dordevic, Goran, and Milan Markovic. "On Optimization of Miller-Rabin Primality Test on TI TMS320C54x Signal Processors." In 2007 14th International Workshop in Systems, Signals and Image Processing and 6th EURASIP Conference focused on Speech and Image Processing, Multimedia Communications and Services - EC-SIPMCS 2007. IEEE, 2007. http://dx.doi.org/10.1109/iwssip.2007.4381195.

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