New Implementations for Tabulating Pseudoprimes and Liars

Whether it is applied to primality test or cryptography, pseudoprimes are one of the most important topics in number theory. Regarding the study of strong pseudoprimes, there are two problems which mathematicians have been working on: 1. Given a, b, find all a-spsp up to b. 2. Given an odd composite n, find all a -n such that n is an a-spsp. where n = a-spsp means n is a strong pseudoprime to base a, and a is a strong liar of n. The two problems are respectively referred to as the tabulation of strong pseudoprimes and the tabulation of strong liars. The main focus of my work in this research project is on the tabulation of strong liars. This can be achieved by the application of the multiplicative group modulo n, denoted by (Z=nZ). Instead of checking each potential candidate, we can actually construct the set of Fermat liars, a \weaker version than strong liars, from the bottom up with the help of the primitive roots of (Z=pZ) for all prime factors p of n. We then sieve out the set of strong liars with Millerwitness() function in NTL library. By implementing the algorithms with appropriate data structures, I verified that in most cases the runtimes have been improved compared to previous algorithms or brute force. This is to be expected, since they have less computational complexities theoretically. All implementations of the algorithms in this research project are in C++. In 2010, Professors Mark Liffiton and Andrew Shallue built a new computer system for research purposes, known as Hyperion. This is essentially a cluster of 8 computers, also referred to as nodes, which are able to finish complex distributed computations. The operating systems installed on all the nodes in the cluster is Linux. By submitting Shell scripts which include instructions from its front end, I can grab a cup of coffee while the programs are running on the cluster. This feature is particularly beneficial for this project since the programs sometimes take tens of minutes to return all the outputs

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

Strengthening the Baillie-PSW primality test

The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the parameters are chosen in an appropriate way. Here, we describe a significant strengthening of this test that comes at almost no additional computational cost. This is achieved by including in the test what we call Lucas-V pseudoprimes, of which there are only five less than $10^{15}$.Comment: 25 page

O desenvolvimento de um algoritmo eficiente em busca de números pseudo-primos fortes

The problem of searching for strictly pseudoprime numbers is relevant in the field of number theory, and it also has a number of applications in cryptography: in particular, with the help of numbers in this class one can strengthen the efficiency of the Miller-Rabin simplicity test by transforming it from probabilistic into deterministic. At the present time, several algorithms for constructing sequences of such numbers are known, but they have a rather high complexity, which makes it impossible to obtain strictly pseudoprime numbers of large magnitude in an acceptable time. The theme of this paper is the construction of strictly pseudoprime numbers of the special form n = pq = (u + 1) (2u + 1), where p, q are prime numbers, u is a natural number. Numbers of this kind are present in the sequence Ψk, used to estimate the number of iterations in the Miller-Rabin simplicity test. We denote by Fk the smallest odd composite number of the above-mentioned type, which successfully passes the Miller-Rabin test with k first prime numbers. The paper proposes a new algorithm for constructing Fk numbers, gives data on its speed and efficiency on the memory used, and specifies the features of the software implementation.El problema de la búsqueda de números estrictamente pseudoprimos es relevante en el campo de la teoría de números, y también tiene varias aplicaciones en criptografía: en particular, con la ayuda de números en esta clase se puede fortalecer la eficiencia de la simplicidad de Miller-Rabin Prueba transformándolo de probabilístico en determinístico. En la actualidad, se conocen varios algoritmos para construir secuencias de tales números, pero tienen una complejidad bastante alta, lo que hace imposible obtener números estrictamente pseudoprimos de gran magnitud en un tiempo aceptable. El tema de este artículo es la construcción de números estrictamente pseudoprime de la forma especial n = pq = (u + 1) (2u + 1), donde p, q son números primos, u es un número natural. Los números de este tipo están presentes en la secuencia Ψk, utilizada para estimar el número de iteraciones en la prueba de simplicidad de Miller-Rabin. Denotamos por Fk el número compuesto impar más pequeño del tipo mencionado anteriormente, que pasa con éxito la prueba de Miller-Rabin con k primeros números primos. El documento propone un nuevo algoritmo para construir números Fk, proporciona datos sobre su velocidad y eficiencia en la memoria utilizada y especifica las características de la implementación del software.O problema de encontrar números pseudoprimos é estritamente relevante no campo da teoria dos números, e também tem muitas aplicações em criptografia: em particular, com a ajuda de números nesta classe pode fortalecer a eficiência da simplicidade de Miller Teste de Rabin transformando-o de probabilístico para determinístico. Atualmente, vários algoritmos são conhecidos por construir seqüências de tais números, mas eles têm uma complexidade bastante alta, o que torna impossível obter números estritamente pseudoprimo de grande magnitude em um tempo aceitável. O assunto deste artigo é a construção de números estritamente pseudoprimo forma especial com n = pq = (L + 1) (2u + 1) em que p, q são números primos, u é um número natural. Números desse tipo estão presentes na seqüência Ψk, usada para estimar o número de iterações no teste de simplicidade de Miller-Rabin. Denotamos por Fk o menor número composto ímpar do tipo mencionado acima, que passa com sucesso no teste de Miller-Rabin com k primeiros números primos. O documento propõe um novo algoritmo para a construção de números Fk, fornece dados sobre sua velocidade e eficiência na memória utilizada e especifica as características da implementação do software

Microcomputer Algorithms for Prime Number Testing

This paper gives a survey of different methods of prime number testing. Emphasis has been given to algorithms based upon Fermat\u27s Theorem: if p is an odd prime number, then p divides ap-a. All of the computer programs described in this paper have been written for use on microcomputers and so the feasibility of using microcomputers is also discussed. Finally, numbers of various forms have been considered for primality with special attention given to Mersenne and Fermat numbers. It is hoped that some of the information contained in this paper may provide worthwhile enrichment ideas for mathematics educators