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

Grained integers and applications to cryptography

To meet the requirements of the modern communication society, cryptographic techniques are of central importance. In modern cryptography, we try to build cryptographic primitives, whose security can be reduced to solving a particular number theoretic problem for which no fast algorithmic method is known by now. Thus, any advance in the understanding of the nature of such problems indirectly gives insight in the analysis of some of the most practical cryptographic techniques. In this work we analyze exactly this aspect much more deeply: How can we use some of the purely theoretical results in number theory to answer very practical questions on the security of widely used cryptographic algorithms and how can we use such results in concrete implementations? While trying to answer these kinds of security-related questions, we always think two-fold: From a cryptographic, security-ensuring perspective and from a cryptanalytic one. After we outlined -- with a special focus on the historical development of these results -- the necessary analytic and algorithmic foundations of number theory, we first delve into the question how point addition on certain elliptic curves can be done efficiently. The resulting formulas have their application in the cryptanalysis of crypto systems that are insecure if factoring integers can be done efficiently. The rest of the thesis is devoted to the study of integers, all of whose prime factors are neither too small nor too large. We show with the help of two applications how one can use the properties of such kinds of integers to answer very practical questions in the design and the analysis of cryptographic primitives: The optimization of a hardware-realization of the cofactorization step of the General Number Field Sieve and the analysis of different standardized key-generation algorithms

LNCS

We argue that the time is ripe to investigate differential monitoring, in which the specification of a program's behavior is implicitly given by a second program implementing the same informal specification. Similar ideas have been proposed before, and are currently implemented in restricted form for testing and specialized run-time analyses, aspects of which we combine. We discuss the challenges of implementing differential monitoring as a general-purpose, black-box run-time monitoring framework, and present promising results of a preliminary implementation, showing low monitoring overheads for diverse programs

IST Austria Technical Report

We argue that the time is ripe to investigate differential monitoring, in which the specification of a program's behavior is implicitly given by a second program implementing the same informal specification. Similar ideas have been proposed before, and are currently implemented in restricted form for testing and specialized run-time analyses, aspects of which we combine. We discuss the challenges of implementing differential monitoring as a general-purpose, black-box run-time monitoring framework, and present promising results of a preliminary implementation, showing low monitoring overheads for diverse programs

Association of Christians in the Mathematical Sciences Proceedings 2017

The conference proceedings of the Association of Christians in the Mathematical Sciences biannual conference, May 31-June 2, 2017 at Charleson Southern University

Primality Tests on Commutator Curves

Das Thema dieser Dissertation sind effiziente Primzahltests. Zunächst wird die Kommutatorkurve eingeführt, die durch einen skalaren Parameter in der zweidimensionalen speziellen linearen Gruppe bestimmt wird. Nach Erforschung der Grundlagen dieser Kurve wird sie in verschiedene Pseudoprimzahltests (z.B. Fermat-Test, Solovay-Strassen-Test) eingebunden. Als wichtigster Pseudoprimzahltest ist dabei der Kommutatorkurventest zu nennen. Es wird bewiesen, dass dieser Test nach einer festen Anzahl von Probedivisionen (alle Primzahlen kleiner 80) das Ergebnis 'wahr' für eine zusammengesetzte Zahl mit einer Wahrscheinlichkeit ausgibt, die kleiner als 1/16 ist. Darüberhinaus wird bewiesen, dass der Miller-Primzahltest unter der Annahme der Korrektheit der Erweiterten Riemannschen Hypothese zur Überprüfung einer Zahl n nur noch für alle Primzahlbasen kleiner als 3/2*ln(n)^2 durchgeführt werden muss. Im Beweis des Primzahltests von G. L. Miller konnte dabei die Notwendigkeit der Erweiterten Riemannschen Hypothese auf nur noch ein Schlüssellemma eingegrenzt werden.This thesis is about efficient primality tests. First, the commutator curve which is described by one scalar parameter in the two-dimensional special linear group will be introduced. After fundamental research of of this curve, it will be included into different compositeness tests (e.g. Fermat's test, Solovay-Strassen test). The most important commutator test is the Commutator Curve Test. Besides, it will be proved that this test after a fixed number of trial divisions (all prime numbers up to 80) returns the result 'true' for a composite number with a probability less than 1/16. Moreover, it will be shown that Miller's test to check a number n only has to be carried out for all prime bases less than 3/2*ln(n)^2. This happens under the assumption that the Extended Riemann Hypothesis is true. The necessity of the Extended Riemann Hypothesis to prove the primality test of G. L. Miller can be reduced to a single key lemma

On a Family of Sequences Related to Chebyshev Polynomials

We consider the appearance of primes in a family of linear recurrence sequences labelled by a positive integer n. The terms of each sequence correspond to a particular class of Lehmer numbers, or (viewing them as polynomials in n) dilated versions of the so-called Chebyshev polynomials of the fourth kind, also known as airfoil polynomials. We prove that when the value of n is given by a dilated Chebyshev polynomial of the first kind evaluated at a suitable integer, either the sequence contains a single prime, or no term is prime. For all other values of n, we conjecture that the sequence contains infinitely many primes, whose distribution has analogous properties to the distribution of Mersenne primes among the Mersenne numbers. Similar results are obtained for the sequences associated with negative integers n, which correspond to Chebyshev polynomials of the third kind, and to another family of Lehmer numbers