Close to Uniform Prime Number Generation With Fewer Random Bits

In this paper, we analyze several variants of a simple method for generating prime numbers with fewer random bits. To generate a prime $p$ less than $x$, the basic idea is to fix a constant $q\propto x^{1-\varepsilon}$, pick a uniformly random $a<q$ coprime to $q$, and choose $p$ of the form $a+t\cdot q$, where only $t$ is updated if the primality test fails. We prove that variants of this approach provide prime generation algorithms requiring few random bits and whose output distribution is close to uniform, under less and less expensive assumptions: first a relatively strong conjecture by H.L. Montgomery, made precise by Friedlander and Granville; then the Extended Riemann Hypothesis; and finally fully unconditionally using the Barban-Davenport-Halberstam theorem. We argue that this approach has a number of desirable properties compared to previous algorithms.Comment: Full version of ICALP 2014 paper. Alternate version of IACR ePrint Report 2011/48

Number Field Sieve with Provable Complexity

In this thesis we give an in-depth introduction to the General Number Field Sieve, as it was used by Buhler, Lenstra, and Pomerance, before looking at one of the modern developments of this algorithm: A randomized version with provable complexity. This version was posited in 2017 by Lee and Venkatesan and will be preceded by ample material from both algebraic and analytic number theory, Galois theory, and probability theory.Comment: MSc Thesis, 113 pages, 1 tabl

Divisibility, Smoothness and Cryptographic Applications

This paper deals with products of moderate-size primes, familiarly known as smooth numbers. Smooth numbers play a crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth numbers play a pivotal role

Тестирование чисел на простоту: теория и практика

Наводиться класифікація та огляд основних алгоритмів тестування чисел на простоту, а також їх порівняльний аналіз та рекомендації з побудови практичних засобів.Classification, review of main primality test algorithms, comparative analysis and recommendation of mean building are given in the article

On Generating Prime Numbers Efficiently

The prime numbers can be considered as the building blocks of natural numbers, having innumerable applications in number theory and cryptography. There exist multiple different sieving algorithms for the generation of prime numbers. In this thesis, an elementary modular result is utilized to construct an analytically useful generator function and its inverse function. The functions are used to generate a (log)log-linear time complexity prime sieving algorithm which is further optimized to be of linear time complexity. The constructed algorithms and their operation are studied and the linear implementations in JS, Python and C++ are compared to other prime sieves.Alkulukuja voidaan pitää luonnollisten lukujen rakennuspalikoina joilla on lukemattomia sovelluksia lukuteoriassa ja kryptografiassa. Alkulukujen luomiseen on olemassa useita erilaisia seulonta-algoritmeja. Tässä opinnäytetyössä käytetään modulaarista perustulosta analyyttisesti hyödyllisten kehitysfunktion ja sen käänteisfunktion luomiseen. Funktioiden avulla luodaan aikakompleksisuudeltaan (log)log-lineaarinen alkulukuseula, joka optimoidaan lineaariseksi. Rakennettuja algoritmeja ja niiden toimintaa tarkastellaan ja lineaarista implementaatiota JS, Python ja C++ ohjelmointikielillä verrataan toisiin alkulukuseuloihin

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

Three Dogmas of First-Order Logic and some Evidence-based Consequences for Constructive Mathematics of differentiating between Hilbertian Theism, Brouwerian Atheism and Finitary Agnosticism

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. We then adopt what may be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences