Beurling Generalized Numbers
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Fast Generation of RSA Keys Using Smooth Integers
1 Fast Generation of RSA Keys using Smooth Integers Vassil Dimitrov, Luigi Vigneri and Vidal Attias Abstract—Primality generation is the cornerstone of several essential cryptographic systems. The problem has been a subject of deep investigations, but there is still a substantial room for improvements. Typically, the algorithms used have two parts – trial divisions aimed at eliminating numbers with small prime factors and primality tests based on an easy-to-compute statement that is valid for primes and invalid for composites. In this paper, we will showcase a technique that will eliminate the first phase of the primality testing algorithms. The computational simulations show a reduction of the primality generation time by about 30% in the case of 1024-bit RSA key pairs. This can be particularly beneficial in the case of decentralized environments for shared RSA keys as the initial trial division part of the key generation algorithms can be avoided at no cost. This also significantly reduces the communication complexity. Another essential contribution of the paper is the introduction of a new one-way function that is computationally simpler than the existing ones used in public-key cryptography. This function can be used to create new random number generators, and it also could be potentially used for designing entirely new public-key encryption systems. Index Terms—Multiple-base Representations, Public-Key Cryptography, Primality Testing, Computational Number Theory, RSA ✦ 1 INTRODUCTION 1.1 Fast generation of prime numbers DDITIVE number theory is a fascinating area of The generation of prime numbers is a cornerstone of A mathematics. In it one can find problems with cryptographic systems such as the RSA cryptosystem. -
Sieving for Twin Smooth Integers with Solutions to the Prouhet-Tarry-Escott Problem
Sieving for twin smooth integers with solutions to the Prouhet-Tarry-Escott problem Craig Costello1, Michael Meyer2;3, and Michael Naehrig1 1 Microsoft Research, Redmond, WA, USA fcraigco,[email protected] 2 University of Applied Sciences Wiesbaden, Germany 3 University of W¨urzburg,Germany [email protected] Abstract. We give a sieving algorithm for finding pairs of consecutive smooth numbers that utilizes solutions to the Prouhet-Tarry-Escott (PTE) problem. Any such solution induces two degree-n polynomials, a(x) and b(x), that differ by a constant integer C and completely split into linear factors in Z[x]. It follows that for any ` 2 Z such that a(`) ≡ b(`) ≡ 0 mod C, the two integers a(`)=C and b(`)=C differ by 1 and necessarily contain n factors of roughly the same size. For a fixed smoothness bound B, restricting the search to pairs of integers that are parameterized in this way increases the probability that they are B-smooth. Our algorithm combines a simple sieve with parametrizations given by a collection of solutions to the PTE problem. The motivation for finding large twin smooth integers lies in their application to compact isogeny-based post-quantum protocols. The recent key exchange scheme B-SIDH and the recent digital signature scheme SQISign both require large primes that lie between two smooth integers; finding such a prime can be seen as a special case of finding twin smooth integers under the additional stipulation that their sum is a prime p. When searching for cryptographic parameters with 2240 ≤ p < 2256, an implementation of our sieve found primes p where p + 1 and p − 1 are 215-smooth; the smoothest prior parameters had a similar sized prime for which p−1 and p+1 were 219-smooth. -
The Prime Number Theorem a PRIMES Exposition
The Prime Number Theorem A PRIMES Exposition Ishita Goluguri, Toyesh Jayaswal, Andrew Lee Mentor: Chengyang Shao TABLE OF CONTENTS 1 Introduction 2 Tools from Complex Analysis 3 Entire Functions 4 Hadamard Factorization Theorem 5 Riemann Zeta Function 6 Chebyshev Functions 7 Perron Formula 8 Prime Number Theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 2 Introduction • Euclid (300 BC): There are infinitely many primes • Legendre (1808): for primes less than 1,000,000: x π(x) ' log x © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 3 Progress on the Distribution of Prime Numbers • Euler: The product formula 1 X 1 Y 1 ζ(s) := = ns 1 − p−s n=1 p so (heuristically) Y 1 = log 1 1 − p−1 p • Chebyshev (1848-1850): if the ratio of π(x) and x= log x has a limit, it must be 1 • Riemann (1859): On the Number of Primes Less Than a Given Magnitude, related π(x) to the zeros of ζ(s) using complex analysis • Hadamard, de la Vallée Poussin (1896): Proved independently the prime number theorem by showing ζ(s) has no zeros of the form 1 + it, hence the celebrated prime number theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 4 Tools from Complex Analysis Theorem (Maximum Principle) Let Ω be a domain, and let f be holomorphic on Ω. (A) jf(z)j cannot attain its maximum inside Ω unless f is constant. (B) The real part of f cannot attain its maximum inside Ω unless f is a constant. Theorem (Jensen’s Inequality) Suppose f is holomorphic on the whole complex plane and f(0) = 1. -
Sieving by Large Integers and Covering Systems of Congruences
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 20, Number 2, April 2007, Pages 495–517 S 0894-0347(06)00549-2 Article electronically published on September 19, 2006 SIEVING BY LARGE INTEGERS AND COVERING SYSTEMS OF CONGRUENCES MICHAEL FILASETA, KEVIN FORD, SERGEI KONYAGIN, CARL POMERANCE, AND GANG YU 1. Introduction Notice that every integer n satisfies at least one of the congruences n ≡ 0(mod2),n≡ 0(mod3),n≡ 1(mod4),n≡ 1(mod6),n≡ 11 (mod 12). A finite set of congruences, where each integer satisfies at least one them, is called a covering system. A famous problem of Erd˝os from 1950 [4] is to determine whether for every N there is a covering system with distinct moduli greater than N.In other words, can the minimum modulus in a covering system with distinct moduli be arbitrarily large? In regards to this problem, Erd˝os writes in [6], “This is perhaps my favourite problem.” It is easy to see that in a covering system, the reciprocal sum of the moduli is at least 1. Examples with distinct moduli are known with least modulus 2, 3, and 4, where this reciprocal sum can be arbitrarily close to 1; see [10], §F13. Erd˝os and Selfridge [5] conjectured that this fails for all large enough choices of the least modulus. In fact, they made the following much stronger conjecture. Conjecture 1. For any number B,thereisanumberNB,suchthatinacovering system with distinct moduli greater than NB, the sum of reciprocals of these moduli is greater than B. A version of Conjecture 1 also appears in [7]. -
Factoring Integers with a Brain-Inspired Computer John V
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS 1 Factoring Integers with a Brain-Inspired Computer John V. Monaco and Manuel M. Vindiola Abstract—The bound to factor large integers is dominated • Constant-time synaptic integration: a single neuron in the by the computational effort to discover numbers that are B- brain may receive electrical potential inputs along synap- smooth, i.e., integers whose largest prime factor does not exceed tic connections from thousands of other neurons. The B. Smooth numbers are traditionally discovered by sieving a polynomial sequence, whereby the logarithmic sum of prime incoming potentials are continuously and instantaneously factors of each polynomial value is compared to a threshold. integrated to compute the neuron’s membrane potential. On a von Neumann architecture, this requires a large block of Like the brain, neuromorphic architectures aim to per- memory for the sieving interval and frequent memory updates, form synaptic integration in constant time, typically by resulting in O(ln ln B) amortized time complexity to check each leveraging physical properties of the underlying device. value for smoothness. This work presents a neuromorphic sieve that achieves a constant-time check for smoothness by reversing Unlike the traditional CPU-based sieve, the factor base is rep- the roles of space and time from the von Neumann architecture resented in space (as spiking neurons) and the sieving interval and exploiting two characteristic properties of brain-inspired in time (as successive time steps). Sieving is performed by computation: massive parallelism and constant time synaptic integration. The effects on sieving performance of two common a population of leaky integrate-and-fire (LIF) neurons whose neuromorphic architectural constraints are examined: limited dynamics are simple enough to be implemented on a range synaptic weight resolution, which forces the factor base to be of current and future architectures. -
On Distribution of Semiprime Numbers
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2014, Vol. 58, No. 8, pp. 43–48. c Allerton Press, Inc., 2014. Original Russian Text c Sh.T. Ishmukhametov, F.F. Sharifullina, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53–59. On Distribution of Semiprime Numbers Sh. T. Ishmukhametov* and F. F. Sharifullina** Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan, 420008 Russia Received January 31, 2013 Abstract—A semiprime is a natural number which is the product of two (possibly equal) prime numbers. Let y be a natural number and g(y) be the probability for a number y to be semiprime. In this paper we derive an asymptotic formula to count g(y) for large y and evaluate its correctness for different y. We also introduce strongly semiprimes, i.e., numbers each of which is a product of two primes of large dimension, and investigate distribution of strongly semiprimes. DOI: 10.3103/S1066369X14080052 Keywords: semiprime integer, strongly semiprime, distribution of semiprimes, factorization of integers, the RSA ciphering method. By smoothness of a natural number n we mean possibility of its representation as a product of a large number of prime factors. A B-smooth number is a number all prime divisors of which are bounded from above by B. The concept of smoothness plays an important role in number theory and cryptography. Possibility of using the concept in cryptography is based on the fact that the procedure of decom- position of an integer into prime divisors (factorization) is a laborious computational process requiring significant calculating resources [1, 2]. -
There Are Infinitely Many Mersenne Primes
Scan to know paper details and author's profile There are Infinitely Many Mersenne Primes Fengsui Liu ZheJiang University Alumnus ABSTRACT From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained. Then we invent a recursive sieve method for exponents of Mersenne primes. This is a novel algorithm on sets of natural numbers. The algorithm mechanically yields a sequence of sets of exponents of almost Mersenne primes, which converge to the set of exponents of all Mersenne primes. The corresponding cardinal sequence is strictly increasing. We capture a particular order topological structure of the set of exponents of all Mersenne primes. The existing theory of this structure allows us to prove that the set of exponents of all Mersenne primes is an infnite set . Keywords and phrases: Mersenne prime conjecture, modulo algorithm, recursive sieve method, limit of sequence of sets. Classification: FOR Code: 010299 Language: English LJP Copyright ID: 925622 Print ISSN: 2631-8490 Online ISSN: 2631-8504 London Journal of Research in Science: Natural and Formal 465U Volume 20 | Issue 3 | Compilation 1.0 © 2020. Fengsui Liu. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncom-mercial 4.0 Unported License http://creativecommons.org/licenses/by-nc/4.0/), permitting all noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited. There are Infinitely Many Mersenne Primes Fengsui Liu ____________________________________________ ABSTRACT From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained. -
Primality Testing and Sub-Exponential Factorization
Primality Testing and Sub-Exponential Factorization David Emerson Advisor: Howard Straubing Boston College Computer Science Senior Thesis May, 2009 Abstract This paper discusses the problems of primality testing and large number factorization. The first section is dedicated to a discussion of primality test- ing algorithms and their importance in real world applications. Over the course of the discussion the structure of the primality algorithms are devel- oped rigorously and demonstrated with examples. This section culminates in the presentation and proof of the modern deterministic polynomial-time Agrawal-Kayal-Saxena algorithm for deciding whether a given n is prime. The second section is dedicated to the process of factorization of large com- posite numbers. While primality and factorization are mathematically tied in principle they are very di⇥erent computationally. This fact is explored and current high powered factorization methods and the mathematical structures on which they are built are examined. 1 Introduction Factorization and primality testing are important concepts in mathematics. From a purely academic motivation it is an intriguing question to ask how we are to determine whether a number is prime or not. The next logical question to ask is, if the number is composite, can we calculate its factors. The two questions are invariably related. If we can factor a number into its pieces then it is obviously not prime, if we can’t then we know that it is prime. The definition of primality is very much derived from factorability. As we progress through the known and developed primality tests and factorization algorithms it will begin to become clear that while primality and factorization are intertwined they occupy two very di⇥erent levels of computational di⇧culty. -
Factorization Techniques, by Elvis Nunez and Chris Shaw
FACTORIZATION TECHNIQUES ELVIS NUNEZ AND CHRIS SHAW Abstract. The security of the RSA public key cryptosystem relies upon the computational difficulty of deriving the factors of a partic- ular semiprime modulus. In this paper we briefly review the history of factorization methods and develop a stable of techniques that will al- low an understanding of Dixon's Algorithm, the procedural basis upon which modern factorization methods such as the Quadratic Sieve and General Number Field Sieve algorithms rest. 1. Introduction During this course we have proven unique factorization in Z. Theorem 1.1. Given n, there exists a unique prime factorization up to order and multiplication by units. However, we have not deeply investigated methods for determining what these factors are for any given integer. We will begin with the most na¨ıve implementation of a factorization method, and refine our toolkit from there. 2. Trial Division p Theorem 2.1. There exists a divisor of n; a such that 1 > a ≤ n. Definition 2.2. π(n) denotes the number of primes less than or equal to n. Proof. Suppose bjn and b ≥ p(n) and ab = n. It follows a = n . Suppose p p p p b pn b = n, then a = n = n. Suppose b > n, then a < n. By theorem 2.1,p we can find a factor of n by dividing n by the numbers in the range (1; n]. By theorem 1.1, we know that we can express n as a productp of primes, and by theorem 2.1 we know there is a factor of npless than n. -
Integer Sequences
UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular -
25 Primes in Arithmetic Progression
b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 SA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 K office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Ribenboim, Paulo. Title: Prime numbers, friends who give problems : a trialogue with Papa Paulo / by Paulo Ribenboim (Queen’s niversity, Canada). Description: New Jersey : World Scientific, 2016. | Includes indexes. Identifiers: LCCN 2016020705| ISBN 9789814725804 (hardcover : alk. paper) | ISBN 9789814725811 (softcover : alk. paper) Subjects: LCSH: Numbers, Prime. Classification: LCC QA246 .R474 2016 | DDC 512.7/23--dc23 LC record available at https://lccn.loc.gov/2016020705 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, SA. In this case permission to photocopy is not required from the publisher. Typeset by Stallion Press Email: [email protected] Printed in Singapore YingOi - Prime Numbers, Friends Who Give Problems.indd 1 22-08-16 9:11:29 AM October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page v Qu’on ne me dise pas que je n’ai rien dit de nouveau; la disposition des mati`eres est nouvelle. -
Prime Factorization and Cryptography a Theoretical Introduction to the General Number Field Sieve
Prime Factorization and Cryptography A theoretical introduction to the General Number Field Sieve Barry van Leeuwen University of Bristol 10 CP Undergraduate Project Supervisor: Dr. Tim Dokchitser February 1, 2019 Acknowledgement of Sources For all ideas taken from other sources (books, articles, internet), the source of the ideas is mentioned in the main text and fully referenced at the end of the report. All material which is quoted essentially word-for-word from other sources is given in quotation marks and referenced. Pictures and diagrams copied from the internet or other sources are labelled with a reference to the web page,book, article etc. Signed: Barry van Leeuwen Dated: February 1, 2019 Abstract From a theoretical puzzle to applications in cryptography and computer sci- ence: The factorization of prime numbers. In this paper we will introduce a historical retrospect by observing different methods of factorizing primes and we will introduce a theoretical approach to the General Number Field Sieve building from a foundation in Algebra and Number Theory. We will in this exclude most considerations of efficiency and practical im- plementation, and instead focus on the mathematical background. In this paper we will introduce the theory of algebraic number fields and Dedekind domains and their importance in understanding the General Number Field Sieve before continuing to explain, step by step, the inner workings of the General Number Field Sieve. Page 1 of 73 Contents Abstract 1 Table of contents 2 1 Introduction 3 2 Prime Numbers and the Algebra of Modern Cryptography 5 2.1 Preliminary Algebra and Number Theory .