There Are Infinitely Many Perrin Pseudoprimes
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Fast Tabulation of Challenge Pseudoprimes Andrew Shallue and Jonathan Webster
THE OPEN BOOK SERIES 2 ANTS XIII Proceedings of the Thirteenth Algorithmic Number Theory Symposium Fast tabulation of challenge pseudoprimes Andrew Shallue and Jonathan Webster msp THE OPEN BOOK SERIES 2 (2019) Thirteenth Algorithmic Number Theory Symposium msp dx.doi.org/10.2140/obs.2019.2.411 Fast tabulation of challenge pseudoprimes Andrew Shallue and Jonathan Webster We provide a new algorithm for tabulating composite numbers which are pseudoprimes to both a Fermat test and a Lucas test. Our algorithm is optimized for parameter choices that minimize the occurrence of pseudoprimes, and for pseudoprimes with a fixed number of prime factors. Using this, we have confirmed that there are no PSW-challenge pseudoprimes with two or three prime factors up to 280. In the case where one is tabulating challenge pseudoprimes with a fixed number of prime factors, we prove our algorithm gives an unconditional asymptotic improvement over previous methods. 1. Introduction Pomerance, Selfridge, and Wagstaff famously offered $620 for a composite n that satisfies (1) 2n 1 1 .mod n/ so n is a base-2 Fermat pseudoprime, Á (2) .5 n/ 1 so n is not a square modulo 5, and j D (3) Fn 1 0 .mod n/ so n is a Fibonacci pseudoprime, C Á or to prove that no such n exists. We call composites that satisfy these conditions PSW-challenge pseudo- primes. In[PSW80] they credit R. Baillie with the discovery that combining a Fermat test with a Lucas test (with a certain specific parameter choice) makes for an especially effective primality test[BW80]. -
Arxiv:1412.5226V1 [Math.NT] 16 Dec 2014 Hoe 11
q-PSEUDOPRIMALITY: A NATURAL GENERALIZATION OF STRONG PSEUDOPRIMALITY JOHN H. CASTILLO, GILBERTO GARC´IA-PULGAR´IN, AND JUAN MIGUEL VELASQUEZ-SOTO´ Abstract. In this work we present a natural generalization of strong pseudoprime to base b, which we have called q-pseudoprime to base b. It allows us to present another way to define a Midy’s number to base b (overpseudoprime to base b). Besides, we count the bases b such that N is a q-probable prime base b and those ones such that N is a Midy’s number to base b. Furthemore, we prove that there is not a concept analogous to Carmichael numbers to q-probable prime to base b as with the concept of strong pseudoprimes to base b. 1. Introduction Recently, Grau et al. [7] gave a generalization of Pocklignton’s Theorem (also known as Proth’s Theorem) and Miller-Rabin primality test, it takes as reference some works of Berrizbeitia, [1, 2], where it is presented an extension to the concept of strong pseudoprime, called ω-primes. As Grau et al. said it is right, but its application is not too good because it is needed m-th primitive roots of unity, see [7, 12]. In [7], it is defined when an integer N is a p-strong probable prime base a, for p a prime divisor of N −1 and gcd(a, N) = 1. In a reading of that paper, we discovered that if a number N is a p-strong probable prime to base 2 for each p prime divisor of N − 1, it is actually a Midy’s number or a overpseu- doprime number to base 2. -
Offprint Provided to the Author by the Publisher
MATHEMATICS OF COMPUTATION Volume 89, Number 321, January 2020, Pages 493–514 https://doi.org/10.1090/mcom/3452 Article electronically published on May 24, 2019 AVERAGE LIAR COUNT FOR DEGREE-2 FROBENIUS PSEUDOPRIMES ANDREW FIORI AND ANDREW SHALLUE Abstract. In this paper we obtain lower and upper bounds on the average number of liars for the Quadratic Frobenius Pseudoprime Test of Grantham [Math. Comp. 70 (2001), pp. 873–891], generalizing arguments of Erd˝os and Pomerance [Math. Comp. 46 (1986), pp. 259–279] and Monier [Theoret. Com- put. Sci. 12 (1980), 97–108]. These bounds are provided for both Jacobi symbol ±1 cases, providing evidence for the existence of several challenge pseudo- primes. 1. Introduction A pseudoprime is a composite number that satisfies some necessary condition for primality. Since primes are necessary building blocks for so many algorithms, and since the most common way to find primes in practice is to apply primality testing algorithms based on such necessary conditions, it is important to gather what information we can about pseudoprimes. In addition to the practical benefits, pseudoprimes have remarkable divisibility properties that make them fascinating objects of study. The most common necessary condition used in practice is that the number has no small divisors. Another common necessary condition follows from a theorem of Fermat, that if n is prime and gcd(a, n)=1,thenan−1 =1(modn). If gcd(a, n)=1andan−1 =1 (modn) for composite n,wecalla a Fermat liar and denote by F (n) the set of Fermat liars with respect to n, or more precisely the set of their residue classes modulo n. -
The RSA Algorithm Clifton Paul Robinson
Bridgewater State University Virtual Commons - Bridgewater State University Honors Program Theses and Projects Undergraduate Honors Program 5-1-2018 The Key to Cryptography: The RSA Algorithm Clifton Paul Robinson Follow this and additional works at: http://vc.bridgew.edu/honors_proj Part of the Computer Sciences Commons Recommended Citation Robinson, Clifton Paul. (2018). The Key ot Cryptography: The RSA Algorithm. In BSU Honors Program Theses and Projects. Item 268. Available at: http://vc.bridgew.edu/honors_proj/268 Copyright © 2018 Clifton Paul Robinson This item is available as part of Virtual Commons, the open-access institutional repository of Bridgewater State University, Bridgewater, Massachusetts. The Key to Cryptography: The RSA Algorithm Clifton Paul Robinson Submitted in Partial Completion of the Requirements for Commonwealth Interdisciplinary Honors in Computer Science and Mathematics Bridgewater State University May 1, 2018 Dr. Jacqueline Anderson Thesis Co-Advisor Dr. Michael Black, Thesis Co-Advisor Dr. Ward Heilman, Committee Member Dr. Haleh Khojasteh, Committee Member BRIDGEWATER STATE UNIVERSITY UNDERGRADUATE THESIS The Key To Cryptography: The RSA Algorithm Author: Advisors: Clifton Paul ROBINSON Dr. Jackie ANDERSON Dr. Michael BLACK Submitted in Partial Completion of the Requirements for Commonwealth Honors in Computer Science and Mathematics Dr. Ward Heilman, Reading Committee Dr. Haleh Khojasteh, Reading Committee ii Dedicated to Mom, Dad, James, and Mimi iii Contents Abstractv 1 Introduction1 1.1 The Project Overview........................1 2 Theorems and Definitions2 2.1 Definitions..............................2 2.2 Theorems...............................5 3 The History of Cryptography6 3.1 Origins................................6 3.2 A Transition.............................6 3.3 Cryptography at War........................7 3.4 The Creation and Uses of RSA...................7 4 The Mathematics9 4.1 What is a Prime Number?.....................9 4.2 Factoring Numbers........................ -
Generating Provable Primes Efficiently on Embedded Devices
Generating Provable Primes Efficiently on Embedded Devices Christophe Clavier1, Benoit Feix1;2, Lo¨ıc Thierry2;?, and Pascal Paillier3 1 XLIM, University of Limoges, [email protected] 2 INSIDE Secure [email protected],[email protected] 3 CryptoExperts [email protected] Abstract. This paper introduces new techniques to generate provable prime numbers efficiently on embedded devices such as smartcards, based on variants of Pocklington's and the Brillhart-Lehmer-Selfridge-Tuckerman- Wagstaff theorems. We introduce two new generators that, combined with cryptoprocessor-specific optimizations, open the way to efficient and tamper-resistant on-board generation of provable primes. We also report practical results from our implementations. Both our theoretical and ex- perimental results show that constructive methods can generate provable primes essentially as efficiently as state-of-the-art generators for probable primes based on Fermat and Miller-Rabin pseudo-tests. We evaluate the output entropy of our two generators and provide techniques to ensure a high level of resistance against physical attacks. This paper intends to provide practitioners with the first practical solutions for fast and secure generation of provable primes in embedded security devices. Keywords: Prime Numbers, Pocklington's theorem, Public Key Cryp- tography, Embedded Software, Modular Exponentiation, Cryptographic Accelerators, Primality Proving. 1 Introduction Large prime numbers are a basic ingredient of keys in several standardized primi- tives such as RSA [21], Digital Signature Algorithm (DSA) [12] or Diffie-Hellman key exchange (DH) [10]. This paper precisely addresses the generation of prov- able prime numbers in embedded, crypto-enabled devices. When it comes to RSA key generation, two approaches coexist: key pairs may be generated off-board (i.e. -
The Efficient Calculation of a Combined Fermat PRP and Lucas
The Efficient Calculation of a Combined Fermat PRP and Lucas PRP Test Paul Underwood November 24, 2014 Abstract By an elementary observation about the computation of the dif- ference of squares for large integers, a deterministic probable prime (PRP) test is given with a running time almost equivalent to that of a Lucas PRP test but with the combined strength of a Fermat PRP test and a Lucas PRP test. 1 Introduction Much has been written about Fermat PRP tests [1, 2, 3], Lucas PRP tests [4, 5], Frobenius PRP tests [6, 7, 8, 9, 10, 11, 12] and combinations of these [13, 14, 15]. These tests provide a probabilistic answer to the question: \Is this integer prime?" Although an affirmative answer is not 100% certain, it is answered fast and reliable enough for \industrial" use [16]. For speed, these PRP tests are usually preceded by factoring methods such as sieving and trial division. The speed of the PRP tests depends on how quickly multiplication and modular reduction can be computed during exponentiation. Techniques such as Karatsuba's algorithm [17, section 9.5.1], Toom-Cook multiplication, Fourier Transform algorithms [17, section 9.5.2] and Montgomery exponen- tiation [17, section 9.2.1] play their roles for different integer sizes. The sizes of the bases used are also critical. Oliver Atkin introduced the concept of a \selfridge" [18], approximately equal to the running time of a Fermat PRP test. Hence a Lucas PRP test is 2 selfridges. The Baillie-PSW test costs 1+2 selfridges, the use of which is very efficient when processing a candidate prime list; There is no known Baillie-PSW pseudoprime but Greene and Chen give a way to construct some examples [19]. -
The Probability That a Random Probable Prime Is Composite*
MATHEMATICS OF COMPUTATION VOLUME 53, NUMBER 188 OCTOBER 1989, PAGES 721-741 The Probability that a Random Probable Prime is Composite* By Su Hee Kim and Carl Pomerance** Abstract. Consider a procedure which (1) chooses a random odd number n < x, (2) chooses a random number 6, 1 < b < n — 1, and (3) accepts n if bn~l = 1 (mod n). Let P(x) denote the probability that this procedure accepts a composite number. It is known from work of Erdös and the second author that P(x) —>0 as x —►oo. In this paper, explicit inequalities are established for P(x)\ For example, it is shown that P(10100) < 2.77 x 10~8 and that P(x) < (logx)"197 for x > 101()5. Introduction. Suppose one wants to produce a random prime p < x, drawn with the uniform distribution. One possible solution is to choose a random number n, 1 < n < x, and apply a test to n that can tell if it is prime or composite. This procedure is repeated independently until a prime is found. By the prime number theorem, the expected number of trials until a prime is drawn is about log x. If one wishes to choose an odd prime, the trials n may be restricted to odd numbers. The expected number of trials is then about ¿ log x. There are many algorithms which can be used to decide if n is prime or com- posite. However, using the Fermât congruence is a very cheap test that is usually recommended as a preliminary procedure before a more time-consuming test is at- tempted. -
Quadratic Frobenius Probable Prime Tests Costing Two Selfridges
Quadratic Frobenius probable prime tests costing two selfridges Paul Underwood June 6, 2017 Abstract By an elementary observation about the computation of the difference of squares for large in- tegers, deterministic quadratic Frobenius probable prime tests are given with running times of approximately 2 selfridges. 1 Introduction Much has been written about Fermat probable prime (PRP) tests [1, 2, 3], Lucas PRP tests [4, 5], Frobenius PRP tests [6, 7, 8, 9, 10, 11, 12] and combinations of these [13, 14, 15]. These tests provide a probabilistic answer to the question: “Is this integer prime?” Although an affirmative answer is not 100% certain, it is answered fast and reliable enough for “industrial” use [16]. For speed, these various PRP tests are usually preceded by factoring methods such as sieving and trial division. The speed of the PRP tests depends on how quickly multiplication and modular reduction can be computed during exponentiation. Techniques such as Karatsuba’s algorithm [17, section 9.5.1], Toom-Cook multiplication, Fourier Transform algorithms [17, section 9.5.2] and Montgomery expo- nentiation [17, section 9.2.1] play their roles for different integer sizes. The sizes of the bases used are also critical. Oliver Atkin introduced the concept of a “Selfridge Unit” [18], approximately equal to the running time of a Fermat PRP test, which is called a selfridge in this paper. The Baillie-PSW test costs 1+3 selfridges, the use of which is very efficient when processing a candidate prime list. There is no known Baillie-PSW pseudoprime but Greene and Chen give a way to construct some similar counterexam- ples [19]. -
Independence of the Miller-Rabin and Lucas Probable Prime Tests
Independence of the Miller-Rabin and Lucas Probable Prime Tests Alec Leng Mentor: David Corwin March 30, 2017 1 Abstract In the modern age, public-key cryptography has become a vital component for se- cure online communication. To implement these cryptosystems, rapid primality test- ing is necessary in order to generate keys. In particular, probabilistic tests are used for their speed, despite the potential for pseudoprimes. So, we examine the commonly used Miller-Rabin and Lucas tests, showing that numbers with many nonwitnesses are usually Carmichael or Lucas-Carmichael numbers in a specific form. We then use these categorizations, through a generalization of Korselt’s criterion, to prove that there are no numbers with many nonwitnesses for both tests, affirming the two tests’ relative independence. As Carmichael and Lucas-Carmichael numbers are in general more difficult for the two tests to deal with, we next search for numbers which are both Carmichael and Lucas-Carmichael numbers, experimentally finding none less than 1016. We thus conjecture that there are no such composites and, using multi- variate calculus with symmetric polynomials, begin developing techniques to prove this. 2 1 Introduction In the current information age, cryptographic systems to protect data have become a funda- mental necessity. With the quantity of data distributed over the internet, the importance of encryption for protecting individual privacy has greatly risen. Indeed, according to [EMC16], cryptography is allows for authentication and protection in online commerce, even when working with vital financial information (e.g. in online banking and shopping). Now that more and more transactions are done through the internet, rather than in person, the importance of secure encryption schemes is only growing. -
Introducing Quaternions to Integer Factorisation
Journal of Physical Science and Application 5 (2) (2015) 101-107 doi: 10.17265/2159-5348/2015.02.003 D DAVID PUBLISHING Introducing Quaternions to Integer Factorisation HuiKang Tong 4500 Ang Mo Kio Avenue 6, 569843, Singapore Abstract: The key purpose of this paper is to open up the concepts of the sum of four squares and the algebra of quaternions into the attempts of factoring semiprimes, the product of two prime numbers. However, the application of these concepts here has been clumsy, and would be better explored by those with a more rigorous mathematical background. There may be real immediate implications on some RSA numbers that are slightly larger than a perfect square. Key words: Integer factorisation, RSA, quaternions, sum of four squares, euler factorisation method. Nomenclature In Section 3, we extend the Euler factoring method to one using the sum of four squares and the algebra p, q: prime factors n: semiprime pq, the product of two primes of quaternions. We comment on the development of P: quaternion with norm p the mathematics in Section 3.1, and introduce the a, b, c, d: components of a quaternion integral quaternions in Section 3.2, and its relationship 1. Introduction with the sum of four squares in Section 3.3. In Section 3.4, we mention an algorithm to generate the sum of We assume that the reader know the RSA four squares. cryptosystem [1]. Notably, the ability to factorise a In Section 4, we propose the usage of concepts of random and large semiprime n (the product of two the algebra of quaternions into the factorisation of prime numbers p and q) efficiently can completely semiprimes. -
Frobenius Pseudoprimes and a Cubic Primality Test
Notes on Number Theory and Discrete Mathematics ISSN 1310–5132 Vol. 20, 2014, No. 4, 11–20 Frobenius pseudoprimes and a cubic primality test Catherine A. Buell1 and Eric W. Kimball2 1 Department of Mathematics, Fitchburg State University 160 Pearl Street, Fitchburg, MA, 01420, USA e-mail: [email protected] 2 Department of Mathematics, Bates College 3 Andrews Rd., Lewiston, ME, 04240, USA e-mail: [email protected] Abstract: An integer, n, is called a Frobenius probable prime with respect to a polynomial when it passes the Frobenius probable prime test. Composite integers that are Frobenius probable primes are called Frobenius pseudoprimes. Jon Grantham developed and analyzed a Frobenius probable prime test with quadratic polynomials. Using the Chinese Remainder Theorem and Frobenius automorphisms, we were able to extend Grantham’s results to some cubic polynomials. This case is computationally similar but more efficient than the quadratic case. Keywords: Frobenius, Pseudoprimes, Cubic, Number fields, Primality. AMS Classification: 11Y11. 1 Introduction Fermat’s Little Theorem is a beloved theorem found in numerous abstract algebra and number theory textbooks. It states, for p an odd prime and (a; p)= 1, ap−1 ≡ 1 mod p. Fermat’s Little Theorem holds for all primes; however, there are some composites that pass this test with a particular base, a. These are called Fermat pseudoprimes. Definition 1.1. A Fermat pseudoprime is a composite n for which an−1 ≡ 1 mod n for some base a and (a; n) = 1. The term pseudoprime is typically used to refer to Fermat pseudoprime. Example 1.2. Consider n = 341 and a = 2. -
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.