Optimized AKS Primality Testing: a Fluctuation Theory Perspective
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The Saddle Point Method in Combinatorics Asymptotic Analysis: Successes and Failures (A Personal View)
Pn The number of inversions in permutations Median versus A (A large) for a Luria-Delbruck-like distribution, with parameter A Sum of positions of records in random permutations Merten's theorem for toral automorphisms Representations of numbers as k=−n "k k The q-Catalan numbers A simple case of the Mahonian statistic Asymptotics of the Stirling numbers of the first kind revisited The Saddle point method in combinatorics asymptotic analysis: successes and failures (A personal view) Guy Louchard May 31, 2011 Guy Louchard The Saddle point method in combinatorics asymptotic analysis: successes and failures (A personal view) Pn The number of inversions in permutations Median versus A (A large) for a Luria-Delbruck-like distribution, with parameter A Sum of positions of records in random permutations Merten's theorem for toral automorphisms Representations of numbers as k=−n "k k The q-Catalan numbers A simple case of the Mahonian statistic Asymptotics of the Stirling numbers of the first kind revisited Outline 1 The number of inversions in permutations 2 Median versus A (A large) for a Luria-Delbruck-like distribution, with parameter A 3 Sum of positions of records in random permutations 4 Merten's theorem for toral automorphisms Pn 5 Representations of numbers as k=−n "k k 6 The q-Catalan numbers 7 A simple case of the Mahonian statistic 8 Asymptotics of the Stirling numbers of the first kind revisited Guy Louchard The Saddle point method in combinatorics asymptotic analysis: successes and failures (A personal view) Pn The number of inversions in permutations Median versus A (A large) for a Luria-Delbruck-like distribution, with parameter A Sum of positions of records in random permutations Merten's theorem for toral automorphisms Representations of numbers as k=−n "k k The q-Catalan numbers A simple case of the Mahonian statistic Asymptotics of the Stirling numbers of the first kind revisited The number of inversions in permutations Let a1 ::: an be a permutation of the set f1;:::; ng. -
Primality Testing and Integer Factorisation
Primality Testing and Integer Factorisation Richard P. Brent, FAA Computer Sciences Laboratory Australian National University Canberra, ACT 2601 Abstract The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the Rivest-Shamir-Adelman (RSA) system, depends on the difficulty of factoring the public keys. In recent years the best known integer factorisation algorithms have improved greatly, to the point where it is now easy to factor a 60-decimal digit number, and possible to factor numbers larger than 120 decimal digits, given the availability of enough computing power. We describe several recent algorithms for primality testing and factorisation, give examples of their use and outline some applications. 1. Introduction It has been known since Euclid’s time (though first clearly stated and proved by Gauss in 1801) that any natural number N has a unique prime power decomposition α1 α2 αk N = p1 p2 ··· pk (1.1) αj (p1 < p2 < ··· < pk rational primes, αj > 0). The prime powers pj are called αj components of N, and we write pj kN. To compute the prime power decomposition we need – 1. An algorithm to test if an integer N is prime. 2. An algorithm to find a nontrivial factor f of a composite integer N. Given these there is a simple recursive algorithm to compute (1.1): if N is prime then stop, otherwise 1. find a nontrivial factor f of N; 2. -
An Analysis of Primality Testing and Its Use in Cryptographic Applications
An Analysis of Primality Testing and Its Use in Cryptographic Applications Jake Massimo Thesis submitted to the University of London for the degree of Doctor of Philosophy Information Security Group Department of Information Security Royal Holloway, University of London 2020 Declaration These doctoral studies were conducted under the supervision of Prof. Kenneth G. Paterson. The work presented in this thesis is the result of original research carried out by myself, in collaboration with others, whilst enrolled in the Department of Mathe- matics as a candidate for the degree of Doctor of Philosophy. This work has not been submitted for any other degree or award in any other university or educational establishment. Jake Massimo April, 2020 2 Abstract Due to their fundamental utility within cryptography, prime numbers must be easy to both recognise and generate. For this, we depend upon primality testing. Both used as a tool to validate prime parameters, or as part of the algorithm used to generate random prime numbers, primality tests are found near universally within a cryptographer's tool-kit. In this thesis, we study in depth primality tests and their use in cryptographic applications. We first provide a systematic analysis of the implementation landscape of primality testing within cryptographic libraries and mathematical software. We then demon- strate how these tests perform under adversarial conditions, where the numbers being tested are not generated randomly, but instead by a possibly malicious party. We show that many of the libraries studied provide primality tests that are not pre- pared for testing on adversarial input, and therefore can declare composite numbers as being prime with a high probability. -
Asymptotic Analysis for Periodic Structures
ASYMPTOTIC ANALYSIS FOR PERIODIC STRUCTURES A. BENSOUSSAN J.-L. LIONS G. PAPANICOLAOU AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island ASYMPTOTIC ANALYSIS FOR PERIODIC STRUCTURES ASYMPTOTIC ANALYSIS FOR PERIODIC STRUCTURES A. BENSOUSSAN J.-L. LIONS G. PAPANICOLAOU AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island M THE ATI A CA M L ΤΡΗΤΟΣ ΜΗ N ΕΙΣΙΤΩ S A O C C I I R E E T ΑΓΕΩΜΕ Y M A F O 8 U 88 NDED 1 2010 Mathematics Subject Classification. Primary 80M40, 35B27, 74Q05, 74Q10, 60H10, 60F05. For additional information and updates on this book, visit www.ams.org/bookpages/chel-374 Library of Congress Cataloging-in-Publication Data Bensoussan, Alain. Asymptotic analysis for periodic structures / A. Bensoussan, J.-L. Lions, G. Papanicolaou. p. cm. Originally published: Amsterdam ; New York : North-Holland Pub. Co., 1978. Includes bibliographical references. ISBN 978-0-8218-5324-5 (alk. paper) 1. Boundary value problems—Numerical solutions. 2. Differential equations, Partial— Asymptotic theory. 3. Probabilities. I. Lions, J.-L. (Jacques-Louis), 1928–2001. II. Papani- colaou, George. III. Title. QA379.B45 2011 515.353—dc23 2011029403 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Primes and Primality Testing
Primes and Primality Testing A Technological/Historical Perspective Jennifer Ellis Department of Mathematics and Computer Science What is a prime number? A number p greater than one is prime if and only if the only divisors of p are 1 and p. Examples: 2, 3, 5, and 7 A few larger examples: 71887 524287 65537 2127 1 Primality Testing: Origins Eratosthenes: Developed “sieve” method 276-194 B.C. Nicknamed Beta – “second place” in many different academic disciplines Also made contributions to www-history.mcs.st- geometry, approximation of andrews.ac.uk/PictDisplay/Eratosthenes.html the Earth’s circumference Sieve of Eratosthenes 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Sieve of Eratosthenes We only need to “sieve” the multiples of numbers less than 10. Why? (10)(10)=100 (p)(q)<=100 Consider pq where p>10. Then for pq <=100, q must be less than 10. By sieving all the multiples of numbers less than 10 (here, multiples of q), we have removed all composite numbers less than 100. -
Definite Integrals in an Asymptotic Setting
A Lost Theorem: Definite Integrals in An Asymptotic Setting Ray Cavalcante and Todor D. Todorov 1 INTRODUCTION We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using Riemann sums. In our axiomatic approach even the proof of the existence of the definite integral (which does use Riemann sums) becomes slightly more elegant than the conventional one. We also discuss an interesting connection between our approach and the history of calculus. The article is written for readers who teach calculus and its applications. It might be accessible to students under a teacher’s supervision and suitable for senior projects on calculus, real analysis, or history of mathematics. Here is a summary of our approach. Let ρ :[a, b] → R be a continuous function and let I :[a, b] × [a, b] → R be the corresponding integral function, defined by y I(x, y)= ρ(t)dt. x Recall that I(x, y) has the following two properties: (A) Additivity: I(x, y)+I (y, z)=I (x, z) for all x, y, z ∈ [a, b]. (B) Asymptotic Property: I(x, x + h)=ρ (x)h + o(h)ash → 0 for all x ∈ [a, b], in the sense that I(x, x + h) − ρ(x)h lim =0. h→0 h In this article we show that properties (A) and (B) are characteristic of the definite integral. More precisely, we show that for a given continuous ρ :[a, b] → R, there is no more than one function I :[a, b]×[a, b] → R with properties (A) and (B). -
Introducing Taylor Series and Local Approximations Using a Historical and Semiotic Approach Kouki Rahim, Barry Griffiths
Introducing Taylor Series and Local Approximations using a Historical and Semiotic Approach Kouki Rahim, Barry Griffiths To cite this version: Kouki Rahim, Barry Griffiths. Introducing Taylor Series and Local Approximations using a Historical and Semiotic Approach. IEJME, Modestom LTD, UK, 2019, 15 (2), 10.29333/iejme/6293. hal- 02470240 HAL Id: hal-02470240 https://hal.archives-ouvertes.fr/hal-02470240 Submitted on 7 Feb 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. INTERNATIONAL ELECTRONIC JOURNAL OF MATHEMATICS EDUCATION e-ISSN: 1306-3030. 2020, Vol. 15, No. 2, em0573 OPEN ACCESS https://doi.org/10.29333/iejme/6293 Introducing Taylor Series and Local Approximations using a Historical and Semiotic Approach Rahim Kouki 1, Barry J. Griffiths 2* 1 Département de Mathématique et Informatique, Université de Tunis El Manar, Tunis 2092, TUNISIA 2 Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA * CORRESPONDENCE: [email protected] ABSTRACT In this article we present the results of a qualitative investigation into the teaching and learning of Taylor series and local approximations. In order to perform a comparative analysis, two investigations are conducted: the first is historical and epistemological, concerned with the pedagogical evolution of semantics, syntax and semiotics; the second is a contemporary institutional investigation, devoted to the results of a review of curricula, textbooks and course handouts in Tunisia and the United States. -
An Introduction to Asymptotic Analysis Simon JA Malham
An introduction to asymptotic analysis Simon J.A. Malham Department of Mathematics, Heriot-Watt University Contents Chapter 1. Order notation 5 Chapter 2. Perturbation methods 9 2.1. Regular perturbation problems 9 2.2. Singular perturbation problems 15 Chapter 3. Asymptotic series 21 3.1. Asymptotic vs convergent series 21 3.2. Asymptotic expansions 25 3.3. Properties of asymptotic expansions 26 3.4. Asymptotic expansions of integrals 29 Chapter 4. Laplace integrals 31 4.1. Laplace's method 32 4.2. Watson's lemma 36 Chapter 5. Method of stationary phase 39 Chapter 6. Method of steepest descents 43 Bibliography 49 Appendix A. Notes 51 A.1. Remainder theorem 51 A.2. Taylor series for functions of more than one variable 51 A.3. How to determine the expansion sequence 52 A.4. How to find a suitable rescaling 52 Appendix B. Exam formula sheet 55 3 CHAPTER 1 Order notation The symbols , o and , were first used by E. Landau and P. Du Bois- Reymond and areOdefined as∼ follows. Suppose f(z) and g(z) are functions of the continuous complex variable z defined on some domain C and possess D ⊂ limits as z z0 in . Then we define the following shorthand notation for the relative!propertiesD of these functions in the limit z z . ! 0 Asymptotically bounded: f(z) = (g(z)) as z z ; O ! 0 means that: there exists constants K 0 and δ > 0 such that, for 0 < z z < δ, ≥ j − 0j f(z) K g(z) : j j ≤ j j We say that f(z) is asymptotically bounded by g(z) in magnitude as z z0, or more colloquially, and we say that f(z) is of `order big O' of g(z). -
Primality Testing for Beginners
STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker http://dx.doi.org/10.1090/stml/070 Primality Testing for Beginners STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell Gerald B. Folland (Chair) Serge Tabachnikov The cover illustration is a variant of the Sieve of Eratosthenes (Sec- tion 1.5), showing the integers from 1 to 2704 colored by the number of their prime factors, including repeats. The illustration was created us- ing MATLAB. The back cover shows a phase plot of the Riemann zeta function (see Appendix A), which appears courtesy of Elias Wegert (www.visual.wegert.com). 2010 Mathematics Subject Classification. Primary 11-01, 11-02, 11Axx, 11Y11, 11Y16. For additional information and updates on this book, visit www.ams.org/bookpages/stml-70 Library of Congress Cataloging-in-Publication Data Rempe-Gillen, Lasse, 1978– author. [Primzahltests f¨ur Einsteiger. English] Primality testing for beginners / Lasse Rempe-Gillen, Rebecca Waldecker. pages cm. — (Student mathematical library ; volume 70) Translation of: Primzahltests f¨ur Einsteiger : Zahlentheorie - Algorithmik - Kryptographie. Includes bibliographical references and index. ISBN 978-0-8218-9883-3 (alk. paper) 1. Number theory. I. Waldecker, Rebecca, 1979– author. II. Title. QA241.R45813 2014 512.72—dc23 2013032423 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. -
Miller–Rabin Test
Lecture 31: Miller–Rabin Test Miller–Rabin Test Recall In the previous lecture we considered an efficient randomized algorithm to generate prime numbers that need n-bits in their binary representation This algorithm sampled a random element in the range f2n−1; 2n−1 + 1;:::; 2n − 1g and test whether it is a prime number or not By the Prime Number Theorem, we are extremely likely to hit a prime number So, all that remains is an algorithm to test whether the random sample we have chosen is a prime number or not Miller–Rabin Test Primality Testing Given an n-bit number N as input, we have to ascertain whether N is a prime number or not in time polynomial in n Only in 2002, Agrawal–Kayal–Saxena constructed a deterministic polynomial time algorithm for primality testing. That is, the algorithm will always run in time polynomial in n. For any input N (that has n-bits in its binary representation), if N is a prime number, the AKS primality testing algorithm will return 1; otherwise (if, the number N is a composite number), the AKS primality testing algorithm will return 0. In practice, this algorithm is not used for primality testing because this turns out to be too slow. In practice, we use a randomized algorithm, namely, the Miller–Rabin Test, that successfully distinguishes primes from composites with very high probability. In this lecture, we will study a basic version of this Miller–Rabin primality test. Miller–Rabin Test Assurance of Miller–Rabin Test Miller–Rabin outputs 1 to indicate that it has classified the input N as a prime. -
M. V. Pedoryuk Asymptotic Analysis Mikhail V
M. V. Pedoryuk Asymptotic Analysis Mikhail V. Fedoryuk Asymptotic Analysis Linear Ordinary Differential Equations Translated from the Russian by Andrew Rodick With 26 Figures Springer-Verlag Berlin Heidelberg GmbH Mikhail V. Fedoryuk t Title of the Russian edition: Asimptoticheskie metody dlya linejnykh obyknovennykh differentsial 'nykh uravnenij Publisher Nauka, Moscow 1983 Mathematics Subject Classification (1991): 34Exx ISBN 978-3-642-63435-2 Library of Congress Cataloging-in-Publication Data. Fedoriuk, Mikhail Vasil'evich. [Asimptoticheskie metody dlia lineinykh obyknovennykh differentsial' nykh uravnenil. English] Asymptotic analysis : linear ordinary differential equations / Mikhail V. Fedoryuk; translated from the Russian by Andrew Rodick. p. cm. Translation of: Asimptoticheskie metody dlia lineinykh obyknovennykh differentsial' nykh uravnenii. Includes bibliographical references and index. ISBN 978-3-642-63435-2 ISBN 978-3-642-58016-1 (eBook) DOI 10.1007/978-3-642-58016-1 1. Differential equations-Asymptotic theory. 1. Title. QA371.F3413 1993 515'.352-dc20 92-5200 This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation. reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplica tion of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current vers ion, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Originally published by Springer-Verlag Berlin Heidelberg New York in 1993 Softcover reprint of the hardcover 1st edition 1993 Typesetting: Springer TEX in-house system 41/3140 - 5 4 3 210 - Printed on acid-free paper Contents Chapter 1. -
15-251: Great Theoretical Ideas in Computer Science Recitation 14 Solutions PRIMES ∈ NP
15-251: Great Theoretical Ideas In Computer Science Recitation 14 Solutions PRIMES 2 NP The set of PRIMES of all primes is in co-NP, because if n is composite and k j n, we can verify this in polynomial time. In fact, the AKS primality test means that PRIMES is in P. We'll just prove PRIMES 2 NP. ∗ (a) We know n is prime iff φ(n) = n−1. Additionally, if n is prime then Zn is cyclic with order n−1. ∗ Given a generator a of Zn and a (guaranteed correct) prime factorization of n − 1, can we verify n is prime? If a has order n − 1, then we are done. So we must verify a has order dividing n − 1: an−1 ≡ 1 mod n and doesn't have smaller order, i.e. for all prime p j n − 1: a(n−1)=p 6≡ 1 mod n a is smaller than n, and there are at most log(n) factors each smaller than n, so our certificate is polynomial in n's representation. Similarly, each arithmetic computation is polynomial in log(n), and there are O(log(n)) of them. 1 (b) What else needs to be verified to use this as a primality certificate? Do we need to add more information? We also need to verify that our factorization (p1; α1); (p2; α2);:::; (pk; αk) of n − 1 is correct, two conditions: α1 α2 αk 1. p1 p2 : : : pk = n − 1 can be verified by multiplying and comparing, polynomial in log(n). 2.