Elementary Number Theory
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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]. -
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. -
Epiglass ® and Other International Paint Products
Epiglass® Multipurpose Epoxy Resin ABOUT THE AUTHORS ROGER MARSHALL For eight years award winning author Roger Marshall has been the technical editor for Soundings magazine where his articles are read by about 250,000 people each month. Marshall’s experience as a writer spans for many years. His work has appeared worldwide as well as the New York Times, Daily Telegraph (UK), Sports illustrated, Sail, Cruising World, Motor Boating and sailing, Yachting and many other newspapers and magazines. Marshall is also the author of twelve marine related books, two of which were translated into Italian and Spanish. His last book All about Powerboats was published by International Marine in the spring of 2002. He has another book Rough Weather Seamanship due in the fall of 2003 and is currently working on a new book, Elements of Powerboat Design for International Marine. But writing is only a small part of Marshall’s talents. He is also a designer of boats, both power and sail. After completing a program in small craft design at Southampton College in England, Marshall, who still holds a British passport, moved to the United States in 1973 to take a position at Sparkman & Stephens, Inc. in New York. He worked there as a designer for nearly 5 years and then left to establish his own yacht design studio in Jamestown, Rhode Island. As an independent designer he has designed a wide range of boats and was project engineer for the Courageous Challenge for the 1987 America’s Cup campaign in Australia. In 1999 one of his cruising yacht designs was selected for inclusion in Ocean Cruising magazine’s American Yacht Review. -
FACTORING COMPOSITES TESTING PRIMES Amin Witno
WON Series in Discrete Mathematics and Modern Algebra Volume 3 FACTORING COMPOSITES TESTING PRIMES Amin Witno Preface These notes were used for the lectures in Math 472 (Computational Number Theory) at Philadelphia University, Jordan.1 The module was aborted in 2012, and since then this last edition has been preserved and updated only for minor corrections. Outline notes are more like a revision. No student is expected to fully benefit from these notes unless they have regularly attended the lectures. 1 The RSA Cryptosystem Sensitive messages, when transferred over the internet, need to be encrypted, i.e., changed into a secret code in such a way that only the intended receiver who has the secret key is able to read it. It is common that alphabetical characters are converted to their numerical ASCII equivalents before they are encrypted, hence the coded message will look like integer strings. The RSA algorithm is an encryption-decryption process which is widely employed today. In practice, the encryption key can be made public, and doing so will not risk the security of the system. This feature is a characteristic of the so-called public-key cryptosystem. Ali selects two distinct primes p and q which are very large, over a hundred digits each. He computes n = pq, ϕ = (p − 1)(q − 1), and determines a rather small number e which will serve as the encryption key, making sure that e has no common factor with ϕ. He then chooses another integer d < n satisfying de % ϕ = 1; This d is his decryption key. When all is ready, Ali gives to Beth the pair (n; e) and keeps the rest secret. -
Primality Test
Primality test A primality test is an algorithm for determining whether (6k + i) for some integer k and for i = −1, 0, 1, 2, 3, or 4; an input number is prime. Amongst other fields of 2 divides (6k + 0), (6k + 2), (6k + 4); and 3 divides (6k mathematics, it is used for cryptography. Unlike integer + 3). So a more efficient method is to test if n is divisible factorization, primality tests do not generally give prime by 2 or 3, then to check through all the numbers of form p factors, only stating whether the input number is prime 6k ± 1 ≤ n . This is 3 times as fast as testing all m. or not. Factorization is thought to be a computationally Generalising further, it can be seen that all primes are of difficult problem, whereas primality testing is compara- the form c#k + i for i < c# where i represents the numbers tively easy (its running time is polynomial in the size of that are coprime to c# and where c and k are integers. the input). Some primality tests prove that a number is For example, let c = 6. Then c# = 2 · 3 · 5 = 30. All prime, while others like Miller–Rabin prove that a num- integers are of the form 30k + i for i = 0, 1, 2,...,29 and k ber is composite. Therefore the latter might be called an integer. However, 2 divides 0, 2, 4,...,28 and 3 divides compositeness tests instead of primality tests. 0, 3, 6,...,27 and 5 divides 0, 5, 10,...,25. -
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Anomalous Primes and Extension of the Korselt Criterion
Anomalous Primes and Extension of the Korselt Criterion Liljana Babinkostova1, Brad Bentz2, Morad Hassan3, André Hernández-Espiet4, Hyun Jong Kim5 1Boise State University, 2Brown University, 3Emory University, 4University of Puerto Rico, 5Massachusetts Institute of Technology Motivation Elliptic Korselt Criteria Lucas Pseudoprimes Cryptosystems in ubiquitous commercial use base their security on the dif- ficulty of factoring. Deployment of these schemes necessitate reliable, ef- Korselt Criteria for Euler and Strong Elliptic Carmichael Numbers Lucas Groups ficient methods of recognizing the primality of a number. A number that ordp(N) D; N L passes a probabilistic test, but is in fact composite is known as a pseu- Let N;p(E) be the exponent of E Z=p Z . Then, N is an Euler Let be coprime integers. The Lucas group Z=NZ is defined on elliptic Carmichael number if and only if, for every prime p dividing N, 2 2 2 doprime. A pseudoprime that passes such test for any base is known as a LZ=NZ = f(x; y) 2 (Z=NZ) j x − Dy ≡ 1 (mod N)g: Carmichael number. The focus of this research is analysis of types of pseu- 2N;p j (N + 1 − aN) : doprimes that arise from elliptic curves and from group structures derived t (N + 1 − a ) N If is the largest odd divisor of N , then is a strong elliptic Algebraic Structure of Lucas Groups from Lucas sequences [2]. We extend the Korselt criterion presented in [3] Carmichael number if and only if, for every prime p dividing N, for two important classes of elliptic pseudoprimes and deduce some of their If p is a prime and D is an integer coprime to p, then L e is a cyclic properties. -
Simple High-Level Code for Cryptographic Arithmetic - with Proofs, Without Compromises
Simple High-Level Code for Cryptographic Arithmetic - With Proofs, Without Compromises The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Erbsen, Andres et al. “Simple High-Level Code for Cryptographic Arithmetic - With Proofs, Without Compromises.” Proceedings - IEEE Symposium on Security and Privacy, May-2019 (May 2019) © 2019 The Author(s) As Published 10.1109/SP.2019.00005 Publisher Institute of Electrical and Electronics Engineers (IEEE) Version Author's final manuscript Citable link https://hdl.handle.net/1721.1/130000 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Simple High-Level Code For Cryptographic Arithmetic – With Proofs, Without Compromises Andres Erbsen Jade Philipoom Jason Gross Robert Sloan Adam Chlipala MIT CSAIL, Cambridge, MA, USA fandreser, jadep, [email protected], [email protected], [email protected] Abstract—We introduce a new approach for implementing where X25519 was the only arithmetic-based crypto primitive cryptographic arithmetic in short high-level code with machine- we need, now would be the time to declare victory and go checked proofs of functional correctness. We further demonstrate home. Yet most of the Internet still uses P-256, and the that simple partial evaluation is sufficient to transform such initial code into the fastest-known C code, breaking the decades- current proposals for post-quantum cryptosystems are far from old pattern that the only fast implementations are those whose Curve25519’s combination of performance and simplicity. instruction-level steps were written out by hand. -
Arxiv:Math/0412262V2 [Math.NT] 8 Aug 2012 Etrgae Tgte Ihm)O Atnscnetr and Conjecture fie ‘Artin’S Number on of Cojoc Me) Domains’
ARTIN’S PRIMITIVE ROOT CONJECTURE - a survey - PIETER MOREE (with contributions by A.C. Cojocaru, W. Gajda and H. Graves) To the memory of John L. Selfridge (1927-2010) Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the lit- erature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contribu- tions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Woj- ciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean domains’. Contents 1. Introduction 2 2. Naive heuristic approach 5 3. Algebraic number theory 5 3.1. Analytic algebraic number theory 6 4. Artin’s heuristic approach 8 5. Modified heuristic approach (`ala Artin) 9 6. Hooley’s work 10 6.1. Unconditional results 12 7. Probabilistic model 13 8. The indicator function 17 arXiv:math/0412262v2 [math.NT] 8 Aug 2012 8.1. The indicator function and probabilistic models 17 8.2. The indicator function in the function field setting 18 9. Some variations of Artin’s problem 20 9.1. Elliptic Artin (by A.C. Cojocaru) 20 9.2. Even order 22 9.3. Order in a prescribed arithmetic progression 24 9.4. Divisors of second order recurrences 25 9.5. Lenstra’s work 29 9.6. -
Introduction to Abstract Algebra “Rings First”
Introduction to Abstract Algebra \Rings First" Bruno Benedetti University of Miami January 2020 Abstract The main purpose of these notes is to understand what Z; Q; R; C are, as well as their polynomial rings. Contents 0 Preliminaries 4 0.1 Injective and Surjective Functions..........................4 0.2 Natural numbers, induction, and Euclid's theorem.................6 0.3 The Euclidean Algorithm and Diophantine Equations............... 12 0.4 From Z and Q to R: The need for geometry..................... 18 0.5 Modular Arithmetics and Divisibility Criteria.................... 23 0.6 *Fermat's little theorem and decimal representation................ 28 0.7 Exercises........................................ 31 1 C-Rings, Fields and Domains 33 1.1 Invertible elements and Fields............................. 34 1.2 Zerodivisors and Domains............................... 36 1.3 Nilpotent elements and reduced C-rings....................... 39 1.4 *Gaussian Integers................................... 39 1.5 Exercises........................................ 41 2 Polynomials 43 2.1 Degree of a polynomial................................. 44 2.2 Euclidean division................................... 46 2.3 Complex conjugation.................................. 50 2.4 Symmetric Polynomials................................ 52 2.5 Exercises........................................ 56 3 Subrings, Homomorphisms, Ideals 57 3.1 Subrings......................................... 57 3.2 Homomorphisms.................................... 58 3.3 Ideals......................................... -
MAS330 FULL NOTES Week 1. Euclidean Algorithm, Linear
MAS330 FULL NOTES Week 1. Euclidean Algorithm, Linear congruences, Chinese Remainder Theorem Elementary Number Theory studies modular arithmetic (i.e. counting modulo an integer n), primes, integers and equations. This week we review modular arithmetic and Euclidean algorithm. 1. Introduction 1.1. What is Number Theory? Not trying to answer this directly, let us mention some typical questions we will deal with in this course. Q. What is the last decimal digit of 31000? A. The last decimal digit of 31000 is 1. This is a simple computation we'll do using Euler's Theorem in Week 4. Q. Is there a formula for prime numbers? A. We don't know if any such formula exists. P. Fermat thought that all numbers of the form 2n Fn = 2 + 1 are prime, but he was mistaken! We study Fermat primes Fn in Week 6. Q. Are there any positive integers (x; y; z) satisfying x2 + y2 = z2? A. There are plenty of solutions of x2 + y2 = z2, e.g. (x; y; z) = (3; 4; 5); (5; 12; 13). In fact, there are infinitely many of them. Ancient Greeks have written the formula for all the solutions! We'll discover this formula in Week 8. Q. How about xn + yn = zn for n ≥ 3? A. There are no solutions of xn + yn = zn in positive integers! This fact is called Fermat's Last Theorem and we discuss it in Week 8. Q. Which terms of the Fibonacci sequence un = f1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233;::: g are divisible by 3? A. -
P-COMPACT GROUPS
HOMOTOPY TYPE AND v1-PERIODIC HOMOTOPY GROUPS OF p-COMPACT GROUPS DONALD M. DAVIS Abstract. We determine the v1-periodic homotopy groups of all irreducible p-compact groups. In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomi- als to these homotopy groups. We show that, with several ex- ceptions, every irreducible p-compact group is a product of ex- plicit spherically-resolved spaces which occur also as factors of p- completed Lie groups. 1. Introduction In [4] and [3], the classification of irreducible p-compact groups was completed. This family of spaces extends the family of (p-completions of) compact simple Lie ¡1 groups. The v1-periodic homotopy groups of any space X, denoted v1 ¼¤(X)(p), are a localization of the portion of the homotopy groups detected by K-theory; they were defined in [20]. In [17] and [16], the author completed the determination of the v1- periodic homotopy groups of all compact simple Lie groups. Here we do the same for all the remaining irreducible p-compact groups.1 Recall that a p-compact group ([22]) is a pair (BX; X) such that BX is p-complete ¤ and X = ΩBX with H (X; Fp) finite. Thus BX determines X and contains more structure than does X. The homotopy type and homotopy groups of X do not take into account this extra structure nor the group structure on X. Date: September 21, 2007. 2000 Mathematics Subject Classification. 55Q52, 57T20, 55N15. Key words and phrases. v1-periodic homotopy groups, p-compact groups, Adams operations, K-theory, invariant theory.