Which Positive Integers Are Interesting?
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The Pseudoprimes to 25 • 109
MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 151 JULY 1980, PAGES 1003-1026 The Pseudoprimes to 25 • 109 By Carl Pomerance, J. L. Selfridge and Samuel S. Wagstaff, Jr. Abstract. The odd composite n < 25 • 10 such that 2n_1 = 1 (mod n) have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Car- michael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened. Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences. 1. Introduction. According to Fermat's "Little Theorem", if p is prime and (a, p) = 1, then ap~1 = 1 (mod p). This theorem provides a "test" for primality which is very often correct: Given a large odd integer p, choose some a satisfying 1 <a <p - 1 and compute ap~1 (mod p). If ap~1 pi (mod p), then p is certainly composite. If ap~l = 1 (mod p), then p is probably prime. Odd composite numbers n for which (1) a"_1 = l (mod«) are called pseudoprimes to base a (psp(a)). (For simplicity, a can be any positive in- teger in this definition. We could let a be negative with little additional work. In the last 15 years, some authors have used pseudoprime (base a) to mean any number n > 1 satisfying (1), whether composite or prime.) It is well known that for each base a, there are infinitely many pseudoprimes to base a. -
Composite Numbers That Give Valid RSA Key Pairs for Any Coprime P
information Article Composite Numbers That Give Valid RSA Key Pairs for Any Coprime p Barry Fagin ID Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA; [email protected]; Tel.: +1-719-339-4514 Received: 13 August 2018; Accepted: 25 August 2018; Published: 28 August 2018 Abstract: RSA key pairs are normally generated from two large primes p and q. We consider what happens if they are generated from two integers s and r, where r is prime, but unbeknownst to the user, s is not. Under most circumstances, the correctness of encryption and decryption depends on the choice of the public and private exponents e and d. In some cases, specific (s, r) pairs can be found for which encryption and decryption will be correct for any (e, d) exponent pair. Certain s exist, however, for which encryption and decryption are correct for any odd prime r - s. We give necessary and sufficient conditions for s with this property. Keywords: cryptography; abstract algebra; RSA; computer science education; cryptography education MSC: [2010] 11Axx 11T71 1. Notation and Background Consider the RSA public-key cryptosystem and its operations of encryption and decryption [1]. Let (p, q) be primes, n = p ∗ q, f(n) = (p − 1)(q − 1) denote Euler’s totient function and (e, d) the ∗ Z encryption/decryption exponent pair chosen such that ed ≡ 1. Let n = Un be the group of units f(n) Z mod n, and let a 2 Un. Encryption and decryption operations are given by: (ae)d ≡ (aed) ≡ (a1) ≡ a mod n We consider the case of RSA encryption and decryption where at least one of (p, q) is a composite number s. -
Monthly Science and Maths Magazine 01
1 GYAN BHARATI SCHOOL QUEST….. Monthly Science and Mathematics magazine Edition: DECEMBER,2019 COMPILED BY DR. KIRAN VARSHA AND MR. SUDHIR SAXENA 2 IDENTIFY THE SCIENTIST She was an English chemist and X-ray crystallographer who made contributions to the understanding of the molecular structures of DNA , RNA, viruses, coal, and graphite. She was never nominated for a Nobel Prize. Her work was a crucial part in the discovery of DNA, for which Francis Crick, James Watson, and Maurice Wilkins were awarded a Nobel Prize in 1962. She died in 1958, and during her lifetime the DNA structure was not considered as fully proven. It took Wilkins and his colleagues about seven years to collect enough data to prove and refine the proposed DNA structure. RIDDLE TIME You measure my life in hours and I serve you by expiring. I’m quick when I’m thin and slow when I’m fat. The wind is my enemy. Hard riddles want to trip you up, and this one works by hitting you with details from every angle. The big hint comes at the end with the wind. What does wind threaten most? I have cities, but no houses. I have mountains, but no trees. I have water, but no fish. What am I? This riddle aims to confuse you and get you to focus on the things that are missing: the houses, trees, and fish. 3 WHY ARE AEROPLANES USUALLY WHITE? The Aeroplanes might be having different logos and decorations. But the colour of the aeroplane is usually white.Painting the aeroplane white is most practical and economical. -
The Distribution of Pseudoprimes
The Distribution of Pseudoprimes Matt Green September, 2002 The RSA crypto-system relies on the availability of very large prime num- bers. Tests for finding these large primes can be categorized as deterministic and probabilistic. Deterministic tests, such as the Sieve of Erastothenes, of- fer a 100 percent assurance that the number tested and passed as prime is actually prime. Despite their refusal to err, deterministic tests are generally not used to find large primes because they are very slow (with the exception of a recently discovered polynomial time algorithm). Probabilistic tests offer a much quicker way to find large primes with only a negligibal amount of error. I have studied a simple, and comparatively to other probabilistic tests, error prone test based on Fermat’s little theorem. Fermat’s little theorem states that if b is a natural number and n a prime then bn−1 ≡ 1 (mod n). To test a number n for primality we pick any natural number less than n and greater than 1 and first see if it is relatively prime to n. If it is, then we use this number as a base b and see if Fermat’s little theorem holds for n and b. If the theorem doesn’t hold, then we know that n is not prime. If the theorem does hold then we can accept n as prime right now, leaving a large chance that we have made an error, or we can test again with a different base. Chances improve that n is prime with every base that passes but for really big numbers it is cumbersome to test all possible bases. -
On Types of Elliptic Pseudoprimes
journal of Groups, Complexity, Cryptology Volume 13, Issue 1, 2021, pp. 1:1–1:33 Submitted Jan. 07, 2019 https://gcc.episciences.org/ Published Feb. 09, 2021 ON TYPES OF ELLIPTIC PSEUDOPRIMES LILJANA BABINKOSTOVA, A. HERNANDEZ-ESPIET,´ AND H. Y. KIM Boise State University e-mail address: [email protected] Rutgers University e-mail address: [email protected] University of Wisconsin-Madison e-mail address: [email protected] Abstract. We generalize Silverman's [31] notions of elliptic pseudoprimes and elliptic Carmichael numbers to analogues of Euler-Jacobi and strong pseudoprimes. We inspect the relationships among Euler elliptic Carmichael numbers, strong elliptic Carmichael numbers, products of anomalous primes and elliptic Korselt numbers of Type I, the former two of which we introduce and the latter two of which were introduced by Mazur [21] and Silverman [31] respectively. In particular, we expand upon the work of Babinkostova et al. [3] on the density of certain elliptic Korselt numbers of Type I which are products of anomalous primes, proving a conjecture stated in [3]. 1. Introduction The problem of efficiently distinguishing the prime numbers from the composite numbers has been a fundamental problem for a long time. One of the first primality tests in modern number theory came from Fermat Little Theorem: if p is a prime number and a is an integer not divisible by p, then ap−1 ≡ 1 (mod p). The original notion of a pseudoprime (sometimes called a Fermat pseudoprime) involves counterexamples to the converse of this theorem. A pseudoprime to the base a is a composite number N such aN−1 ≡ 1 mod N. -
Eureka Issue 61
Eureka 61 A Journal of The Archimedeans Cambridge University Mathematical Society Editors: Philipp Legner and Anja Komatar © The Archimedeans (see page 94 for details) Do not copy or reprint any parts without permission. October 2011 Editorial Eureka Reinvented… efore reading any part of this issue of Eureka, you will have noticed The Team two big changes we have made: Eureka is now published in full col- our, and printed on a larger paper size than usual. We felt that, with Philipp Legner Design and Bthe internet being an increasingly large resource for mathematical articles of Illustrations all kinds, it was necessary to offer something new and exciting to keep Eu- reka as successful as it has been in the past. We moved away from the classic Anja Komatar Submissions LATEX-look, which is so common in the scientific community, to a modern, more engaging, and more entertaining design, while being conscious not to Sean Moss lose any of the mathematical clarity and rigour. Corporate Ben Millwood To make full use of the new design possibilities, many of this issue’s articles Publicity are based around mathematical images: from fractal modelling in financial Lu Zou markets (page 14) to computer rendered pictures (page 38) and mathemati- Subscriptions cal origami (page 20). The Showroom (page 46) uncovers the fundamental role pictures have in mathematics, including patterns, graphs, functions and fractals. This issue includes a wide variety of mathematical articles, problems and puzzles, diagrams, movie and book reviews. Some are more entertaining, such as Bayesian Bets (page 10), some are more technical, such as Impossible Integrals (page 80), or more philosophical, such as How to teach Physics to Mathematicians (page 42). -
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. -
Idempotent Integers: the Complete Class of Numbers N = ¯P¯Q That Work
Idempotent Integers: The complete class of numbers n =p ¯q¯ that work correctly in RSA A Mathematical Curiosity Barry Fagin Senior Associate Dean of the Faculty Professor of Computer Science United States Air Force Academy 2021 Conference on Geometry, Algebraic Number Theory and Applications Barry Fagin (A Mathematical Curiosity) Take two positive numbers p¯; q¯. Compute n =p ¯q¯. Pick e; d such that ed ≡ 1. (¯p−1)(¯q−1) What are the conditions on p¯; q¯ such that s.t. 1 ed ? 8a 2 Zn; 8e; d ed ≡ ; a ≡ a (¯p−1)(¯q−1) n In other words, what are the conditions on p¯; q¯ such that RSA works correctly? Note: not asking about security. Barry Fagin (A Mathematical Curiosity) 70's: p¯; q¯ prime meets the conditions. If suciently large, system is believed to be secure due to computational intractability of factoring. 90's: Some Carmichael numbers C also can be factored into p¯; q¯ that meet the required conditions. This work (since 2018) gives necessary and sucient conditions, shows examples. Barry Fagin (A Mathematical Curiosity) Cut to the chase Answer is: p¯; q¯ square free, gcd(p¯; q¯) = 1 such that (¯p − 1)(¯q − 1) ≡ 0 λ(n) where λ denotes the Carmichael lambda function. Barry Fagin (A Mathematical Curiosity) From n's point of view Equivalently, a square-free n can be factored into n =p ¯q¯ such that (¯p − 1)(¯q − 1) ≡ 0 λ(n) In that case, we say that n has an idempotent factorization, and that n is an idempotent integer. Barry Fagin (A Mathematical Curiosity) The rst 8 square-free n with m ≥ 3 factors that admit idempotent factorizations are shown below: n p or p¯ q¯ 30 5 6 42 7 6 66 11 6 78 13 6 102 17 6 105 7 15 114 19 6 130 13 10 Table: Values of n that admit idempotent factorizations Barry Fagin (A Mathematical Curiosity) Idempotent factorizations where one of p¯; q¯ is prime and one is composite are semi-composite. -
Fermat Pseudoprimes
1 TWO HUNDRED CONJECTURES AND ONE HUNDRED AND FIFTY OPEN PROBLEMS ON FERMAT PSEUDOPRIMES (COLLECTED PAPERS) Education Publishing 2013 Copyright 2013 by Marius Coman Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 Peer-Reviewers: Dr. A. A. Salama, Faculty of Science, Port Said University, Egypt. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Mohamed Eisa, Dept. of Computer Science, Port Said Univ., Egypt. EAN: 9781599732572 ISBN: 978-1-59973-257-2 1 INTRODUCTION Prime numbers have always fascinated mankind. For mathematicians, they are a kind of “black sheep” of the family of integers by their constant refusal to let themselves to be disciplined, ordered and understood. However, we have at hand a powerful tool, insufficiently investigated yet, which can help us in understanding them: Fermat pseudoprimes. It was a night of Easter, many years ago, when I rediscovered Fermat’s "little" theorem. Excited, I found the first few Fermat absolute pseudoprimes (561, 1105, 1729, 2465, 2821, 6601, 8911…) before I found out that these numbers are already known. Since then, the passion for study these numbers constantly accompanied me. Exceptions to the above mentioned theorem, Fermat pseudoprimes seem to be more malleable than prime numbers, more willing to let themselves to be ordered than them, and their depth study will shed light on many properties of the primes, because it seems natural to look for the rule studying it’s exceptions, as a virologist search for a cure for a virus studying the organisms that have immunity to the virus. -
The 2005 Exam, Problems Only
UNC Charlotte 2005 Comprehensive March 7, 2005 1. The numbers x and y satisfy 2x = 15 and 15y = 32. What is the value xy? (A) 3 (B) 4 (C) 5 (D) 6 (E) none of A, B, C or D 2. Suppose x, y, z, and w are real numbers satisfying x/y = 4/7, y/z = 14/3, and z/w = 3/11. When (x + y + z)/w is written in the form m/n where m and n are positive integers with no common divisors bigger than 1, what is m + n? (A) 20 (B) 26 (C) 32 (D) 36 (E) 37 3. Let m be an integer such that 1 ≤ m ≤ 1000. Find the probability of selecting at random an integer m such that the quadratic equation 6x2 − 5mx + m2 = 0 has at least one integer solution. (A) 0.333 (B) 0.5 (C) 0.667 (D) 0.778 (E) 0.883 4. Let z = x + iy be a complex number, where x and y are real numbers. Let A and B be the sets defined by A{z | |z| ≤ 2} and B = {z| (1 − i)z + (1 + i)¯z ≥ 4}. √ Recall that z = x − iy is the conjugate of z and that |z| = x2 + y2. Find the area of the region A ∩ B. (A) π/4 (B) π − 2 (C) (π − 2)/4 (D) 4π (E) π − 4 UNC Charlotte 2005 Comprehensive 6 5. The area of the a × b rectangle shown in the picture below is 5π of the area of b the circle. Assuming b > a, what is the value of a ? .......................... -
Improving the Speed and Accuracy of the Miller-Rabin Primality Test
Improving the Speed and Accuracy of the Miller-Rabin Primality Test Shyam Narayanan Mentor: David Corwin MIT PRIMES-USA Abstract Currently, even the fastest deterministic primality tests run slowly, with the Agrawal- Kayal-Saxena (AKS) Primality Test runtime O~(log6(n)), and probabilistic primality tests such as the Fermat and Miller-Rabin Primality Tests are still prone to false re- sults. In this paper, we discuss the accuracy of the Miller-Rabin Primality Test and the number of nonwitnesses for a composite odd integer n. We also extend the Miller- '(n) Rabin Theorem by determining when the number of nonwitnesses N(n) equals 4 5 and by proving that for all n, if N(n) > · '(n) then n must be of one of these 3 32 forms: n = (2x + 1)(4x + 1), where x is an integer, n = (2x + 1)(6x + 1), where x is an integer, n is a Carmichael number of the form pqr, where p, q, r are distinct primes congruent to 3 (mod 4). We then find witnesses to certain forms of composite numbers with high rates of nonwitnesses and find that Jacobi nonresidues and 2 are both valuable bases for the Miller-Rabin test. Finally, we investigate the frequency of strong pseudoprimes further and analyze common patterns using MATLAB. This work is expected to result in a faster and better primality test for large integers. 1 1 Introduction Data is growing at an astoundingly rapid rate, and better information security is re- quired to protect increasing quantities of data. Improved data protection requires more sophisticated cryptographic methods. -
On $ K $-Lehmer Numbers
ON k-LEHMER NUMBERS JOSE´ MAR´IA GRAU AND ANTONIO M. OLLER-MARCEN´ Abstract. Lehmer’s totient problem consists of determining the set of pos- itive integers n such that ϕ(n)|n − 1 where ϕ is Euler’s totient function. In this paper we introduce the concept of k-Lehmer number. A k-Lehmer num- ber is a composite number such that ϕ(n)|(n − 1)k. The relation between k-Lehmer numbers and Carmichael numbers leads to a new characterization of Carmichael numbers and to some conjectures related to the distribution of Carmichael numbers which are also k-Lehmer numbers. AMS 2000 Mathematics Subject Classification: 11A25,11B99 1. Introduction Lehmer’s totient problem asks about the existence of a composite number such that ϕ(n)|(n − 1), where ϕ is Euler’s totient function. Some authors denote these numbers by Lehmer numbers. In 1932, Lehmer (see [13]) showed that every Lemher numbers n must be odd and square-free, and that the number of distinct prime factors of n, d(n), must satisfy d(n) > 6. This bound was subsequently extended to d(n) > 10. The current best result, due to Cohen and Hagis (see [9]), is that n must have at least 14 prime factors and the biggest lower bound obtained for such numbers is 1030 (see [17]). It is known that there are no Lehmer numbers in certain sets, such as the Fibonacci sequence (see [15]), the sequence of repunits in base g for any g ∈ [2, 1000] (see [8]) or the Cullen numbers (see [11]).