FERMAT VERSUS WILSON CONGRUENCES, ARITHMETIC DERIVATIVES and ZETA VALUES 1. Introduction Since the Finite Fields Occur As the Re
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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. -
Results for Wieferich Primes 2
Results for Wieferich Primes N. A. Carella Abstract: Let v 2 be a fixed integer, and let x 1 and z x be large numbers. ≥ ≥ ≥ The exact asymptotic formula for the number of Wieferich primes p, defined by vp−1 ≡ 1 mod p2, in the short interval [x,x + z] is proposed in this note. The search conducted on the last 100 years have produced two primes p<x = 1015 such that 2p−1 1 mod p2. ≡ The probabilistic and theoretical information within predicts the existence of another base v = 2 prime on the interval [1015, 1040]. Furthermore, a result for the upper bound on the number of Wieferich primes is used to demonstrate that the subset of nonWieferich primes has density 1. AMS Mathematical Subjects Classification: Primary 11A41; Secondary 11B25. Keywords: Distribution of Prime, Wieferiech prime, Finite Rings. Contents 1 Introduction 3 1.1 SummaryofHeuristics. .. .. .. .. .. .. .. .. 3 1.2 ResultsInShortIntervals . ... 4 1.3 AverageOrder .................................. 4 1.4 Guide ....................................... 5 2 Basic Analytic Results 6 2.1 SumsAndProductsOverThePrimes . 6 2.2 TotientsFunctions ............................... 7 2.3 Sums Of Totients Functions Over The Integers . ...... 7 2.4 Sums Of Totients Functions Over The Primes . ..... 9 2.5 Sums Of Totients Functionsc Over Subsets Of Integers . ......... 9 arXiv:1712.08166v2 [math.GM] 5 May 2018 2.6 Problems ..................................... 11 3 Finite Cyclic Groups 13 3.1 MultiplicativeOrders. 13 3.2 MaximalCyclicSubgroups . 14 4 Characteristic Functions 15 4.1 Characteristic Functions Modulo Prime Powers . ........ 15 4.2 Characteristic Functions Modulo n ....................... 16 4.3 Problems ..................................... 17 5 Equivalent Exponential Sums 17 1 results for wieferich primes 2 6 Upper Bound For The Main Term 20 7 Evaluations Of The Main Terms 21 7.1 SumsOverThePrimes............................. -
Wieferich Primes and Period Lengths for the Expansions of Fractions
314 MATHEMATICSMAGAZINE Wieferich Primesand PeriodLengths for the Expansionsof Fractions GENE GARZA JEFF YOUNG University of Montevallo Montevallo, Al 35115 [email protected] It is well known that some decimal expansions terminate, while others repeat, at least eventually, in patterns,which may be short or lengthy (we shall call this repeating pattern the period of the expansion). Here we will extend some known results while exploring expansions of fractions in any base. Our goal will be to find a formula for the length of the period of such expansions. The interested reader is referred to the recent award-winning article by Jones and Pearce, who show how to display such decimal expansions graphically [3]. We will consider both the expansions of (the reciprocals of) primes and of com- posites. It would seem that the easier part of this problem would be that of primes. However, there are difficulties/anomalies among primes that make it hard to find a for- mula that works in all cases. The most interesting such case is that of Wieferich primes, whose reciprocals are characterizedby expansions whose periods are the same length as the periods of their squares. For example, the length of the period of 1/1093 is 1092 which is the same as that of 1/10932. This, as we shall see, is not normally the case. For someone seeking a simple formula, this is bad news. However, as our table at the end shows, Wieferich primes are quite rare. Preliminaries Let's review what is meant by the expansion of a fraction and, in particular,the decimal expansion of a fraction. -
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. -
Prime Divisors of Sparse Values of Cyclotomic Polynomials and Wieferich Primes
Journal of Number Theory 201 (2019) 1–22 Contents lists available at ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt General Section Prime divisors of sparse values of cyclotomic ✩ polynomials and Wieferich primes M. Ram Murty ∗, François Séguin Department of Mathematics, Queen’s University, Kingston, Ontario K7L 3N6, Canada a r t i c l e i n f o a b s t r a c t Article history: Bang (1886), Zsigmondy (1892) and Birkhoff and Vandiver Received 17 September 2018 (1904) initiated the study of the largest prime divisors of Received in revised form 31 sequences of the form an − bn, denoted P (an − bn), by December 2018 essentially proving that for integers a >b > 0, P (an − bn) ≥ Accepted 25 February 2019 n +1 for every n > 2. Since then, the problem of finding Available online 20 March 2019 bounds on the largest prime factor of Lehmer sequences, Lucas Communicated by F. Pellarin sequences or special cases thereof has been studied by many, MSC: most notably by Schinzel (1962), and Stewart (1975, 2013). 11B39 In 2002, Murty and Wong proved, conditionally upon the abc − 11N69 conjecture, that P (an − bn) n2 for any > 0. In this 11D45 article, we improve this result for the specific case where b =1. 11A41 Specifically, we obtain a more precise result, and one that is Keywords: dependent on a condition we believe to be weaker than the Lucas sequences abc conjecture. Our result actually concerns the largest prime Wieferich primes factor of the nth cyclotomic polynomial evaluated at a fixed integer a, P (Φn(a)), as we let n grow. -
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. -
Overpseudoprimes, Mersenne Numbers and Wieferich Primes 2
OVERPSEUDOPRIMES, MERSENNE NUMBERS AND WIEFERICH PRIMES VLADIMIR SHEVELEV Abstract. We introduce a new class of pseudoprimes - so-called “overpseu- doprimes” which is a special subclass of super-Poulet pseudoprimes. De- noting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime if and only if the value of h(n) is invariant of all divisors d > 1 of n. In particular, we prove that all composite Mersenne numbers 2p − 1, where p is prime, and squares of Wieferich primes are overpseudoprimes. 1. Introduction n Sometimes the numbers Mn =2 − 1, n =1, 2,..., are called Mersenne numbers, although this name is usually reserved for numbers of the form p (1) Mp =2 − 1 where p is prime. In our paper we use the latter name. In this form numbers Mp at the first time were studied by Marin Mersenne (1588-1648) at least in 1644 (see in [1, p.9] and a large bibliography there). We start with the following simple observation. Let n be odd and h(n) denote the multiplicative order of 2 modulo n. arXiv:0806.3412v9 [math.NT] 15 Mar 2012 Theorem 1. Odd d> 1 is a divisor of Mp if and only if h(d)= p. Proof. If d > 1 is a divisor of 2p − 1, then h(d) divides prime p. But h(d) > 1. Thus, h(d)= p. The converse statement is evident. Remark 1. This observation for prime divisors of Mp belongs to Max Alek- seyev ( see his comment to sequence A122094 in [5]). -
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]. -
Arxiv:1206.0606V1 [Math.NT]
OVERPSEUDOPRIMES, AND MERSENNE AND FERMAT NUMBERS AS PRIMOVER NUMBERS VLADIMIR SHEVELEV, GILBERTO GARC´IA-PULGAR´IN, JUAN MIGUEL VELASQUEZ-SOTO,´ AND JOHN H. CASTILLO Abstract. We introduce a new class of pseudoprimes-so called “overpseu- doprimes to base b”, which is a subclass of strong pseudoprimes to base b. Denoting via |b|n the multiplicative order of b modulo n, we show that a composite n is overpseudoprime if and only if |b|d is invariant for all divisors d> 1 of n. In particular, we prove that all composite Mersenne numbers 2p − 1, where p is prime, are overpseudoprime to base 2 and squares of Wieferich primes are overpseudoprimes to base 2. Finally, we show that some kinds of well known numbers are overpseudoprime to a base b. 1. Introduction First and foremost, we recall some definitions and fix some notation. Let b an integer greater than 1 and N a positive integer relatively prime to b. Throughout, we denote by |b|N the multiplicative order of b modulo N. For a prime p, νp(N) means the greatest exponent of p in the prime factorization of N. Fermat’s little theorem implies that 2p−1 ≡ 1 (mod p), where p is an odd prime p. An odd prime p, is called a Wieferich prime if 2p−1 ≡ 1 (mod p2), We recall that a Poulet number, also known as Fermat pseudoprime to base 2, is a composite number n such that 2n−1 ≡ 1 (mod n). A Poulet number n which verifies that d divides 2d −2 for each divisor d of n, is called a Super-Poulet pseudoprime. -
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.