Composite Numbers That Give Valid RSA Key Pairs for Any Coprime P
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Idempotent Factorizations of Square-Free Integers
information Article Idempotent Factorizations of Square-Free Integers Barry Fagin Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA; [email protected]; Tel.: +1-719-333-7377 Received: 20 June 2019; Accepted: 3 July 2019; Published: 6 July 2019 Abstract: We explore the class of positive integers n that admit idempotent factorizations n = p¯q¯ such that l(n)¶(p¯ − 1)(q¯ − 1), where l is the Carmichael lambda function. Idempotent factorizations with p¯ and q¯ prime have received the most attention due to their cryptographic advantages, but there are an infinite number of n with idempotent factorizations containing composite p¯ and/or q¯. Idempotent factorizations are exactly those p¯ and q¯ that generate correctly functioning keys in the Rivest–Shamir–Adleman (RSA) 2-prime protocol with n as the modulus. While the resulting p¯ and q¯ have no cryptographic utility and therefore should never be employed in that capacity, idempotent factorizations warrant study in their own right as they live at the intersection of multiple hard problems in computer science and number theory. We present some analytical results here. We also demonstrate the existence of maximally idempotent integers, those n for which all bipartite factorizations are idempotent. We show how to construct them, and present preliminary results on their distribution. Keywords: cryptography; abstract algebra; Rivest–Shamir–Adleman (RSA); computer science education; cryptography education; number theory; factorization MSC: [2010] 11Axx 11T71 1. Introduction Certain square-free positive integers n can be factored into two numbers (p¯, q¯) such that l(n)¶ (p¯ − 1)(q¯ − 1), where l is the Carmichael lambda function. -
The Ray Attack on RSA Cryptosystems’, in R
The ray attack, an inefficient trial to break RSA cryptosystems∗ Andreas de Vries† FH S¨udwestfalen University of Applied Sciences, Haldener Straße 182, D-58095 Hagen Abstract The basic properties of RSA cryptosystems and some classical attacks on them are described. Derived from geometric properties of the Euler functions, the Euler function rays, a new ansatz to attack RSA cryptosystemsis presented. A resulting, albeit inefficient, algorithm is given. It essentially consists of a loop with starting value determined by the Euler function ray and with step width given by a function ωe(n) being a multiple of the order ordn(e), where e denotes the public key exponent and n the RSA modulus. For n = pq and an estimate r < √pq for the smaller prime factor p, the running time is given by T (e,n,r)= O((r p)lnelnnlnr). − Contents 1 Introduction 1 2 RSA cryptosystem 2 2.1 Properties of an RSA key system . ... 4 2.2 ClassicalRSAattacks. 5 3 The Euler function ray attack 7 3.1 The ω-function and the order of a number modulo n ............... 7 3.2 Properties of composed numbers n = pq ..................... 10 3.3 Thealgorithm................................... 13 arXiv:cs/0307029v1 [cs.CR] 11 Jul 2003 4 Discussion 15 A Appendix 15 A.1 Euler’sTheorem.................................. 15 A.2 The Carmichael function and Carmichael’s Theorem . ......... 16 1 Introduction Since the revolutionary idea of asymmetric cryptosystems was born in the 1970’s, due to Diffie and Hellman [4] and Rivest, Shamir and Adleman [9], public key technology became an in- dispensable part of contemporary electronically based communication. -
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. -
On Terms of Generalized Fibonacci Sequences Which Are Powers of Their Indexes
mathematics Article On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes Pavel Trojovský Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic, [email protected]; Tel.: +42-049-333-2860 Received: 29 June 2019; Accepted: 31 July 2019; Published: 3 August 2019 (k) Abstract: The k-generalized Fibonacci sequence (Fn )n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k ≥ 2, is defined by the values 0, 0, ... , 0, 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the (k) t (2) 2 (3) 2 Diophantine equation Fm = m , with t > 1 and m > k + 1, has only solutions F12 = 12 and F9 = 9 . Keywords: k-generalized Fibonacci sequence; Diophantine equation; linear form in logarithms; continued fraction MSC: Primary 11J86; Secondary 11B39. 1. Introduction The well-known Fibonacci sequence (Fn)n≥0 is given by the following recurrence of the second order Fn+2 = Fn+1 + Fn, for n ≥ 0, with the initial terms F0 = 0 and F1 = 1. Fibonacci numbers have a lot of very interesting properties (see e.g., book of Koshy [1]). One of the famous classical problems, which has attracted a attention of many mathematicians during the last thirty years of the twenty century, was the problem of finding perfect powers in the sequence of Fibonacci numbers. Finally in 2006 Bugeaud et al. [2] (Theorem 1), confirmed these expectations, as they showed that 0, 1, 8 and 144 are the only perfect powers in the sequence of Fibonacci numbers. -
Number Systems and Radix Conversion
Number Systems and Radix Conversion Sanjay Rajopadhye, Colorado State University 1 Introduction These notes for CS 270 describe polynomial number systems. The material is not in the textbook, but will be required for PA1. We humans are comfortable with numbers in the decimal system where each po- sition has a weight which is a power of 10: units have a weight of 1 (100), ten’s have 10 (101), etc. Even the fractional apart after the decimal point have a weight that is a (negative) power of ten. So the number 143.25 has a value that is 1 ∗ 102 + 4 ∗ 101 + 3 ∗ 0 −1 −2 2 5 10 + 2 ∗ 10 + 5 ∗ 10 , i.e., 100 + 40 + 3 + 10 + 100 . You can think of this as a polyno- mial with coefficients 1, 4, 3, 2, and 5 (i.e., the polynomial 1x2 + 4x + 3 + 2x−1 + 5x−2+ evaluated at x = 10. There is nothing special about 10 (just that humans evolved with ten fingers). We can use any radix, r, and write a number system with digits that range from 0 to r − 1. If our radix is larger than 10, we will need to invent “new digits”. We will use the letters of the alphabet: the digit A represents 10, B is 11, K is 20, etc. 2 What is the value of a radix-r number? Mathematically, a sequence of digits (the dot to the right of d0 is called the radix point rather than the decimal point), : : : d2d1d0:d−1d−2 ::: represents a number x defined as follows X i x = dir i 2 1 0 −1 −2 = ::: + d2r + d1r + d0r + d−1r + d−2r + ::: 1 Example What is 3 in radix 3? Answer: 0.1. -
Cheat Sheet of SSE/AVX Intrinsics, for Doing Arithmetic Ll F(Int Ind, Int K) { Return Dp[Ind][K]; } on Several Numbers at Once
University of Bergen Garbage Collectors Davide Pallotti, Jan Soukup, Olav Røthe Bakken NWERC 2017 Nov 8, 2017 UiB template .bashrc .vimrc troubleshoot 1 tan v + tan w Contest (1) Any possible infinite recursion? tan(v + w) = Invalidated pointers or iterators? 1 − tan v tan w template.cpp Are you using too much memory? v + w v − w 15 lines Debug with resubmits (e.g. remapped signals, see Various). sin v + sin w = 2 sin cos #include <bits/stdc++.h> 2 2 using namespace std; Time limit exceeded: v + w v − w Do you have any possible infinite loops? cos v + cos w = 2 cos cos #define rep(i, a, b) for(int i = a; i < (b); ++i) What is the complexity of your algorithm? 2 2 #define trav(a, x) for(auto& a : x) Are you copying a lot of unnecessary data? (References) #define all(x) x.begin(), x.end() How big is the input and output? (consider scanf) (V + W ) tan(v − w)=2 = (V − W ) tan(v + w)=2 #define sz(x) (int)(x).size() Avoid vector, map. (use arrays/unordered_map) typedef long long ll; What do your team mates think about your algorithm? where V; W are lengths of sides opposite angles v; w. typedef pair<int, int> pii; typedef vector<int> vi; Memory limit exceeded: a cos x + b sin x = r cos(x − φ) What is the max amount of memory your algorithm should need? int main() { Are you clearing all datastructures between test cases? a sin x + b cos x = r sin(x + φ) cin.sync_with_stdio(0); cin.tie(0); cin.exceptions(cin.failbit); p 2 2 } Mathematics (2) where r = a + b ; φ = atan2(b; a). -
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. -
Arxiv:1608.06086V1 [Math.NT]
POWER OF TWO AS SUMS OF THREE PELL NUMBERS JHON J. BRAVO, BERNADETTE FAYE AND FLORIAN LUCA Abstract. In this paper, we find all the solutions of the Diophantine equation a Pℓ +Pm +Pn =2 , in nonnegative integer variables (n,m,ℓ,a) where Pk is the k-th term of the Pell sequence Pn n≥0 given by P0 = 0, P1 = 1 and Pn+1 =2Pn+Pn−1 for all n 1. { } ≥ MSC: 11D45, 11B39; 11A25 Keywords: Diophantine equations, Pell numbers, Linear forms in logarithm, reduction method. 1. Introduction The Pell sequence P is the binary reccurent sequence given by P = 0, P =1 { n}n≥0 0 1 and Pn+1 = 2Pn + Pn−1 for all n 0. There are many papers in the literature dealing with Diophantine equations≥ obtained by asking that members of some fixed binary recurrence sequence be squares, factorials, triangular, or belonging to some other interesting sequence of positive integers. For example, in 2008, A. Peth˝o[18] found all the perfect powers (of exponent larger than 1) in the Pell sequence. His result is the following. Theorem 1 (A. Peth˝o, [18]). The only positive integer solutions (n, q, x) with q 2 of the Diophantine equation ≥ q Pn = x are (n, q, x) = (1, q, 1) and (7, 2, 13). That is, the only perfect powers of exponent larger than 1 in the Pell numbers are 2 P1 =1 and P7 = 13 . The case q = 2 had been treated earlier by Ljunggren [13]. Peth˝o’s result was rediscovered by J. H. -
Rapid Multiplication Modulo the Sum and Difference of Highly Composite Numbers
MATHEMATICS OF COMPUTATION Volume 72, Number 241, Pages 387{395 S 0025-5718(02)01419-9 Article electronically published on March 5, 2002 RAPID MULTIPLICATION MODULO THE SUM AND DIFFERENCE OF HIGHLY COMPOSITE NUMBERS COLIN PERCIVAL Abstract. We extend the work of Richard Crandall et al. to demonstrate how the Discrete Weighted Transform (DWT) can be applied to speed up multiplication modulo any number of the form a b where p is small. In ± p ab particular this allows rapid computation modulo numbers of thej form k 2n 1. Q · ± In addition, we prove tight bounds on the rounding errors which naturally occur in floating-point implementations of FFT and DWT multiplications. This makes it possible for FFT multiplications to be used in situations where correctness is essential, for example in computer algebra packages. 1. Introduction In their seminal paper of 1994, Richard Crandall and Barry Fagin introduced the Discrete Weighted Transform (DWT) as a means of eliminating zero-padding when performing integer multiplication modulo Mersenne numbers [2]. While this does not give any improvement in the order of the multiplication, it nevertheless cuts the transform length (and thus time and memory) in half. For these reasons the DWT has become a fixture of the search for Mersenne primes [10]. By using the same form of irrational-base representation as is used for Mersenne numbers, we can in fact eliminate the zero-padding when working modulo a b ± provided that p ab p is sufficiently small that we have enough precision. Essen- tially, as with Mersennej numbers, we choose the base so that the reduction modulo xn 1 implicitQ in the FFT multiplication turns into a reduction modulo a b. -
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. -
Arithmetic Properties of Φ(N)/Λ(N) and the Structure of the Multiplicative Group Modulo N
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Missouri: MOspace Arithmetic Properties of '(n)/λ(n) and the Structure of the Multiplicative Group Modulo n William D. Banks Department of Mathematics, University of Missouri Columbia, MO 65211 USA [email protected] Florian Luca Instituto de Matem´aticas Universidad Nacional Aut´onoma de M´exico C.P. 58089, Morelia, Michoac´an, M´exico [email protected] Igor E. Shparlinski Department of Computing Macquarie University Sydney, NSW 2109, Australia [email protected] Abstract For a positive integer n, we let '(n) and λ(n) denote the Euler function and the Carmichael function, respectively. We define ξ(n) as the ratio '(n)/λ(n) and study various arithmetic properties of ξ(n). AMS-classification: 11A25 Keywords: Euler function, Carmichael function. 1 1 Introduction and Notation Let '(n) denote the Euler function, which is defined as usual by '(n) = #(Z=nZ)× = pν−1(p − 1); n ≥ 1: ν pYk n The Carmichael function λ(n) is defined for all n ≥ 1 as the largest order of any element in the multiplicative group (Z=nZ)×. More explicitly, for any prime power pν, one has pν−1(p − 1) if p ≥ 3 or ν ≤ 2; λ(pν) = 2ν−2 if p = 2 and ν ≥ 3; and for an arbitrary integer n ≥ 2, ν1 νk λ(n) = lcm λ(p1 ); : : : ; λ(pk ) ; ν1 νk where n = p1 : : : pk is the prime factorization of n. Clearly, λ(1) = 1. Despite their many similarities, the functions '(n) and λ(n) often exhibit remarkable differences in their arithmetic behavior, and a vast number of results about the growth rate and various arithmetical properties of '(n) and λ(n) have been obtained; see for example [4, 5, 7, 8, 9, 11, 15]. -
A Diophantine Equation in K–Generalized Fibonacci Numbers and Repdigits
A Diophantine equation in k{generalized Fibonacci numbers and repdigits Jhon J. Bravo Departamento de Matem´aticas Universidad del Cauca Calle 5 No 4{70, Popay´an,Colombia E-mail: [email protected] Carlos Alexis G´omezRuiz Departamento de Matem´aticas Universidad del Valle 25360 Calle 13 No 100-00, Cali, Colombia E-mail: [email protected] Florian Luca School of Mathematics University of the Witwatersrand Private Bag X3 Wits 2050 Johannesburg, South Africa Max Planck Institute for Mathematics Vivatsgasse 7 53111 Bonn, Germany Department of Mathematics Faculty of Sciences University of Ostrava 30 Dubna 22 701 03 Ostrava 1, Czech Republic E-mail: [email protected] Abstract 2010 Mathematics Subject Classification: 11B39; 11J86. Key words and phrases: Generalized Fibonacci numbers, lower bounds for nonzero linear forms in logarithms of algebraic numbers, repdigits. 1 2 J. J. Bravo, C. A. G´omezand F. Luca (k) The k−generalized Fibonacci sequence fFn gn starts with the value 0;:::; 0; 1 (a total of k terms) and each term afterwards is the sum of the k preceding terms. In the present paper, we study on members of k{generalized Fibonacci sequence which are sum of two repdigts, extending a result of D´ıazand Luca [5] regarding Fibonacci numbers with the above property. 1 Introduction Given an integer k ≥ 2, we consider the k{generalized Fibonacci sequence (k) (k) or, for simplicity, the k{Fibonacci sequence F := fFn gn≥2−k given by the recurrence (k) (k) (k) (k) (1.1) Fn = Fn−1 + Fn−2 + ··· + Fn−k for all n ≥ 2; (k) (k) (k) (k) with the initial conditions F−(k−2) = F−(k−3) = ··· = F0 = 0 and F1 = 1.