Prime Factorization and Cryptography a Theoretical Introduction to the General Number Field Sieve
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On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function
mathematics Article On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function ŠtˇepánHubálovský 1,* and Eva Trojovská 2 1 Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic 2 Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +420-49-333-2704 Received: 3 October 2020; Accepted: 14 October 2020; Published: 16 October 2020 Abstract: Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk ≡ 0 (mod n). In this paper, we shall find all positive solutions of the Diophantine equation z(j(n)) = n, where j is the Euler totient function. Keywords: Fibonacci numbers; order of appearance; Euler totient function; fixed points; Diophantine equations MSC: 11B39; 11DXX 1. Introduction Let (Fn)n≥0 be the sequence of Fibonacci numbers which is defined by 2nd order recurrence Fn+2 = Fn+1 + Fn, with initial conditions Fi = i, for i 2 f0, 1g. These numbers (together with the sequence of prime numbers) form a very important sequence in mathematics (mainly because its unexpectedly and often appearance in many branches of mathematics as well as in another disciplines). We refer the reader to [1–3] and their very extensive bibliography. We recall that an arithmetic function is any function f : Z>0 ! C (i.e., a complex-valued function which is defined for all positive integer). -
Lecture 3: Chinese Remainder Theorem 25/07/08 to Further
Department of Mathematics MATHS 714 Number Theory: Lecture 3: Chinese Remainder Theorem 25/07/08 To further reduce the amount of computations in solving congruences it is important to realise that if m = m1m2 ...ms where the mi are pairwise coprime, then a ≡ b (mod m) if and only if a ≡ b (mod mi) for every i, 1 ≤ i ≤ s. This leads to the question of simultaneous solution of a system of congruences. Theorem 1 (Chinese Remainder Theorem). Let m1,m2,...,ms be pairwise coprime integers ≥ 2, and b1, b2,...,bs arbitrary integers. Then, the s congruences x ≡ bi (mod mi) have a simultaneous solution that is unique modulo m = m1m2 ...ms. Proof. Let ni = m/mi; note that (mi, ni) = 1. Every ni has an inversen ¯i mod mi (lecture 2). We show that x0 = P1≤j≤s nj n¯j bj is a solution of our system of s congruences. Since mi divides each nj except for ni, we have x0 = P1≤j≤s njn¯j bj ≡ nin¯ibj (mod mi) ≡ bi (mod mi). Uniqueness: If x is any solution of the system, then x − x0 ≡ 0 (mod mi) for all i. This implies that m|(x − x0) i.e. x ≡ x0 (mod m). Exercise 1. Find all solutions of the system 4x ≡ 2 (mod 6), 3x ≡ 5 (mod 7), 2x ≡ 4 (mod 11). Exercise 2. Using the Chinese Remainder Theorem (CRT), solve 3x ≡ 11 (mod 2275). Systems of linear congruences in one variable can often be solved efficiently by combining inspection (I) and Euclid’s algorithm (EA) with the Chinese Remainder Theorem (CRT). -
Fast Generation of RSA Keys Using Smooth Integers
1 Fast Generation of RSA Keys using Smooth Integers Vassil Dimitrov, Luigi Vigneri and Vidal Attias Abstract—Primality generation is the cornerstone of several essential cryptographic systems. The problem has been a subject of deep investigations, but there is still a substantial room for improvements. Typically, the algorithms used have two parts – trial divisions aimed at eliminating numbers with small prime factors and primality tests based on an easy-to-compute statement that is valid for primes and invalid for composites. In this paper, we will showcase a technique that will eliminate the first phase of the primality testing algorithms. The computational simulations show a reduction of the primality generation time by about 30% in the case of 1024-bit RSA key pairs. This can be particularly beneficial in the case of decentralized environments for shared RSA keys as the initial trial division part of the key generation algorithms can be avoided at no cost. This also significantly reduces the communication complexity. Another essential contribution of the paper is the introduction of a new one-way function that is computationally simpler than the existing ones used in public-key cryptography. This function can be used to create new random number generators, and it also could be potentially used for designing entirely new public-key encryption systems. Index Terms—Multiple-base Representations, Public-Key Cryptography, Primality Testing, Computational Number Theory, RSA ✦ 1 INTRODUCTION 1.1 Fast generation of prime numbers DDITIVE number theory is a fascinating area of The generation of prime numbers is a cornerstone of A mathematics. In it one can find problems with cryptographic systems such as the RSA cryptosystem. -
Sieving for Twin Smooth Integers with Solutions to the Prouhet-Tarry-Escott Problem
Sieving for twin smooth integers with solutions to the Prouhet-Tarry-Escott problem Craig Costello1, Michael Meyer2;3, and Michael Naehrig1 1 Microsoft Research, Redmond, WA, USA fcraigco,[email protected] 2 University of Applied Sciences Wiesbaden, Germany 3 University of W¨urzburg,Germany [email protected] Abstract. We give a sieving algorithm for finding pairs of consecutive smooth numbers that utilizes solutions to the Prouhet-Tarry-Escott (PTE) problem. Any such solution induces two degree-n polynomials, a(x) and b(x), that differ by a constant integer C and completely split into linear factors in Z[x]. It follows that for any ` 2 Z such that a(`) ≡ b(`) ≡ 0 mod C, the two integers a(`)=C and b(`)=C differ by 1 and necessarily contain n factors of roughly the same size. For a fixed smoothness bound B, restricting the search to pairs of integers that are parameterized in this way increases the probability that they are B-smooth. Our algorithm combines a simple sieve with parametrizations given by a collection of solutions to the PTE problem. The motivation for finding large twin smooth integers lies in their application to compact isogeny-based post-quantum protocols. The recent key exchange scheme B-SIDH and the recent digital signature scheme SQISign both require large primes that lie between two smooth integers; finding such a prime can be seen as a special case of finding twin smooth integers under the additional stipulation that their sum is a prime p. When searching for cryptographic parameters with 2240 ≤ p < 2256, an implementation of our sieve found primes p where p + 1 and p − 1 are 215-smooth; the smoothest prior parameters had a similar sized prime for which p−1 and p+1 were 219-smooth. -
Sequences of Primes Obtained by the Method of Concatenation
SEQUENCES OF PRIMES OBTAINED BY THE METHOD OF CONCATENATION (COLLECTED PAPERS) Copyright 2016 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: 9781599734668 ISBN: 978-1-59973-466-8 1 INTRODUCTION The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (as, for instance, the prime terms in Smarandache concatenated odd -
Number Theory Course Notes for MA 341, Spring 2018
Number Theory Course notes for MA 341, Spring 2018 Jared Weinstein May 2, 2018 Contents 1 Basic properties of the integers 3 1.1 Definitions: Z and Q .......................3 1.2 The well-ordering principle . .5 1.3 The division algorithm . .5 1.4 Running times . .6 1.5 The Euclidean algorithm . .8 1.6 The extended Euclidean algorithm . 10 1.7 Exercises due February 2. 11 2 The unique factorization theorem 12 2.1 Factorization into primes . 12 2.2 The proof that prime factorization is unique . 13 2.3 Valuations . 13 2.4 The rational root theorem . 15 2.5 Pythagorean triples . 16 2.6 Exercises due February 9 . 17 3 Congruences 17 3.1 Definition and basic properties . 17 3.2 Solving Linear Congruences . 18 3.3 The Chinese Remainder Theorem . 19 3.4 Modular Exponentiation . 20 3.5 Exercises due February 16 . 21 1 4 Units modulo m: Fermat's theorem and Euler's theorem 22 4.1 Units . 22 4.2 Powers modulo m ......................... 23 4.3 Fermat's theorem . 24 4.4 The φ function . 25 4.5 Euler's theorem . 26 4.6 Exercises due February 23 . 27 5 Orders and primitive elements 27 5.1 Basic properties of the function ordm .............. 27 5.2 Primitive roots . 28 5.3 The discrete logarithm . 30 5.4 Existence of primitive roots for a prime modulus . 30 5.5 Exercises due March 2 . 32 6 Some cryptographic applications 33 6.1 The basic problem of cryptography . 33 6.2 Ciphers, keys, and one-time pads . -
Introduction to Number Theory CS1800 Discrete Math; Notes by Virgil Pavlu; Updated November 5, 2018
Introduction to Number Theory CS1800 Discrete Math; notes by Virgil Pavlu; updated November 5, 2018 1 modulo arithmetic All numbers here are integers. The integer division of a at n > 1 means finding the unique quotient q and remainder r 2 Zn such that a = nq + r where Zn is the set of all possible remainders at n : Zn = f0; 1; 2; 3; :::; n−1g. \mod n" = remainder at division with n for n > 1 (n it has to be at least 2) \a mod n = r" means mathematically all of the following : · r is the remainder of integer division a to n · a = n ∗ q + r for some integer q · a; r have same remainder when divided by n · a − r = nq is a multiple of n · n j a − r, a.k.a n divides a − r EXAMPLES 21 mod 5 = 1, because 21 = 5*4 +1 same as saying 5 j (21 − 1) 24 = 10 = 3 = -39 mod 7 , because 24 = 7*3 +3; 10=7*1+3; 3=7*0 +3; -39=7*(-6)+3. Same as saying 7 j (24 − 10) or 7 j (3 − 10) or 7 j (10 − (−39)) etc LEMMA two numbers a; b have the same remainder mod n if and only if n divides their difference. We can write this in several equivalent ways: · a mod n = b mod n, saying a; b have the same remainder (or modulo) · a = b( mod n) · n j a − b saying n divides a − b · a − b = nk saying a − b is a multiple of n (k is integer but its value doesnt matter) 1 EXAMPLES 21 = 11 (mod 5) = 1 , 5 j (21 − 11) , 21 mod 5 = 11 mod 5 86 mod 10 = 1126 mod 10 , 10 j (86 − 1126) , 86 − 1126 = 10k proof: EXERCISE. -
On the Number Field Sieve: Polynomial Selection and Smooth Elements in Number Fields
On the Number Field Sieve: Polynomial Selection and Smooth Elements in Number Fields Nicholas Vincent Coxon BSc (hons) A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in June 2012 School of Mathematics and Physics Abstract The number field sieve is the asymptotically fastest known algorithm for factoring large integers that are free of small prime factors. Two aspects of the algorithm are considered in this thesis: polynomial selection and smooth elements in number fields. The contributions to polynomial selection are twofold. First, existing methods of polynomial generation, namely those based on Montgomery's method, are extended and tools developed to aid in their analysis. Second, a new approach to polynomial generation is developed and realised. The development of the approach is driven by results obtained on the divisibility properties of univariate resultants. Examples from the literature point toward the utility of applying decoding algorithms for algebraic error-correcting codes to problems of finding elements in a ring with a smooth representation. In this thesis, the problem of finding algebraic integers in a number field with smooth norm is reformulated as a decoding problem for a family of error-correcting codes called NF-codes. An algorithm for solving the weighted list decoding problem for NF-codes is provided. The algorithm is then used to find algebraic integers with norm containing a large smooth factor. Bounds on the existence of such numbers are derived using algorithmic and combinatorial methods. ii Declaration by the Author This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. -
Sieving by Large Integers and Covering Systems of Congruences
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 20, Number 2, April 2007, Pages 495–517 S 0894-0347(06)00549-2 Article electronically published on September 19, 2006 SIEVING BY LARGE INTEGERS AND COVERING SYSTEMS OF CONGRUENCES MICHAEL FILASETA, KEVIN FORD, SERGEI KONYAGIN, CARL POMERANCE, AND GANG YU 1. Introduction Notice that every integer n satisfies at least one of the congruences n ≡ 0(mod2),n≡ 0(mod3),n≡ 1(mod4),n≡ 1(mod6),n≡ 11 (mod 12). A finite set of congruences, where each integer satisfies at least one them, is called a covering system. A famous problem of Erd˝os from 1950 [4] is to determine whether for every N there is a covering system with distinct moduli greater than N.In other words, can the minimum modulus in a covering system with distinct moduli be arbitrarily large? In regards to this problem, Erd˝os writes in [6], “This is perhaps my favourite problem.” It is easy to see that in a covering system, the reciprocal sum of the moduli is at least 1. Examples with distinct moduli are known with least modulus 2, 3, and 4, where this reciprocal sum can be arbitrarily close to 1; see [10], §F13. Erd˝os and Selfridge [5] conjectured that this fails for all large enough choices of the least modulus. In fact, they made the following much stronger conjecture. Conjecture 1. For any number B,thereisanumberNB,suchthatinacovering system with distinct moduli greater than NB, the sum of reciprocals of these moduli is greater than B. A version of Conjecture 1 also appears in [7]. -
Factoring Integers with a Brain-Inspired Computer John V
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS 1 Factoring Integers with a Brain-Inspired Computer John V. Monaco and Manuel M. Vindiola Abstract—The bound to factor large integers is dominated • Constant-time synaptic integration: a single neuron in the by the computational effort to discover numbers that are B- brain may receive electrical potential inputs along synap- smooth, i.e., integers whose largest prime factor does not exceed tic connections from thousands of other neurons. The B. Smooth numbers are traditionally discovered by sieving a polynomial sequence, whereby the logarithmic sum of prime incoming potentials are continuously and instantaneously factors of each polynomial value is compared to a threshold. integrated to compute the neuron’s membrane potential. On a von Neumann architecture, this requires a large block of Like the brain, neuromorphic architectures aim to per- memory for the sieving interval and frequent memory updates, form synaptic integration in constant time, typically by resulting in O(ln ln B) amortized time complexity to check each leveraging physical properties of the underlying device. value for smoothness. This work presents a neuromorphic sieve that achieves a constant-time check for smoothness by reversing Unlike the traditional CPU-based sieve, the factor base is rep- the roles of space and time from the von Neumann architecture resented in space (as spiking neurons) and the sieving interval and exploiting two characteristic properties of brain-inspired in time (as successive time steps). Sieving is performed by computation: massive parallelism and constant time synaptic integration. The effects on sieving performance of two common a population of leaky integrate-and-fire (LIF) neurons whose neuromorphic architectural constraints are examined: limited dynamics are simple enough to be implemented on a range synaptic weight resolution, which forces the factor base to be of current and future architectures. -
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......................................... -
Arxiv:1606.08690V5 [Math.NT] 27 Apr 2021 on Prime Factors of Mersenne
On prime factors of Mersenne numbers Ady Cambraia Jr,∗ Michael P. Knapp,† Ab´ılio Lemos∗, B. K. Moriya∗ and Paulo H. A. Rodrigues‡ [email protected] [email protected] [email protected] [email protected] paulo [email protected] April 29, 2021 Abstract n Let (Mn)n≥0 be the Mersenne sequence defined by Mn = 2 − 1. Let ω(n) be the number of distinct prime divisors of n. In this short note, we present a description of the Mersenne numbers satisfying ω(Mn) ≤ 3. Moreover, we prove that the inequality, (1−ǫ) log log n given ǫ> 0, ω(Mn) > 2 − 3 holds for almost all positive integers n. Besides, a we present the integer solutions (m, n, a) of the equation Mm+Mn = 2p with m,n ≥ 2, p an odd prime number and a a positive integer. 2010 Mathematics Subject Classification: 11A99, 11K65, 11A41. Keywords: Mersenne numbers, arithmetic functions, prime divisors. 1 Introduction arXiv:1606.08690v5 [math.NT] 27 Apr 2021 n Let (Mn)n≥0 be the Mersenne sequence defined by Mn = 2 − 1, for n ≥ 0. A simple argument shows that if Mn is a prime number, then n is a prime number. When Mn is a prime number, it is called a Mersenne prime. Throughout history, many researchers sought to find Mersenne primes. Some tools are very important for the search for Mersenne primes, mainly the Lucas-Lehmer test. There are papers (see for example [1, 5, 21]) that seek to describe the prime factors of Mn, where Mn is a composite number and n is a prime number.