The Quadratic Sieve Factoring Algorithm
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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]. -
The Quadratic Sieve - Introduction to Theory with Regard to Implementation Issues
The Quadratic Sieve - introduction to theory with regard to implementation issues RNDr. Marian Kechlibar, Ph.D. April 15, 2005 Contents I The Quadratic Sieve 3 1 Introduction 4 1.1 The Quadratic Sieve - short description . 5 1.1.1 Polynomials and relations . 5 1.1.2 Smooth and partial relations . 7 1.1.3 The Double Large Prime Variation . 8 1.1.4 Problems to solve . 10 2 Quadratic Sieve Implementation 12 2.1 The Factor Base . 12 2.2 The sieving process . 15 2.2.1 Interval sieving and solution of polynomials . 16 2.2.2 Practical implementation . 16 2.3 Generation of polynomials . 17 2.3.1 Desirable properties of polynomials . 17 2.3.2 Assessment of magnitude of coecients . 18 2.3.3 MPQS - The Silverman Method . 20 2.3.4 SIQS principle . 21 2.3.5 Desirable properties of b . 22 2.3.6 SIQS - Generation of the Bi's . 23 2.3.7 Generation of b with Gray code formulas . 24 2.3.8 SIQS - General remarks on a determination . 26 2.3.9 SIQS - The bit method for a coecient . 27 2.3.10 SIQS - The Carrier-Wagsta method for a coecient . 28 2.4 Combination of the relations, partial relations and linear algebra 30 2.5 Linear algebra step . 31 2.6 The Singleton Gap . 32 1 3 Experimental Results 36 3.1 Sieving speed - dependence on FB size . 36 3.2 Sieving speed - dependence on usage of 1-partials . 38 3.3 Singletons - dependence on log(N) and FB size . 39 3.4 Properties of the sieving matrices . -
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. -
Sieve Algorithms for the Discrete Logarithm in Medium Characteristic Finite Fields Laurent Grémy
Sieve algorithms for the discrete logarithm in medium characteristic finite fields Laurent Grémy To cite this version: Laurent Grémy. Sieve algorithms for the discrete logarithm in medium characteristic finite fields. Cryptography and Security [cs.CR]. Université de Lorraine, 2017. English. NNT : 2017LORR0141. tel-01647623 HAL Id: tel-01647623 https://tel.archives-ouvertes.fr/tel-01647623 Submitted on 24 Nov 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. AVERTISSEMENT Ce document est le fruit d'un long travail approuvé par le jury de soutenance et mis à disposition de l'ensemble de la communauté universitaire élargie. Il est soumis à la propriété intellectuelle de l'auteur. Ceci implique une obligation de citation et de référencement lors de l’utilisation de ce document. D'autre part, toute contrefaçon, plagiat, reproduction illicite encourt une poursuite pénale. Contact : [email protected] LIENS Code de la Propriété Intellectuelle. articles L 122. 4 Code de la Propriété Intellectuelle. articles L 335.2- L 335.10 http://www.cfcopies.com/V2/leg/leg_droi.php -
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........................ -
Integer Factorization and Computing Discrete Logarithms in Maple
Integer Factorization and Computing Discrete Logarithms in Maple Aaron Bradford∗, Michael Monagan∗, Colin Percival∗ [email protected], [email protected], [email protected] Department of Mathematics, Simon Fraser University, Burnaby, B.C., V5A 1S6, Canada. 1 Introduction As part of our MITACS research project at Simon Fraser University, we have investigated algorithms for integer factorization and computing discrete logarithms. We have implemented a quadratic sieve algorithm for integer factorization in Maple to replace Maple's implementation of the Morrison- Brillhart continued fraction algorithm which was done by Gaston Gonnet in the early 1980's. We have also implemented an indexed calculus algorithm for discrete logarithms in GF(q) to replace Maple's implementation of Shanks' baby-step giant-step algorithm, also done by Gaston Gonnet in the early 1980's. In this paper we describe the algorithms and our optimizations made to them. We give some details of our Maple implementations and present some initial timings. Since Maple is an interpreted language, see [7], there is room for improvement of both implementations by coding critical parts of the algorithms in C. For example, one of the bottle-necks of the indexed calculus algorithm is finding and integers which are B-smooth. Let B be a set of primes. A positive integer y is said to be B-smooth if its prime divisors are all in B. Typically B might be the first 200 primes and y might be a 50 bit integer. ∗This work was supported by the MITACS NCE of Canada. 1 2 Integer Factorization Starting from some very simple instructions | \make integer factorization faster in Maple" | we have implemented the Quadratic Sieve factoring al- gorithm in a combination of Maple and C (which is accessed via Maple's capabilities for external linking). -
Cryptology and Computational Number Theory (Boulder, Colorado, August 1989) 41 R
http://dx.doi.org/10.1090/psapm/042 Other Titles in This Series 50 Robert Calderbank, editor, Different aspects of coding theory (San Francisco, California, January 1995) 49 Robert L. Devaney, editor, Complex dynamical systems: The mathematics behind the Mandlebrot and Julia sets (Cincinnati, Ohio, January 1994) 48 Walter Gautschi, editor, Mathematics of Computation 1943-1993: A half century of computational mathematics (Vancouver, British Columbia, August 1993) 47 Ingrid Daubechies, editor, Different perspectives on wavelets (San Antonio, Texas, January 1993) 46 Stefan A. Burr, editor, The unreasonable effectiveness of number theory (Orono, Maine, August 1991) 45 De Witt L. Sumners, editor, New scientific applications of geometry and topology (Baltimore, Maryland, January 1992) 44 Bela Bollobas, editor, Probabilistic combinatorics and its applications (San Francisco, California, January 1991) 43 Richard K. Guy, editor, Combinatorial games (Columbus, Ohio, August 1990) 42 C. Pomerance, editor, Cryptology and computational number theory (Boulder, Colorado, August 1989) 41 R. W. Brockett, editor, Robotics (Louisville, Kentucky, January 1990) 40 Charles R. Johnson, editor, Matrix theory and applications (Phoenix, Arizona, January 1989) 39 Robert L. Devaney and Linda Keen, editors, Chaos and fractals: The mathematics behind the computer graphics (Providence, Rhode Island, August 1988) 38 Juris Hartmanis, editor, Computational complexity theory (Atlanta, Georgia, January 1988) 37 Henry J. Landau, editor, Moments in mathematics (San Antonio, Texas, January 1987) 36 Carl de Boor, editor, Approximation theory (New Orleans, Louisiana, January 1986) 35 Harry H. Panjer, editor, Actuarial mathematics (Laramie, Wyoming, August 1985) 34 Michael Anshel and William Gewirtz, editors, Mathematics of information processing (Louisville, Kentucky, January 1984) 33 H. Peyton Young, editor, Fair allocation (Anaheim, California, January 1985) 32 R. -
Factoring and Discrete Log
Factoring and Discrete Log Nadia Heninger University of Pennsylvania June 1, 2015 Textbook RSA [Rivest Shamir Adleman 1977] Public Key Private Key N = pq modulus p; q primes e encryption exponent d decryption exponent (d = e−1 mod (p − 1)(q − 1)) Encryption public key = (N; e) ciphertext = messagee mod N message = ciphertextd mod N Textbook RSA [Rivest Shamir Adleman 1977] Public Key Private Key N = pq modulus p; q primes e encryption exponent d decryption exponent (d = e−1 mod (p − 1)(q − 1)) Signing public key = (N; e) signature = messaged mod N message = signaturee mod N Computational problems Factoring Problem: Given N, compute its prime factors. I Computationally equivalent to computing private key d. I Factoring is in NP and coNP ! not NP-complete (unless P=NP or similar). Computational problems eth roots mod N Problem: Given N, e, and c, compute x such that xe ≡ c mod N. I Equivalent to decrypting an RSA-encrypted ciphertext. I Equivalent to selective forgery of RSA signatures. I Conflicting results about whether it reduces to factoring: I \Breaking RSA may not be equivalent to factoring" [Boneh Venkatesan 1998] \an algebraic reduction from factoring to breaking low-exponent RSA can be converted into an efficient factoring algorithm" I \Breaking RSA generically is equivalent to factoring" [Aggarwal Maurer 2009] \a generic ring algorithm for breaking RSA in ZN can be converted into an algorithm for factoring" I \RSA assumption": This problem is hard. A garden of attacks on textbook RSA Unpadded RSA encryption is homomorphic under multiplication. Let's have some fun! Attack: Malleability Given a ciphertext c = Enc(m) = me mod N, attacker can forge ciphertext Enc(ma) = cae mod N for any a. -
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. -
Note to Users
NOTE TO USERS This reproduction is the best copy available. UMI A SURVEY OF RESULTS ON GIUGA'S CONJECTURE AND RELATED CONJECTURES by Joseph R. Hobart BSc., University of Northern British Columbia, 2004 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in MATHEMATICAL, COMPUTER AND PHYSICAL SCIENCES (MATHEMATICS) THE UNIVERSITY OF NORTHERN BRITISH COLUMBIA July 2005 © Joseph R. Hobart, 2005 Library and Bibliothèque et 1 ^ 1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l'édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre référence ISBN: 978-0-494-28392-9 Our file Notre référence ISBN: 978-0-494-28392-9 NOTICE: AVIS: The author has granted a non L'auteur a accordé une licence non exclusive exclusive license allowing Library permettant à la Bibliothèque et Archives and Archives Canada to reproduce,Canada de reproduire, publier, archiver, publish, archive, preserve, conserve,sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l'Internet, prêter, telecommunication or on the Internet,distribuer et vendre des thèses partout dans loan, distribute and sell theses le monde, à des fins commerciales ou autres, worldwide, for commercial or non sur support microforme, papier, électronique commercial purposes, in microform,et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriété du droit d'auteur ownership and moral rights in et des droits moraux qui protège cette thèse. this thesis. Neither the thesis Ni la thèse ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent être imprimés ou autrement may be printed or otherwise reproduits sans son autorisation. -
Tables of Fibonacci and Lucas Factorizations
MATHEMATICS OF COMPUTATION VOLUME 50, NUMBER 181 JANUARY 1988, PAGES 251-260 Tables of Fibonacci and Lucas Factorizations By John Brillhart, Peter L. Montgomery, and Robert D. Silverman Dedicated to Dov Jarden Abstract. We list the known prime factors of the Fibonacci numbers Fn for n < 999 and Lucas numbers Ln for n < 500. We discuss the various methods used to obtain these factorizations, and primality tests, and give some history of the subject. 1. Introduction. In the Supplements section at the end of this issue we give in two tables the known prime factors of the Fibonacci numbers Fn, 3 < n < 999, n odd, and the Lucas numbers Ln, 2 < n < 500. The sequences Fn and Ln are defined recursively by the formulas . ^n+2 = Fn+X + Fn, Fo = 0, Fi = 1, Ln+2 = Ln+i + Ln, in = 2, L\ = 1. The use of a different subscripting destroys the divisibility properties of these numbers. We also have the formulas an - 3a (1-2) Fn = --£-, Ln = an + ßn, a —ß where a = (1 + \/B)/2 and ß — (1 - v^5)/2. This paper is concerned with the multiplicative structure of Fn and Ln. It includes both theoretical and numerical results. 2. Multiplicative Structure of Fn and Ln. The identity (2.1) F2n = FnLn follows directly from (1.2). Although the Fibonacci and Lucas numbers are defined additively, this is one of many multiplicative identities relating these sequences. The identities in this paper are derived from the familiar polynomial factorization (2.2) xn-yn = ]\*d(x,y), n>\, d\n where $d(x,y) is the dth cyclotomic polynomial in homogeneous form. -
Primality Testing and Sub-Exponential Factorization
Primality Testing and Sub-Exponential Factorization David Emerson Advisor: Howard Straubing Boston College Computer Science Senior Thesis May, 2009 Abstract This paper discusses the problems of primality testing and large number factorization. The first section is dedicated to a discussion of primality test- ing algorithms and their importance in real world applications. Over the course of the discussion the structure of the primality algorithms are devel- oped rigorously and demonstrated with examples. This section culminates in the presentation and proof of the modern deterministic polynomial-time Agrawal-Kayal-Saxena algorithm for deciding whether a given n is prime. The second section is dedicated to the process of factorization of large com- posite numbers. While primality and factorization are mathematically tied in principle they are very di⇥erent computationally. This fact is explored and current high powered factorization methods and the mathematical structures on which they are built are examined. 1 Introduction Factorization and primality testing are important concepts in mathematics. From a purely academic motivation it is an intriguing question to ask how we are to determine whether a number is prime or not. The next logical question to ask is, if the number is composite, can we calculate its factors. The two questions are invariably related. If we can factor a number into its pieces then it is obviously not prime, if we can’t then we know that it is prime. The definition of primality is very much derived from factorability. As we progress through the known and developed primality tests and factorization algorithms it will begin to become clear that while primality and factorization are intertwined they occupy two very di⇥erent levels of computational di⇧culty.