Factoring Integers with a Brain-Inspired Computer John V
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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. -
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. -
The Quadratic Sieve Factoring Algorithm
The Quadratic Sieve Factoring Algorithm Eric Landquist MATH 488: Cryptographic Algorithms December 14, 2001 1 1 Introduction Mathematicians have been attempting to find better and faster ways to fac- tor composite numbers since the beginning of time. Initially this involved dividing a number by larger and larger primes until you had the factoriza- tion. This trial division was not improved upon until Fermat applied the factorization of the difference of two squares: a2 b2 = (a b)(a + b). In his method, we begin with the number to be factored:− n. We− find the smallest square larger than n, and test to see if the difference is square. If so, then we can apply the trick of factoring the difference of two squares to find the factors of n. If the difference is not a perfect square, then we find the next largest square, and repeat the process. While Fermat's method is much faster than trial division, when it comes to the real world of factoring, for example factoring an RSA modulus several hundred digits long, the purely iterative method of Fermat is too slow. Sev- eral other methods have been presented, such as the Elliptic Curve Method discovered by H. Lenstra in 1987 and a pair of probabilistic methods by Pollard in the mid 70's, the p 1 method and the ρ method. The fastest algorithms, however, utilize the− same trick as Fermat, examples of which are the Continued Fraction Method, the Quadratic Sieve (and it variants), and the Number Field Sieve (and its variants). The exception to this is the El- liptic Curve Method, which runs almost as fast as the Quadratic Sieve. -
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). -
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 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. -
On Distribution of Semiprime Numbers
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2014, Vol. 58, No. 8, pp. 43–48. c Allerton Press, Inc., 2014. Original Russian Text c Sh.T. Ishmukhametov, F.F. Sharifullina, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53–59. On Distribution of Semiprime Numbers Sh. T. Ishmukhametov* and F. F. Sharifullina** Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan, 420008 Russia Received January 31, 2013 Abstract—A semiprime is a natural number which is the product of two (possibly equal) prime numbers. Let y be a natural number and g(y) be the probability for a number y to be semiprime. In this paper we derive an asymptotic formula to count g(y) for large y and evaluate its correctness for different y. We also introduce strongly semiprimes, i.e., numbers each of which is a product of two primes of large dimension, and investigate distribution of strongly semiprimes. DOI: 10.3103/S1066369X14080052 Keywords: semiprime integer, strongly semiprime, distribution of semiprimes, factorization of integers, the RSA ciphering method. By smoothness of a natural number n we mean possibility of its representation as a product of a large number of prime factors. A B-smooth number is a number all prime divisors of which are bounded from above by B. The concept of smoothness plays an important role in number theory and cryptography. Possibility of using the concept in cryptography is based on the fact that the procedure of decom- position of an integer into prime divisors (factorization) is a laborious computational process requiring significant calculating resources [1, 2]. -
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. -
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.