An Investigation on Integer Factorization Applied to Public Key Cryptography
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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 Number Field Sieve for Discrete Logarithms
The Number Field Sieve for Discrete Logarithms Henrik Røst Haarberg Master of Science Submission date: June 2016 Supervisor: Kristian Gjøsteen, MATH Norwegian University of Science and Technology Department of Mathematical Sciences Abstract We present two general number field sieve algorithms solving the discrete logarithm problem in finite fields. The first algorithm pre- sented deals with discrete logarithms in prime fields Fp, while the second considers prime power fields Fpn . We prove, using the standard heuristic, that these algorithms will run in sub-exponential time. We also give an overview of different index calculus algorithms solving the discrete logarithm problem efficiently for different possible relations between the characteristic and the extension degree. To be able to give a good introduction to the algorithms, we present theory necessary to understand the underlying algebraic structures used in the algorithms. This theory is largely algebraic number theory. 1 Contents 1 Introduction 4 1.1 Discrete logarithms . .4 1.2 The general number field sieve and L-notation . .4 2 Theory 6 2.1 Number fields . .6 2.1.1 Dedekind domains . .7 2.1.2 Module structure . .9 2.1.3 Norm of ideals . .9 2.1.4 Units . 10 2.2 Prime ideals . 10 2.3 Smooth numbers . 13 2.3.1 Density . 13 2.3.2 Exponent vectors . 13 3 The number field sieve in prime fields 15 3.1 Overview . 15 3.2 Calculating logarithms . 15 3.3 Sieving . 17 3.4 Schirokauer maps . 18 3.5 Linear algebra . 20 3.5.1 A note about smooth t and g .............. 22 3.6 Run time . -
Bound Estimation for Divisors of RSA Modulus with Small Divisor-Ratio
International Journal of Network Security, Vol.23, No.3, PP.412-425, May 2021 (DOI: 10.6633/IJNS.202105 23(3).06) 412 Bound Estimation for Divisors of RSA Modulus with Small Divisor-ratio Xingbo Wang (Corresponding author: Xingbo Wang) Department of Mechatronic Engineering, Foshan University Guangdong Engineering Center of Information Security for Intelligent Manufacturing System, China Email: [email protected]; [email protected] (Received Nov. 16, 2019; Revised and Accepted Mar. 8, 2020; First Online Apr. 17, 2021) Abstract make it easier to find the small divisor; articles [2, 10] found out the distribution of an odd integer's square-root Through subtle analysis on relationships among ances- in the T3 tree; articles [11, 15, 17] investigated divisors' tors, symmetric brothers, and the square root of a node distribution of an RSA number, presenting in detail how on the T3 tree, the article puts a method forwards to cal- two divisors distribute on the levels of the T3 tree in terms culate an interval that contains a divisor of a semiprime of their divisor-ratio. whose divisor-ratio is less than 3/2. Concrete mathemat- This paper, following the studies in [11, 15, 17], and ical reasonings to derive the method and programming based on the inequalities proved in [12] as well as the procedure from realizing the calculations are shown in theorems proved in [13, 16], gives in detail a bound esti- detail. Numerical experiments are made by applying the mation to the divisors of an RSA number. The results in method on both ordinary small semiprimes and the RSA this paper are helpful to design algorithm to search the numbers. -
Enhancing the Securing RSA Algorithm from Attack
Journal of Basic and Applied Engineering Research Print ISSN: 2350-0077; Online ISSN: 2350-0255; Volume 1, Number 10; October, 2014 pp. 48-63 © Krishi Sanskriti Publications http://www.krishisanskriti.org/jbaer.html Enhancing the Securing RSA Algorithm from Attack Swati Srivastava 1, Meenu 2 1M.Tech Student, Department of CSE, Madan Mohan Malaviya University of Technology, Gorakhpur, U.P. 2Department of CSE, Madan MohanMalaviya University of Technology, Gorakhpur, U.P. Abstract: The RSA public key and signature scheme is often used authenticity, integrity, and limited access to data. In in modern communications technologies; it is one of the firstly Cryptography we differentiate between private key defined public key cryptosystem that enable secure cryptographic systems (also known as conventional communicating over public unsecure communication channels. cryptography systems) and public key cryptographic systems. In praxis many protocols and security standards use the RSA, Private Key Cryptography, also known as secret-key or thus the security of the RSA is critical because any weaknesses in the RSA crypto system may lead the whole system to become symmetric-key encryption, has an old history, and is based on vulnerable against attacks. This paper introduce a security using one shared secret key for encryption and decryption. enhancement on the RSA cryptosystem, it suggests the use of The development of fast computers and communication randomized parameters in the encryption process to make RSA technologies did allow us to define many -
Factorization of RSA-180
Factorization of RSA-180 S. A. Danilov, I. A. Popovyan Moscow State University, Russia May 9, 2010∗ Abstract We present a brief report on the factorization of RSA-180, currently smallest unfactored RSA number. We show that the numbers of similar size could be factored in a reasonable time at home using open source factoring software running on a few Intel Core i7 PCs. 1 Introduction In 1991 RSA Labs published a list of semiprime numbers of different size and announced a reward for their factorization. The numbers from that list called RSA numbers became a measure of the quality of the factorization tools. We began working on our factorization project on November 2009. We started with the smallest unfactored RSA number for that moment, RSA-170, written in 170 decimal digits. The factorization was finished on 31 December 2009, then we found out that Dominik Bonenberger and Martin Krone [1] were ahead of us for two days and had already presented the RSA-170 prime de- compostion. Meanwhile the new world record in factorization was set [2] – the international team of scientists managed to factor the RSA-768, a 232 decimal digits long RSA number. On January 2010 after a short break we decided to continue the project and took the number RSA-180 = 191147927718986609689229466631454649812986246 276667354864188503638807260703436799058776201 365135161278134258296128109200046702912984568 752800330221777752773957404540495707851421041, the next smallest RSA number with unknown factorization. ∗Revised at 13.04.2010 1 2 Factorization of RSA-180 Our tools for factorization are essentially based on two open source implemen- tations of General Numebr Field Sieve (GNFS) algorithm – the community maintained GGNFS suite [3] and Jason Papadopoulos’s msieve [4]. -
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. -
The Factoring Dead: Preparing for the Cryptopocalypse
THE FACTORING DEAD: PREPARING FOR THE CRYPTOPOCALYPSE Javed Samuel — javed[at]isecpartners[dot]com iSEC Partners, Inc 123 Mission Street, Suite 1020 San Francisco, CA 94105 https://www.isecpartners.com March 20, 2014 Abstract This paper will explain the latest breakthroughs in the academic cryptography community and look ahead at what practical issues could arise for popular cryptosystems. Specifically, we will focus on the recent major devel- opments in discrete mathematics and their potential ability to undermine our trust in the most basic asymmetric primitives, including RSA. We will explain the basic theories behind RSA and the state-of-the-art in large number- ing factoring, and how several recent papers may point the way to massive improvements in this area. The paper will then switch to the practical aspects of the doomsday scenario, and will answer the question “What happens the day after RSA is broken?” We will point out the many obvious and hidden uses of RSA and related algorithms and outline how software engineers and security teams can operate in a post-RSA world. We will also discuss the results of our survey of popular products and software, and point out the ways in which individuals can prepare for the “zombie cryptopocalypse”. 1 INTRODUCTION Over the past few years, there have been numerous attacks on the current SSL infrastructure. These have ranged from BEAST [97], CRIME [88], Lucky 13 [2][86], RC4 bias attacks [1][91] and BREACH [42]. These attacks all show the fragility of the current SSL architecture as vulnerabilities have been found in a variety of features ranging from compression, timing and padding [90]. -
Use of SIMD-Based Data Parallelism to Speed up Sieving in Integer-Factoring Algorithms ?
Use of SIMD-Based Data Parallelism to Speed up Sieving in Integer-Factoring Algorithms ? Binanda Sengupta and Abhijit Das Department of Computer Science and Engineering Indian Institute of Technology Kharagpur, West Bengal, PIN: 721302, India binanda.sengupta,[email protected] Abstract. Many cryptographic protocols derive their security from the appar- ent computational intractability of the integer factorization problem. Currently, the best known integer-factoring algorithms run in subexponential time. Effi- cient parallel implementations of these algorithms constitute an important area of practical research. Most reported implementations use multi-core and/or dis- tributed parallelization. In this paper, we use SIMD-based parallelization to speed up the sieving stage of integer-factoring algorithms. We experiment on the two fastest variants of factoring algorithms: the number-field sieve method and the multiple-polynomial quadratic sieve method. Using Intel’s SSE2 and AVX in- trinsics, we have been able to speed up index calculations in each core during sieving. This performance enhancement is attributed to a reduction in the pack- ing and unpacking overheads associated with SIMD registers. We handle both line sieving and lattice sieving. We also propose improvements to make our im- plementations cache-friendly. We obtain speedup figures in the range 5–40%. To the best of our knowledge, no public discussions on SIMD parallelization in the context of integer-factoring algorithms are available in the literature. Keywords: Integer Factorization, Sieving, Multiple-Polynomial Quadratic Sieve Method, Number-Field Sieve Method, Single Instruction Multiple Data, Stream- ing SIMD Extensions, Advanced Vector Extensions 1 Introduction Let n be a large composite integer having the factorization k v vp1 vp2 pk vpi n = p1 p2 ··· pk = ∏ pi : i=1 The integer factorization problem deals with the determination of all the prime divisors p1; p2;:::; pk of n and their corresponding multiplicities vp1 ;vp2 ;:::;vpk . -
Question 1.1. What Is the RSA Laboratories' Frequently Asked
Copyright © 1996, 1998 RSA Data Security, Inc. All rights reserved. RSA BSAFE Crypto-C, RSA BSAFE Crypto-J, PKCS, S/WAN, RC2, RC4, RC5, MD2, MD4, and MD5 are trade- marks or registered trademarks of RSA Data Security, Inc. Other products and names are trademarks or regis- tered trademarks of their respective owners. For permission to reprint or redistribute in part or in whole, send e-mail to [email protected] or contact your RSA representative. RSA Laboratories’ Frequently Asked Questions About Today’s Cryptography, v4.0 2 Table of Contents Table of Contents............................................................................................ 3 Foreword......................................................................................................... 8 Section 1: Introduction .................................................................................... 9 Question 1.1. What is the RSA Laboratories’ Frequently Asked Questions About Today’s Cryptography? ................................................................................................................ 9 Question 1.2. What is cryptography? ............................................................................................10 Question 1.3. What are some of the more popular techniques in cryptography? ................... 11 Question 1.4. How is cryptography applied? ............................................................................... 12 Question 1.5. What are cryptography standards? ...................................................................... -
Parallel Strategy to Factorize Fermat Numbers with Implementation in Maple Software
Journal of Software Parallel Strategy to Factorize Fermat Numbers with Implementation in Maple Software Jianhui Li*, Manlan Liu Foshan Polytechnic, Foshan City, PRC, 528000, China. * Corresponding author. Tel.: +86075787263015; email: [email protected] Manuscript submitted October 08, 2020; accepted December 11, 2020. doi: 10.17706/jsw.16.4.167-173 Abstract: In accordance with the traits of parallel computing, the paper proposes a parallel algorithm to factorize the Fermat numbers through parallelization of a sequential algorithm. The kernel work to parallelize a sequential algorithm is presented by subdividing the computing interval into subintervals that are assigned to the parallel processes to perform the parallel computing. Maple experiments show that the parallelization increases the computational efficiency of factoring the Fermat numbers, especially to the Fermat number with big divisors. Key words: Integer factorization, fermat number, parallel computing, algorithm. 1. Introduction Factorization of big integer has been a hard problem and has been paid attention to in mathematics and cryptography. Any factoring algorithm is both scientifically valuable and practically applicable, as stated in [1]. Historically, factorization of big integer always took huge computing resources and involved in parallel computing. For example, it took half a year and 80 4-cored CPUs of AMD Opteron @ 2.2GHz to factorize RSA768 [2]. Actually, Brent R P began to study the possibility of parallel computing early in 1990 since Pollard Rho algorithm came into being [3]. And then parallel approaches for the elliptic curve method (ECM), continued fraction, quadric sieve (QS) and number field sieve (NFS) were developed respectively [4]-[7]. Now the NFS has been regarded to be the most efficient method. -
Algorithms for Factoring Integers of the Form N = P * Q
Algorithms for Factoring Integers of the Form n = p * q Robert Johnson Math 414 Final Project 12 March 2010 1 Table of Contents 1. Introduction 3 2. Trial Division Algorithm 3 3. Pollard (p – 1) Algorithm 5 4. Lenstra Elliptic Curve Method 9 5. Continued Fraction Factorization Method 13 6. Conclusion 15 2 Introduction This paper is a brief overview of four methods used to factor integers. The premise is that the integers are of the form n = p * q, i.e. similar to RSA numbers. The four methods presented are Trial Division, the Pollard (p -1) algorithm, the Lenstra Elliptic Curve Method, and the Continued Fraction Factorization Method. In each of the sections, we delve into how the algorithm came about, some background on the machinery that is implemented in the algorithm, the algorithm itself, as well as a few eXamples. This paper assumes the reader has a little eXperience with group theory and ring theory and an understanding of algebraic functions. This paper also assumes the reader has eXperience with programming so that they might be able to read and understand the code for the algorithms. The math introduced in this paper is oriented in such a way that people with a little eXperience in number theory may still be able to read and understand the algorithms and their implementation. The goal of this paper is to have those readers with a little eXperience in number theory be able to understand and use the algorithms presented in this paper after they are finished reading. Trial Division Algorithm Trial division is the simplest of all the factoring algorithms as well as the easiest to understand. -
Factorization of a 512–Bit RSA Modulus?
Factorization of a 512{Bit RSA Modulus? Stefania Cavallar3,BruceDodson8,ArjenK.Lenstra1,WalterLioen3, Peter L. Montgomery10, Brian Murphy2, Herman te Riele3, Karen Aardal13, 4 11 9 5 View metadata, citation and similar papersJeff at core.ac.uk Gilchrist ,G´erard Guillerm ,PaulLeyland ,Jo¨el Marchand , brought to you by CORE 6 12 14 Fran¸cois Morain , Alec Muffett , Chris and Craig Putnamprovided,and by Infoscience - École polytechnique fédérale de Lausanne Paul Zimmermann7 1 Citibank, 1 North Gate Road, Mendham, NJ 07945–3104, USA [email protected] 2 Computer Sciences Laboratory, ANU, Canberra ACT 0200, Australia [email protected] 3 CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands {cavallar,walter,herman}@cwi.nl 4 Entrust Technologies Ltd., 750 Heron Road, Suite E08, Ottawa, ON, K1V 1A7, Canada [email protected] 5 Laboratoire Gage, Ecole´ Polytechnique/CNRS, Palaiseau, France [email protected] 6 Laboratoire d’Informatique, Ecole´ Polytechnique, Palaiseau, France [email protected] 7 Inria Lorraine and Loria, Nancy, France [email protected] 8 Lehigh University, Bethlehem, PA, USA [email protected] 9 Microsoft Research Ltd, Cambridge, UK [email protected] 10 780 Las Colindas Road, San Rafael, CA 94903–2346 USA Microsoft Research and CWI [email protected] 11 SITX (Centre of IT resources), Ecole´ Polytechnique, Palaiseau, France [email protected] 12 Sun Microsystems, Riverside Way, Watchmoor Park, Camberley, UK [email protected] 13 Dept. of Computer Science,