# Factorization of a 1061-bit number by the Special Number Field Sieve

Save this PDF as:

Size: px
Start display at page:

Download "Factorization of a 1061-bit number by the Special Number Field Sieve"

## Transcription

1 Factorization of a 1061-bit number by the Special Number Field Sieve Greg Childers California State University Fullerton Fullerton, CA August 4, 2012 Abstract I provide the details of the factorization of the Mersenne number by the Special Number Field Sieve. Although this factorization is easier than the completed factorization of RSA-768, it represents a new milestone for factorization using publicly available software. 1 Introduction The Number Field Sieve (NFS) is currently the fastest classical algorithm for factoring a large integer into its prime cofactors [8]. Continued study of the practical implementations of the NFS is of significant interest for the security assessment of common public-key cryptosystems, chief among them being the RSA algorithm. The security of the RSA encryption algorithm relies on the fact that integer factorization is difficult. Improvements to the NFS algorithm are of significant practical importance, and factoring milestones are followed by the applied cryptography community. Significant milestones include the factoring of a 512-bit RSA modulus by the general NFS (GNFS) in 2000 [4], by the special NFS (SNFS) in 2007 [2], and a 768-bit RSA modulus in 2010 [7]. Using the SNFS, the complete factorization of the Mersenne number, , has been determined. Prior this this effort, this number had no known factors. Although easier than the factorization of RSA-768, this represents a new largest factorization using SNFS, and the largest factorization to date using publicly available software. NFS is comprised of five basic steps: polynomial selection, sieving for relations, filtering of relations, linear algebra, and square root. Each of these steps will be detailed below. 1

2 2 Polynomial selection Polynomial selection consists of finding two polynomials that share a common root modulo the number being factored. For this SNFS factorization, selection of appropriate polynomials is trivial. The polynomials f(x) = x 6 2 and g(x) = x share the common root modulo Sieving Sieving was by far the most computationally intensive step in this factorization. Although sieving is somewhat memory intensive, requiring one to two gigabytes per process, the individual sieving tasks do not need to interact. Taking advantage of the increasing amount of memory in today s typical home computer, I administer a distributed computing project to provide the capacity computing required for the sieving. Using the Berkeley Open Infrastructure for Network Computing (BOINC) [1], the project distributes the NFS sieving tasks to volunteer computers and collects the results [5]. The sieving binaries used are based on the gnfs-lasieve lattice sieve code written by Jens Franke and Thorsten Kleinjung [6] adapted for use in the BOINC framework. Factor base bounds of 250 million were used on both the algebraic and rational sides to reduce the amount of memory required for the computation. A large prime bound of 2 33 was used on both sides with up to two large primes allowed on the algebraic side and up to three large primes on the rational side. Sieving was performed over a rectangular region of size using the gnfs-lasieve4i16e binary. These sieving parameters turned out to be suboptimal, requiring sieving over a significantly larger range of q than initially expected. Most of the special-q between 20 million and 1.45 billion on the rational side, and between 20 million and 1.07 billion on the algebraic side were sieved. The total computational effort expended on sieving was approximately 3 CPU-centuries and yielded 671, 385, 523 unique relations. 4 Filtering Filtering was performed using the MSIEVE software library [10]. Filtering started with the set of approximately 671 million unique relations. Following the singleton and clique removal steps, the matrix had approximate 282 million rows. Following the merge phase, a final matrix of 90.3 million rows and columns with an average weight of per column was produced. Filtering required approximately 50 CPU-hours and 40 gigabytes of memory to complete, and the final matrix in binary form requires approximately 40 gigabytes of 2

3 storage. 5 Linear algebra The MSIEVE software library uses the block Lanczos algorithm [9] for the linear algebra. In collaboration with Jason Papadopoulos, the author of the MSIEVE library, and Ilya Popovyan [11], a parallel version of MSIEVE utilizing MPI has been written and continues to be optimized. The linear algebra was performed using a MPI grid (576 total cores), and was split between the Trestles cluster at the San Diego Supercomputing Center and the Lonestar cluster at the Texas Advanced Computing Center due to allocation details. On both clusters the nodes use DDR3 memory, and communication between nodes within the cluster uses Infiniband interconnects. The linear algebra required approximately 35 CPU-years, and yielded 32 non-trivial dependencies. 6 Square root MSIEVE uses a straightforward, but memory-intensive, algorithm for the algebraic square root. This involves multiplying all the relations involved in a non-trivial dependency from the linear algebra modulo the monic algebraic polynomial f(x). Because of the size of the products, FFT-based multiplication is used. For , this product completed in 9 CPU-hours, required 14 gigabytes of memory, and produced a fifth degree polynomial with coefficients 4.03 gigabits in size. The algebraic square root, calculated using q-adic Newton iteration, required 13.5 CPU-hours. Each dependency has a 50% chance of yielding the factors, and the factors were found on the second dependency. The square root step required a total of 45 CPU-hours. 7 Results The number is the product of 143-digit and 177-digit prime numbers, P 143 P 177, where P 143 =

4 P 177 = The factorizations of P 143 ± 1 and P 177 ± 1 are P = P = P = P = These factors will be published in the fourth edition of the Cunningham book [3] and are currently available in the factorization tables of the Cunningham project website [12]. 8 Acknowledgments This calculation could not have been performed without the free, publicly available lattice sieve of Franke and Kleinjung, MSIEVE by Jason Papadopoulos, and the BOINC infrastructure. Computational time on the Trestles and Lonestar clusters is provided by National Science Foundation XSEDE grant number TG-DMS And finally, this effort would not have been possible without the over 1,000 individuals who donate time on their personal computers to the project. 4

5 References [1] D. P. Anderson. BOINC: a system for public-resource computing and storage. In 5th IEEE/ACM International Workshop on Grid Computing, pages 4 10, [2] K. Aoki, J. Franke, T. Kleinjung, A. K. Lenstra, and D. A. Osvik. A kilobit special number field sieve factorization. In Advances in Cryptology ASIACRYPT Springer, Berlin / Heidelberg, [3] J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr. Factorizations of b n ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers. American Mathematical Society, Providence, Rhode Island, third edition, Available online at [4] S. Cavallar, B. Dodson, A. K. Lenstra, W. M. Lioen, P. L. Montgomery, B. Murphy, H. J. J. te Riele, K. Aardal, J. Gilchrist, G. Guillerm, P. C. Leyland, J. Marchand, F. Morain, A. Muffett, C. Putnam, C. Putnam, and P. Zimmermann. Factorization of a 512-bit RSA modulus. In Advances in Cryptology EUROCRYPT Springer, Berlin / Heidelberg, [5] G. Childers [6] J. Franke and T. Kleinjung. Continued fractions and lattice sieving. Proceedings of SHARCS 2005, Source available for download from [7] T. Kleinjung, K. Aoki, J. Franke, A. Lenstra, E. Thomé, J. Bos, P. Gaudry, A. Kruppa, P. Montgomery, D. A. Osvik, H. te Riele, A. Timofeev, and P. Zimmermann. Factorization of a 768-bit RSA modulus. Cryptology eprint Archive, Report 2010/006, [8] A. K. Lenstra and H. W. Lenstra, Jr. The development of the number field sieve. In Lecture Notes in Mathematics, Volume Springer-Verlag, Berlin / Heidelberg, [9] P. L. Montgomery. A block Lanczos algorithm for finding dependencies over GF(2). In EUROCRYPT 95: Proceedings of the 14th annual international conference on theory and application of cryptographic techniques, pages , Berlin / Heidelberg, Springer-Verlag. [10] J. Papadopoulos. MSIEVE, [11] I. Popovyan. Efficient parallelization of Lanczos type algorithms. Cryptology eprint Archive, Report 2011/416, [12] S. S. Wagstaff, Jr. The Cunningham Project, ssw/cun/. 5

### The filtering step of discrete logarithm and integer factorization algorithms

The filtering step of discrete logarithm and integer factorization algorithms Cyril Bouvier To cite this version: Cyril Bouvier. The filtering step of discrete logarithm and integer factorization algorithms.

### A kilobit special number field sieve factorization

A kilobit special number field sieve factorization Kazumaro Aoki 1, Jens Franke 2, Thorsten Kleinjung 2, Arjen K. Lenstra 3, and Dag Arne Osvik 3 1 NTT, 1-1 Hikarinooka, Yokosuka-shi, Kanagawa-ken, 239-0847

### A Kilobit Special Number Field Sieve Factorization

A Kilobit Special Number Field Sieve Factorization Kazumaro Aoki 1,JensFranke 2, Thorsten Kleinjung 2, Arjen K. Lenstra 3,4,andDagArneOsvik 3 1 NTT, 3-9-11 Midori-cho, Musashino-shi, Tokyo, 180-8585 Japan

### Factorization of a 512 bit RSA Modulus

Factorization of a 512 bit RSA Modulus Stefania Cavallar 3, Bruce Dodson 8, Arjen K. Lenstra 1, Walter Lioen 3, Peter L. Montgomery 10, Brian Murphy 2, Herman te Riele 3, Karen Aardal 13, Jeff Gilchrist

### The Quadratic Sieve Factoring Algorithm

The Quadratic Sieve Factoring Algorithm Eric Landquist MATH 488: Cryptographic Algorithms December 14, 2001 1 Introduction Mathematicians have been attempting to find better and faster ways to factor composite

### Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give

### On the coefficients of the polynomial in the number field sieve

On the coefficients of the polynomial in the number field sieve Yang Min a, Meng Qingshu b,, Wang Zhangyi b, Li Li a, Zhang Huanguo b a International School of Software, Wuhan University, Hubei, China,

### Primality Testing and Factorization Methods

Primality Testing and Factorization Methods Eli Howey May 27, 2014 Abstract Since the days of Euclid and Eratosthenes, mathematicians have taken a keen interest in finding the nontrivial factors of integers,

### Factorization of RSA-140 using the number field sieve

Centrum voor Wiskunde en Informatica Factorization of RSA-140 using the number field sieve S. Cavallar, B. Dodson, A.K. Lenstra, P.C. Leyland, W.M. Lioen, P.L. Montgomery, B. Murphy, H.J.J. te Riele, P.

### Factoring integers and Producing primes

Factoring integers,..., RSA Erbil, Kurdistan 0 Lecture in Number Theory College of Sciences Department of Mathematics University of Salahaddin Debember 4, 2014 Factoring integers and Producing primes Francesco

### Optimization of the MPQS-factoring algorithm on the Cyber 205 and the NEC SX-2

Optimization of the MPQS-factoring algorithm on the Cyber 205 and the NEC SX-2 Walter Lioen, Herman te Riele, Dik Winter CWI P.O. Box 94079, 1090 GB Amsterdam, The Netherlands ABSTRACT This paper describes

### Factorization of a 768-Bit RSA Modulus

Factorization of a 768-Bit RSA Modulus Thorsten Kleinjung 1, Kazumaro Aoki 2,JensFranke 3,ArjenK.Lenstra 1, Emmanuel Thomé 4,JoppeW.Bos 1, Pierrick Gaudry 4, Alexander Kruppa 4, Peter L. Montgomery 5,6,DagArneOsvik

### Factorizations of a n ± 1, 13 a < 100

Factorizations of a n ± 1, 13 a < 100 Richard P. Brent Computer Sciences Laboratory, Australian National University GPO Box 4, Canberra, ACT 2601, Australia e-mail: rpb@cslab.anu.edu.au and Herman J. J.

### Evaluation Report on the Factoring Problem

Evaluation Report on the Factoring Problem Jacques Stern 1 Introduction This document is an evaluation of the factoring problem, as a basis for designing cryptographic schemes. It relies on the analysis

### Factoring as a Service

Factoring as a Service Luke Valenta, Shaanan Cohney, Alex Liao, Joshua Fried, Satya Bodduluri, Nadia Heninger University of Pennsylvania Abstract The difficulty of integer factorization is fundamental

### ' DEC SRC, 130 Lytton Avenue, Palo Alto, CA 94301, U.S.A

On the factorization of RSA-120 T. Denny1, B. Dodson2, A. K. Lenstra3, M. S. Manasse4 * Lehrstuhl Prof. Buchmann, Fachbereich Informatik, Universitit des Saarlandes, Postfach 1150, 66041 Saarbriicken,

### Public-Key Cryptanalysis 1: Introduction and Factoring

Public-Key Cryptanalysis 1: Introduction and Factoring Nadia Heninger University of Pennsylvania July 21, 2013 Adventures in Cryptanalysis Part 1: Introduction and Factoring. What is public-key crypto

### Factoring Algorithms

Institutionen för Informationsteknologi Lunds Tekniska Högskola Department of Information Technology Lund University Cryptology - Project 1 Factoring Algorithms The purpose of this project is to understand

### International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013

FACTORING CRYPTOSYSTEM MODULI WHEN THE CO-FACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II Mohammedia-Casablanca,

### Factoring with the Quadratic Sieve on Large Vector Computers

Factoring with the Quadratic Sieve on Large Vector Computers Herman te Riele, Walter Lioen and Dik Winter Centre for Mathematics and Computer Science P.O. Box 4079, 1009 AB Amsterdam, The Netherlands The

### Runtime and Implementation of Factoring Algorithms: A Comparison

Runtime and Implementation of Factoring Algorithms: A Comparison Justin Moore CSC290 Cryptology December 20, 2003 Abstract Factoring composite numbers is not an easy task. It is classified as a hard algorithm,

### SHARK A Realizable Special Hardware Sieving Device for Factoring 1024-bit Integers

SHARK A Realizable Special Hardware Sieving Device for Factoring 1024-bit Integers Jens Franke 1, Thorsten Kleinjung 1, Christof Paar 2, Jan Pelzl 2, Christine Priplata 3, Colin Stahlke 3 1 University

### CGHub Client Security Guide Documentation

CGHub Client Security Guide Documentation Release 3.1 University of California, Santa Cruz April 16, 2014 CONTENTS 1 Abstract 1 2 GeneTorrent: a secure, client/server BitTorrent 2 2.1 GeneTorrent protocols.....................................

### Square Root Algorithms for the Number Field Sieve

Square Root Algorithms for the Number Field Sieve Emmanuel Thomé INRIA Nancy, Villers-lès-Nancy, France Abstract. We review several methods for the square root step of the Number Field Sieve, and present

### The number field sieve

The number field sieve A.K. Lenstra Bellcore, 435 South Street, Morristown, NJ 07960 H.W. Lenstra, Jr. Department of Mathematics, University of California, Berkeley, CA 94720 M.S. Manasse DEC SRC, 130

### Factoring pq 2 with Quadratic Forms: Nice Cryptanalyses

Factoring pq 2 with Quadratic Forms: Nice Cryptanalyses Phong Nguyễn http://www.di.ens.fr/~pnguyen & ASIACRYPT 2009 Joint work with G. Castagnos, A. Joux and F. Laguillaumie Summary Factoring A New Factoring

### Comparative Analysis for Performance acceleration of Modern Asymmetric Crypto Systems

J. of Comp. and I.T. Vol. 3(1&2), 1-6 (2012). Comparative Analysis for Performance acceleration of Modern Asymmetric Crypto Systems RAJ KUMAR 1 and V.K. SARASWAT 2 1,2 Department of Computer Science, ICIS

### Arithmetic algorithms for cryptology 5 October 2015, Paris. Sieves. Razvan Barbulescu CNRS and IMJ-PRG. R. Barbulescu Sieves 0 / 28

Arithmetic algorithms for cryptology 5 October 2015, Paris Sieves Razvan Barbulescu CNRS and IMJ-PRG R. Barbulescu Sieves 0 / 28 Starting point Notations q prime g a generator of (F q ) X a (secret) integer

### Factoring N = p r q for Large r

Factoring N = p r q for Large r Dan Boneh 1,GlennDurfee 1, and Nick Howgrave-Graham 2 1 Computer Science Department, Stanford University, Stanford, CA 94305-9045 {dabo,gdurf}@cs.stanford.edu 2 Mathematical

### Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR

Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices A Biswas, IT, BESU SHIBPUR McGraw-Hill The McGraw-Hill Companies, Inc., 2000 Set of Integers The set of integers, denoted by Z,

### Mathematics of Cryptography Part I

CHAPTER 2 Mathematics of Cryptography Part I (Solution to Odd-Numbered Problems) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinity to positive infinity.

### Factorization Methods: Very Quick Overview

Factorization Methods: Very Quick Overview Yuval Filmus October 17, 2012 1 Introduction In this lecture we introduce modern factorization methods. We will assume several facts from analytic number theory.

### The van Hoeij Algorithm for Factoring Polynomials

The van Hoeij Algorithm for Factoring Polynomials Jürgen Klüners Abstract In this survey we report about a new algorithm for factoring polynomials due to Mark van Hoeij. The main idea is that the combinatorial

### Solinas primes of small weight for fixed sizes

Solinas primes of small weight for fixed sizes José de Jesús Angel Angel and Guillermo Morales-Luna Computer Science Department, CINVESTAV-IPN, Mexico jjangel@computacion.cs.cinvestav.mx gmorales@cs.cinvestav.mx

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Special-Purpose Hardware for Factoring: the NFS Sieving Step

Special-Purpose Hardware for Factoring: the NFS Sieving Step Adi Shamir Eran Tromer Weizmann Institute of Science {shamir, tromer}@wisdom.weizmann.ac.il Abstract In the quest for factorization of larger

### Factoring integers, Producing primes and the RSA cryptosystem

Factoring integers,..., RSA Erbil, Kurdistan 0 Lecture in Number Theory College of Sciences Department of Mathematics University of Salahaddin Debember 1, 2014 Factoring integers, Producing primes and

### FACTORING WITH TWO LARGE PRIMES

mathematics of computation volume 63, number 208 october 1994, pages 785-798 FACTORING WITH TWO LARGE PRIMES A. K. LENSTRA AND M. S. MANASSE Abstract. We describe a modification to the well-known large

### Prime Numbers The generation of prime numbers is needed for many public key algorithms:

CA547: CRYPTOGRAPHY AND SECURITY PROTOCOLS 1 7 Number Theory 2 7.1 Prime Numbers Prime Numbers The generation of prime numbers is needed for many public key algorithms: RSA: Need to find p and q to compute

### RSA Question 2. Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true?

RSA Question 2 Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true? Bob chooses a random e (1 < e < Φ Bob ) such that gcd(e,φ Bob )=1. Then, d = e -1

### 2 Primality and Compositeness Tests

Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 33, 1635-1642 On Factoring R. A. Mollin Department of Mathematics and Statistics University of Calgary, Calgary, Alberta, Canada, T2N 1N4 http://www.math.ucalgary.ca/

### A Factoring and Discrete Logarithm based Cryptosystem

Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 11, 511-517 HIKARI Ltd, www.m-hikari.com A Factoring and Discrete Logarithm based Cryptosystem Abdoul Aziz Ciss and Ahmed Youssef Ecole doctorale de Mathematiques

### Two Integer Factorization Methods

Two Integer Factorization Methods Christopher Koch April 22, 2014 Abstract Integer factorization methods are algorithms that find the prime divisors of any positive integer. Besides studying trial division

### Cryptography. Course 2: attacks against RSA. Jean-Sébastien Coron. September 26, Université du Luxembourg

Course 2: attacks against RSA Université du Luxembourg September 26, 2010 Attacks against RSA Factoring Equivalence between factoring and breaking RSA? Mathematical attacks Attacks against plain RSA encryption

### On the Cost of Factoring RSA-1024

On the Cost of Factoring RSA-1024 Adi Shamir Eran Tromer Weizmann Institute of Science shamir,tromer @wisdom.weizmann.ac.il Abstract As many cryptographic schemes rely on the hardness of integer factorization,

### The Mathematical Cryptography of the RSA Cryptosystem

The Mathematical Cryptography of the RSA Cryptosystem Abderrahmane Nitaj Laboratoire de Mathématiques Nicolas Oresme Université de Caen, France abderrahmanenitaj@unicaenfr http://wwwmathunicaenfr/~nitaj

### The RSA Algorithm: A Mathematical History of the Ubiquitous Cryptological Algorithm

The RSA Algorithm: A Mathematical History of the Ubiquitous Cryptological Algorithm Maria D. Kelly December 7, 2009 Abstract The RSA algorithm, developed in 1977 by Rivest, Shamir, and Adlemen, is an algorithm

### Update # 5 to Factorizations of b n ± 1

Update # 5 to Factorizations of b n ± 1 Samuel S. Wagstaff, Jr. The following tables present the updates made to Factorizations of b n ± 1from October 23, 1982, when the first edition went to press, to

### Faster deterministic integer factorisation

David Harvey (joint work with Edgar Costa, NYU) University of New South Wales 25th October 2011 The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers

### Finding Small Roots of Bivariate Integer Polynomial Equations Revisited

Finding Small Roots of Bivariate Integer Polynomial Equations Revisited Jean-Sébastien Coron Gemplus Card International 34 rue Guynemer, 92447 Issy-les-Moulineaux, France jean-sebastien.coron@gemplus.com

### When e-th Roots Become Easier Than Factoring

When e-th Roots Become Easier Than Factoring Antoine Joux 1, David Naccache 2, and Emmanuel Thomé 3 1 dga and Université de Versailles, uvsq prism 45 avenue des États-Unis, f-78035 Versailles cedex, France

### Determining the Optimal Combination of Trial Division and Fermat s Factorization Method

Determining the Optimal Combination of Trial Division and Fermat s Factorization Method Joseph C. Woodson Home School P. O. Box 55005 Tulsa, OK 74155 Abstract The process of finding the prime factorization

### Factoring. Factoring 1

Factoring Factoring 1 Factoring Security of RSA algorithm depends on (presumed) difficulty of factoring o Given N = pq, find p or q and RSA is broken o Rabin cipher also based on factoring Factoring like

### An Efficient Hardware Architecture for Factoring Integers with the Elliptic Curve Method

An Efficient Hardware Architecture for Factoring Integers with the Elliptic Curve Method Jens Franke 1, Thorsten Kleinjung 1, Christof Paar 2, Jan Pelzl 2, Christine Priplata 3, Martin Šimka4, Colin Stahlke

### Factoring Cubic Polynomials

Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than

### Factoring a semiprime n by estimating φ(n)

Factoring a semiprime n by estimating φ(n) Kyle Kloster May 7, 2010 Abstract A factoring algorithm, called the Phi-Finder algorithm, is presented that factors a product of two primes, n = pq, by determining

### 3. Applications of Number Theory

3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a

### Integer factorization is in P

Integer factorization is in P Yuly Shipilevsky Toronto, Ontario, Canada E-mail address: yulysh2000@yahoo.ca Abstract A polynomial-time algorithm for integer factorization, wherein integer factorization

### March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### PUBLIC KEY ENCRYPTION

PUBLIC KEY ENCRYPTION http://www.tutorialspoint.com/cryptography/public_key_encryption.htm Copyright tutorialspoint.com Public Key Cryptography Unlike symmetric key cryptography, we do not find historical

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### Cryptosystem. Diploma Thesis. Mol Petros. July 17, 2006. Supervisor: Stathis Zachos

s and s and Diploma Thesis Department of Electrical and Computer Engineering, National Technical University of Athens July 17, 2006 Supervisor: Stathis Zachos ol Petros (Department of Electrical and Computer

### Index Calculation Attacks on RSA Signature and Encryption

Index Calculation Attacks on RSA Signature and Encryption Jean-Sébastien Coron 1, Yvo Desmedt 2, David Naccache 1, Andrew Odlyzko 3, and Julien P. Stern 4 1 Gemplus Card International {jean-sebastien.coron,david.naccache}@gemplus.com

### Factoring Report. MEC Consulting (communicated via RSA Security) Dr.Preda Mihailescu

Factoring Report 2001 12 4 MEC Consulting (communicated via RSA Security) Dr.Preda Mihailescu Factoring Report Dr. Preda Mihailescu MEC Consulting Seestr. 78, 8700 Erlenbach Zürich Email: preda@upb.de

### In this paper a new signature scheme and a public key cryptotsystem are proposed. They can be seen as a compromise between the RSA and ElGamal-type sc

Digital Signature and Public Key Cryptosystem in a Prime Order Subgroup of Z n Colin Boyd Information Security Research Centre, School of Data Communications Queensland University of Technology, Brisbane

### Chapter 9 Public Key Cryptography and RSA

Chapter 9 Public Key Cryptography and RSA Cryptography and Network Security: Principles and Practices (3rd Ed.) 2004/1/15 1 9.1 Principles of Public Key Private-Key Cryptography traditional private/secret/single

### Study of algorithms for factoring integers and computing discrete logarithms

Study of algorithms for factoring integers and computing discrete logarithms First Indo-French Workshop on Cryptography and Related Topics (IFW 2007) June 11 13, 2007 Paris, France Dr. Abhijit Das Department

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

### Mathematics of Internet Security. Keeping Eve The Eavesdropper Away From Your Credit Card Information

The : Keeping Eve The Eavesdropper Away From Your Credit Card Information Department of Mathematics North Dakota State University 16 September 2010 Science Cafe Introduction Disclaimer: is not an internet

### MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins

MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins The RSA encryption scheme works as follows. In order to establish the necessary public

### Mathematics of Computation, Vol. 41, No. 163. (Jul., 1983), pp. 287-294.

Factoring Large Numbers with a Quadratic Sieve Joseph L. Gerver Mathematics of Computation, Vol. 41, No. 163. (Jul., 1983), pp. 287-294. Stable URL: http://links.jstor.org/sici?sici=0025-5718%28198307%2941%3a163%3c287%3aflnwaq%3e2.0.co%3b2-4

### Public-Key Cryptography. Oregon State University

Public-Key Cryptography Çetin Kaya Koç Oregon State University 1 Sender M Receiver Adversary Objective: Secure communication over an insecure channel 2 Solution: Secret-key cryptography Exchange the key

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### RSA Attacks. By Abdulaziz Alrasheed and Fatima

RSA Attacks By Abdulaziz Alrasheed and Fatima 1 Introduction Invented by Ron Rivest, Adi Shamir, and Len Adleman [1], the RSA cryptosystem was first revealed in the August 1977 issue of Scientific American.

### A New Generic Digital Signature Algorithm

Groups Complex. Cryptol.? (????), 1 16 DOI 10.1515/GCC.????.??? de Gruyter???? A New Generic Digital Signature Algorithm Jennifer Seberry, Vinhbuu To and Dongvu Tonien Abstract. In this paper, we study

### Implementing Network Security Protocols

Implementing Network Security Protocols based on Elliptic Curve Cryptography M. Aydos, E. Savaş, and Ç. K. Koç Electrical & Computer Engineering Oregon State University Corvallis, Oregon 97331, USA {aydos,savas,koc}@ece.orst.edu

### RSA Cryptosystem. Yufei Tao. Department of Computer Science and Engineering Chinese University of Hong Kong. RSA Cryptosystem

Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong In this lecture, we will discuss the RSA cryptosystem, which is widely adopted as a way to encrypt a message, or

### MATH 168: FINAL PROJECT Troels Eriksen. 1 Introduction

MATH 168: FINAL PROJECT Troels Eriksen 1 Introduction In the later years cryptosystems using elliptic curves have shown up and are claimed to be just as secure as a system like RSA with much smaller key

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

### The Advanced Encryption Standard (AES)

The Advanced Encryption Standard (AES) Conception - Why A New Cipher? Conception - Why A New Cipher? DES had outlived its usefulness Vulnerabilities were becoming known 56-bit key was too small Too slow

### SECURITY IMPROVMENTS TO THE DIFFIE-HELLMAN SCHEMES

www.arpapress.com/volumes/vol8issue1/ijrras_8_1_10.pdf SECURITY IMPROVMENTS TO THE DIFFIE-HELLMAN SCHEMES Malek Jakob Kakish Amman Arab University, Department of Computer Information Systems, P.O.Box 2234,

### UNIVERSITY OF MASSACHUSETTS Dept. of Electrical & Computer Engineering. Introduction to Cryptography ECE 597XX/697XX

UNIVERSITY OF MASSACHUSETTS Dept. of Electrical & Computer Engineering Introduction to Cryptography ECE 597XX/697XX Part 6 Introduction to Public-Key Cryptography Israel Koren ECE597/697 Koren Part.6.1

### An Overview of Integer Factoring Algorithms. The Problem

An Overview of Integer Factoring Algorithms Manindra Agrawal IITK / NUS The Problem Given an integer n, find all its prime divisors as efficiently as possible. 1 A Difficult Problem No efficient algorithm

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### A Tool Kit for Finding Small Roots of Bivariate Polynomials over the Integers

A Tool Kit for Finding Small Roots of Bivariate Polynomials over the Integers Johannes Blömer, Alexander May Faculty of Computer Science, Electrical Engineering and Mathematics University of Paderborn

### Basic Algorithms In Computer Algebra

Basic Algorithms In Computer Algebra Kaiserslautern SS 2011 Prof. Dr. Wolfram Decker 2. Mai 2011 References Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, 1993. Cox, D.; Little,

### Principles of Public Key Cryptography. Applications of Public Key Cryptography. Security in Public Key Algorithms

Principles of Public Key Cryptography Chapter : Security Techniques Background Secret Key Cryptography Public Key Cryptography Hash Functions Authentication Chapter : Security on Network and Transport

### ECE 842 Report Implementation of Elliptic Curve Cryptography

ECE 842 Report Implementation of Elliptic Curve Cryptography Wei-Yang Lin December 15, 2004 Abstract The aim of this report is to illustrate the issues in implementing a practical elliptic curve cryptographic

### Factoring Algorithms

Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

### Breaking 128-bit Secure Supersingular Binary Curves

Breaking 128-bit Secure Supersingular Binary Curves (or how to solve discrete logarithms in F 2 4 1223 and F 2 12 367) Robert Granger 1, Thorsten Kleinjung 1, and Jens Zumbrägel 2 1 Laboratory for Cryptologic

### Enhancing Advanced Encryption Standard S-Box Generation Based on Round Key

Enhancing Advanced Encryption Standard S-Box Generation Based on Round Key Julia Juremi Ramlan Mahmod Salasiah Sulaiman Jazrin Ramli Faculty of Computer Science and Information Technology, Universiti Putra

### Modular arithmetic. x ymodn if x = y +mn for some integer m. p. 1/??

p. 1/?? Modular arithmetic Much of modern number theory, and many practical problems (including problems in cryptography and computer science), are concerned with modular arithmetic. While this is probably

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### Ron was wrong, Whit is right

Ron was wrong, Whit is right Arjen K. Lenstra, James P. Hughes 2, Maxime Augier, Joppe W. Bos, Thorsten Kleinjung, and Christophe Wachter EPFL IC LACAL, Station 4, CH-05 Lausanne, Switzerland 2 Self, Palo

### Prime and Composite Terms in Sloane s Sequence A056542

1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.3 Prime and Composite Terms in Sloane s Sequence A056542 Tom Müller Institute for Cusanus-Research University and Theological

### STUDY ON ELLIPTIC AND HYPERELLIPTIC CURVE METHODS FOR INTEGER FACTORIZATION. Takayuki Yato. A Senior Thesis. Submitted to

STUDY ON ELLIPTIC AND HYPERELLIPTIC CURVE METHODS FOR INTEGER FACTORIZATION by Takayuki Yato A Senior Thesis Submitted to Department of Information Science Faculty of Science The University of Tokyo on