Basic Algorithms In Computer Algebra


 Jemima Little
 3 years ago
 Views:
Transcription
1 Basic Algorithms In Computer Algebra Kaiserslautern SS 2011 Prof. Dr. Wolfram Decker 2. Mai 2011
2
3 References Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Cox, D.; Little, J.; O Shea, D.: Ideals, Varieties, and Algorithms. Springer, Forster, O.: Algorithmische Zahlentheorie, Vieweg, Gerhard, J.; von zur Gathen, J.: Modern Computer Algebra. Cambridge Press, Greuel, F.M.; Pfister, G.: A Singular introduction to Commutative Algebra. 2nd ed., Springer, Kaplan, M.: Computer Algebra. Springer, Knuth, D.E.: The Art of Computer Programming. Volumes 1,2,3. AddisonWesley, Lidl, H.; Niederreiter, H.: Introduction to Finite Fields. Cambridge University Press, PYTHON documentation, SINGULAR Online Manual,
4
5 0 Introduction Computer algebra is a discipline between mathematics and computer science which deals with designing, analyzing, implementing, and applying algebraic algorithms. We comment on this in more detail in what follows. 0.1 Application Areas Computer algebra is interdisciplinary in nature, with links to quite a number of areas in mathematics, with applications in mathematics, other branches of science, and engineering: Through computer algebra methods, a number of mathematical disciplines become accessible to experiments. This is in particular true for various parts of algebra, number theory, and geometry. See and SwinnertonDyer Conjecture/ for a famous conjecture in number theory which is based on computer experiments. Algebraic algorithms open up new ways of computing in classical application areas of mathematics in science (physics, chemistry, biology). A pioneering and prominent example is work by Veltman and t Hooft who won a Nobel price in physics in 1999 (awarded for having placed particle physics theory on a firmer mathematical foundation ). Modern application areas of mathematics such as cryptography, coding theory, CAD, robotics, algebraic statistics, and algebraic biology heavily rely on computer algebra. 0.2 Implementations There is a large variety of computer algebra systems suiting different needs. See and the slides from the first lecture of this course at decker/lehre/ss11/pramasymbol/index.html. The practical sessions accompanying the course feature the modern programming language Python (see and the computer algebra system Singular (see 5
6 0 INTRODUCTION 0.3 Analyzing Algorithms One way of measuring the efficiency of an algorithm is to give assymptotic bounds on its running time which depend on the size of the input. For this, we will use the bigoh notation. 0.4 Designing Algorithms When designing algorithms, we will describe them in a somewhat informal way which makes use of the structural conventions of a programming language. We refer to such a description as pseudocode. Here is a classical example which should be familiar from the beginner s course Algebraische Strukturen: Algorithm 1 Euclid s Algorithm Input: m, n R, n 0, where R is an Euclidean domain Output: gcd(m, n) a := m, b := n while (b 0) do r := rem(a, b) // division with remainder a := b, b := r a := n(a) // normal form return (a) Some comments on this are in order. In any Euclidean domain R, with Euclidean function v, if a, b R with b 0, and if we write a = q b + r, with r = 0 or v(r) < v(b), quo(a, b) = q and rem(a, b) = r for the quotient q and the remainder r, respectively. Note, however, that q and r need not be unique. If R = Z, with v given by the absolute value, uniqueness is achieved by the additional requirement that r 0. In the case where R = K[x] is the polynomial ring over a field K, with v given by the degree, q and r are unique without further requirement. Remark 0.5. If R is any integral domain, and if a greatest common divisor c of two elements a, b R exists, all such divisors are obtained by multyplying c with a unit of R. That is, the greatest common divisors form an equivalence class under being associated (see the first sheet of exercises). In this lecture, we always assume that in each such equivalence class a normal form is selected. If the class is represented by a R, we write n(a) for the normal form. Here, we set n(0) = 0 and n(1) = 1 (the normal form of the units in R), and require that n(a b) = n(a) n(b). If a 0, then a = u(a) n(a) for a uniquely determined unit u(a) R. By convention, u(0) = 0. Given two elements a, b R for which greatest common divisors exist, gcd(a, b) will denote the greatest common divisor in normal form. Similarly, for the least common multiple lcm(a, b). In 6
7 0.4 DESIGNING ALGORITHMS a unique factorization domain (UFD), greatest common divisors and lowest common multiples always exist, as we know from the course Algebraische Strukturen. Basic examples of UFDs are principal ideal domains such as Z and K[x]. Recall that every Euclideam domain is a principal ideal domain. Example 0.6. forms. (i) In Z, only ±1 are units. We choose the integers 0 as normal (ii) In a field K, all nonzero elements are units. Their normal form is, thus, 1. (iii) Given any ring R in which normal forms are distinguished, the normal forms in the polynomial ring R[x] are the polynomials whose leading coefficients are normal forms in R. If R = K is a field, we get the monic polynomials. Now, we give an example of Euclid s algorithm at work: Example 0.7. In Z, we have gcd(18, 30) = 6: r a b The computation of greatest common divisors in Z[x] can, in principle, be reduced to that in Q[x] using Gauss lemma (we will see this later). A problem with this approach is intermediate coefficient swell, see the slide from the second lecture of the course at decker/lehre/ss11/pramasymbol/index.html. Modular algorithms provide a solution to this problem. Their basic idea is to reduce the given polynomials modulo one (or several) prime numbers, compute their greatest common divisors in the corresponding prime fields Z/pZ, and lift the results to Z[x] (for example, by Chinese remaindering). To demonstrate the general use of modular computations, we give two examples from elementary number theory: Example 0.8. We show that the equation x 2 + y 2 = 4z + 3 has no integer solutions. The idea is to reduce the equation modulo 4: x 2 + y 2 = 3. In Z/4Z = {0, 1, 2, 3}, only 0, 1 are squares: a a Hence, the possible values for x 2 + y 2 in Z/4Z are 0, 1, 2, but not 3. 7
8 0 INTRODUCTION Example 0.9. In 1732, Euler proved that the 5th Fermat number F 5 = is not prime. In fact, Euler showed that 641 is a factor of F 5. His idea was to show that F 5 modulo 641 is zero. For this, we first write 641 = = = = Then and This implies and, thus, = mod 641, hence mod 641, mod = 2 32 mod 641 F 5 = mod What are Algebraic Algorithms? Algebraic algorithms deal with algebraic objects, make us of algebraic methods, and are based on algebraic theorems. Objects are represented exactly and calculations are carried through exactly (no approximation is applied at any step) What is an Algorithm? An algorithm is a set of instructions for solving a particular problem in finitely many, well defined steps. Starting from a given input, the instructions describe a computation which eventually will produce an output and terminate. The transition from one step to the next one is not necessarily deterministic: probabilistic algorithms incorporate random input, which may lead to random performance and random output A First Motivating Example: The RSA Cryptosystem Cryptology is the science of secure communication. Its basic scenario is: 8
9 0.12 A FIRST MOTIVATING EXAMPLE: THE RSA CRYPTOSYSTEM Eve eavesdropper m e s s a g e Bob sender Alice receiver Its basic idea is to encrypt the message: sender PSfrag replacements key K algorithm encryption ε transmitted ciphertext plaintext receiver key L PSfrag replacements transmitted cipher text algorithm decryption δ decrypted text In a symmetric cryptosystem, K = L. Example 0.13 (Caesar Shift). Replace each letter of the plain text by a fixed number of positions down the alphabet. For instance: To decrypt, reverse the shift. A B C X Y Z shift 3 D E F A B C 9
10 0 INTRODUCTION Example The German Enigma is an electromechanical rotor machine which was used for the encryption and decryption of secret messages during World War II. See for a demonstration. A major problem coming with symmetric cryptosystems is the initial exchange of one or more keys. Modern public key cryptosystems are asymmetric and avoid, thus, this problem: Alice creates both a public key and an associated private key, and publishes the public key. Bob uses the public key to encrypt his plain text, and Alice her private key to decrypt the transmitted cypher text. Example 0.15 (RSA Cryptosystem). Two basic facts and one conjecture prepare the way: Fact 1: It is easy to find a random prime number of a given size. We can use modern probabilistic algorithms. Fact 2: Multiplying large numbers is easy. There are many efficient ways for multiplication. Conjecture 3: Given a large integer, it is hard to find its prime factors. Algorithms for factorizing are still expensive. Based on all this, Alice proceeds as follows: 1. Generate two large, roughly equal random primes p q (1024 bits each). 2. Compute the modulus N = p q. 3. Select a (small) public exponent e satisfying gcd(e, (p 1)(q 1)) = Use the extended Euclidean algorithm to compute a private exponent d satisfying de 1 mod (p 1)(q 1). 5. Publish (N, e) as the public key and hide (N, d) as the private key. Before exchanging messages, Bob and Alice agree on a way of encoding pieces of text as integers between 0 and N 1 (long messages are broken into pieces). Using, say A 0, B 1,, Z 25 and the 26 ary number system, CAESAR is encoded as = Then Bob encrypts a piece of text x as ε(x) := y := x e (using repeated squaring). Alice decrypts y as δ(y) := y d mod N mod N (also using repeated squaring). This makes sense since (as we will see in the exercises). δ(y) = x 10
11 0.17 A SECOND MOTIVATING EXAMPLE: ROBOTICS Remark Among others, the RSA system can be used to solve the initial key exchange problem for a modern, computer based symmetric cryptosystem such as AES. See Good Privacy and Encryption Standard A Second Motivating Example: Robotics The inverse kinematic problem is to find all joint angles that place the hand of a robot in a desired position. Example 0.18 (A Planar Robot). Consider the following robot arm which only moves in the plane: θ 2 length l 2 length l 3 θ 3 PSfrag replacements θ 1 length l 1 If (a, b) is the point to be reached by the robot s hand, the corresponding angles satisfy the polynomial system of equations below (to simplify the system, we ignore the orientation of the hand, that is, the angle θ 3 ): where a = l 3 (c 1 c 2 s 1 s 2 ) + l 2 c 1, b = l 3 (c 1 s 2 + c 2 s 1 + l 2 s 1, 0 = c s2 1 1, 0 = c s2 2 1, c i = cos θ 1, s i = sin θ 1. Computing a Gröbner Basis as in the Singular example on the slides from the first lecture, we get an equivalent system which can be easily solved for the c i and s i (to simplify the presentation of the result, we suppose that l 2 = l 3 = 1). 11
12 0 INTRODUCTION See Cox, Little, and O Shea for details. c 2 a2 +b 2 2 2, s 2 + a2 +b 2 a s 1 a2 b+b 3 2a, c 1 + b a s 1 a2 +b 2 2a, s a2 b+b 3 (a 2 +b 2 ) s 1 + (a2 +b 2 ) 2 4a 2 4(a 2 +b 2 ). We refer to the homepage of Charles Wampler (see cwample1/) for movies featuring Robonaut2 and for the slides from a lecture on Kinematics and Numerical Algebraic Geometry which in particular include a discussion of possible robot joints. 12
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationDiscrete Mathematics, Chapter 4: Number Theory and Cryptography
Discrete Mathematics, Chapter 4: Number Theory and Cryptography Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 4 1 / 35 Outline 1 Divisibility
More information3. 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
More informationNumber Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may
Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition
More informationBreaking 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
More informationElementary 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,
More information9 Modular Exponentiation and Cryptography
9 Modular Exponentiation and Cryptography 9.1 Modular Exponentiation Modular arithmetic is used in cryptography. In particular, modular exponentiation is the cornerstone of what is called the RSA system.
More information(x + a) n = x n + a Z n [x]. Proof. If n is prime then the map
22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find
More informationDiscrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University
Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called
More informationRSA and Primality Testing
and Primality Testing Joan Boyar, IMADA, University of Southern Denmark Studieretningsprojekter 2010 1 / 81 Correctness of cryptography cryptography Introduction to number theory Correctness of with 2
More informationAdvanced Cryptography
Family Name:... First Name:... Section:... Advanced Cryptography Final Exam July 18 th, 2006 Start at 9:15, End at 12:00 This document consists of 12 pages. Instructions Electronic devices are not allowed.
More informationCIS 5371 Cryptography. 8. Encryption 
CIS 5371 Cryptography p y 8. Encryption  Asymmetric Techniques Textbook encryption algorithms In this chapter, security (confidentiality) is considered in the following sense: Allornothing secrecy.
More informationCryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur
Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Module No. # 01 Lecture No. # 05 Classic Cryptosystems (Refer Slide Time: 00:42)
More informationCryptography. Helmer Aslaksen Department of Mathematics National University of Singapore
Cryptography Helmer Aslaksen Department of Mathematics National University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/sfm/ 1 Basic Concepts There are many situations in life where
More informationRSA 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.
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationLecture 13  Basic Number Theory.
Lecture 13  Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are nonnegative integers. We say that A divides B, denoted
More informationLecture 13: Factoring Integers
CS 880: Quantum Information Processing 0/4/0 Lecture 3: Factoring Integers Instructor: Dieter van Melkebeek Scribe: Mark Wellons In this lecture, we review order finding and use this to develop a method
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationPublic Key Cryptography: RSA and Lots of Number Theory
Public Key Cryptography: RSA and Lots of Number Theory Public vs. PrivateKey Cryptography We have just discussed traditional symmetric cryptography: Uses a single key shared between sender and receiver
More informationCHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
More informationPROBLEM SET 6: POLYNOMIALS
PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other
More informationThe 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
More informationThe Mathematics of the RSA PublicKey Cryptosystem
The Mathematics of the RSA PublicKey Cryptosystem Burt Kaliski RSA Laboratories ABOUT THE AUTHOR: Dr Burt Kaliski is a computer scientist whose involvement with the security industry has been through
More informationThe application of prime numbers to RSA encryption
The application of prime numbers to RSA encryption Prime number definition: Let us begin with the definition of a prime number p The number p, which is a member of the set of natural numbers N, is considered
More informationModern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)
Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)
More informationFactoring & Primality
Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount
More informationArithmetic algorithms for cryptology 5 October 2015, Paris. Sieves. Razvan Barbulescu CNRS and IMJPRG. R. Barbulescu Sieves 0 / 28
Arithmetic algorithms for cryptology 5 October 2015, Paris Sieves Razvan Barbulescu CNRS and IMJPRG R. Barbulescu Sieves 0 / 28 Starting point Notations q prime g a generator of (F q ) X a (secret) integer
More informationMATH 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
More informationUnique Factorization
Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon
More informationMathematics 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
More informationPOLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS
POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).
More informationMaths delivers! A guide for teachers Years 11 and 12. RSA Encryption
1 Maths delivers! 2 3 4 5 6 7 8 9 10 11 12 A guide for teachers Years 11 and 12 RSA Encryption Maths delivers! RSA Encryption Dr Michael Evans AMSI Editor: Dr Jane Pitkethly, La Trobe University Illustrations
More informationThe 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
More information12 Greatest Common Divisors. The Euclidean Algorithm
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to
More informationThe Mathematics of RSA
The Mathematics of RSA Dimitri Papaioannou May 24, 2007 1 Introduction Cryptographic systems come in two flavors. Symmetric or Private key encryption and Asymmetric or Public key encryption. Strictly speaking,
More informationNumber Theory and Cryptography using PARI/GP
Number Theory and Cryptography using Minh Van Nguyen nguyenminh2@gmail.com 25 November 2008 This article uses to study elementary number theory and the RSA public key cryptosystem. Various commands will
More informationCryptography and Network Security Chapter 9
Cryptography and Network Security Chapter 9 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 9 Public Key Cryptography and RSA Every Egyptian received two names,
More informationIntroduction to Modern Algebra
Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write
More informationPublic Key Cryptography and RSA. Review: Number Theory Basics
Public Key Cryptography and RSA Murat Kantarcioglu Based on Prof. Ninghui Li s Slides Review: Number Theory Basics Definition An integer n > 1 is called a prime number if its positive divisors are 1 and
More informationMA2C03 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
More informationFactorization Algorithms for Polynomials over Finite Fields
Degree Project Factorization Algorithms for Polynomials over Finite Fields Sajid Hanif, Muhammad Imran 20110503 Subject: Mathematics Level: Master Course code: 4MA11E Abstract Integer factorization is
More informationPrinciples 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
More information9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.
9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n1 x n1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role
More informationComputer Algebra for Computer Engineers
p.1/14 Computer Algebra for Computer Engineers Preliminaries Priyank Kalla Department of Electrical and Computer Engineering University of Utah, Salt Lake City p.2/14 Notation R: Real Numbers Q: Fractions
More informationPrimality  Factorization
Primality  Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions
More informationNumber Theory and the RSA Public Key Cryptosystem
Number Theory and the RSA Public Key Cryptosystem Minh Van Nguyen nguyenminh2@gmail.com 05 November 2008 This tutorial uses to study elementary number theory and the RSA public key cryptosystem. A number
More informationCryptography and Network Security
Cryptography and Network Security Fifth Edition by William Stallings Chapter 9 Public Key Cryptography and RSA PrivateKey Cryptography traditional private/secret/single key cryptography uses one key shared
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationElements of Applied Cryptography Public key encryption
Network Security Elements of Applied Cryptography Public key encryption Public key cryptosystem RSA and the factorization problem RSA in practice Other asymmetric ciphers Asymmetric Encryption Scheme Let
More informationMathematics 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 McGrawHill The McGrawHill Companies, Inc., 2000 Set of Integers The set of integers, denoted by Z,
More informationComputer and Network Security
MIT 6.857 Computer and Networ Security Class Notes 1 File: http://theory.lcs.mit.edu/ rivest/notes/notes.pdf Revision: December 2, 2002 Computer and Networ Security MIT 6.857 Class Notes by Ronald L. Rivest
More information7. Some irreducible polynomials
7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of
More information1. The RSA algorithm In this chapter, we ll learn how the RSA algorithm works.
MATH 13150: Freshman Seminar Unit 18 1. The RSA algorithm In this chapter, we ll learn how the RSA algorithm works. 1.1. Bob and Alice. Suppose that Alice wants to send a message to Bob over the internet
More informationMathematics 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
More informationNumber Theory Hungarian Style. Cameron Byerley s interpretation of Csaba Szabó s lectures
Number Theory Hungarian Style Cameron Byerley s interpretation of Csaba Szabó s lectures August 20, 2005 2 0.1 introduction Number theory is a beautiful subject and even cooler when you learn about it
More informationCryptography and Network Security. Prof. D. Mukhopadhyay. Department of Computer Science and Engineering. Indian Institute of Technology, Kharagpur
Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 12 Block Cipher Standards
More informationNotes on Network Security Prof. Hemant K. Soni
Chapter 9 Public Key Cryptography and RSA PrivateKey Cryptography traditional private/secret/single key cryptography uses one key shared by both sender and receiver if this key is disclosed communications
More informationFactoring integers, Producing primes and the RSA cryptosystem HarishChandra Research Institute
RSA cryptosystem HRI, Allahabad, February, 2005 0 Factoring integers, Producing primes and the RSA cryptosystem HarishChandra Research Institute Allahabad (UP), INDIA February, 2005 RSA cryptosystem HRI,
More informationFACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY
FACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY LINDSEY R. BOSKO I would like to acknowledge the assistance of Dr. Michael Singer. His guidance and feedback were instrumental in completing this
More informationPrime Numbers and Irreducible Polynomials
Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.
More informationOn Generalized Fermat Numbers 3 2n +1
Applied Mathematics & Information Sciences 4(3) (010), 307 313 An International Journal c 010 Dixie W Publishing Corporation, U. S. A. On Generalized Fermat Numbers 3 n +1 Amin Witno Department of Basic
More informationA 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
More informationSection 4.2: The Division Algorithm and Greatest Common Divisors
Section 4.2: The Division Algorithm and Greatest Common Divisors The Division Algorithm The Division Algorithm is merely long division restated as an equation. For example, the division 29 r. 20 32 948
More informationOutline. Computer Science 418. Digital Signatures: Observations. Digital Signatures: Definition. Definition 1 (Digital signature) Digital Signatures
Outline Computer Science 418 Digital Signatures Mike Jacobson Department of Computer Science University of Calgary Week 12 1 Digital Signatures 2 Signatures via Public Key Cryptosystems 3 Provable 4 Mike
More informationOutline. Cryptography. Bret Benesh. Math 331
Outline 1 College of St. Benedict/St. John s University Department of Mathematics Math 331 2 3 The internet is a lawless place, and people have access to all sorts of information. What is keeping people
More informationThe Division Algorithm for Polynomials Handout Monday March 5, 2012
The Division Algorithm for Polynomials Handout Monday March 5, 0 Let F be a field (such as R, Q, C, or F p for some prime p. This will allow us to divide by any nonzero scalar. (For some of the following,
More informationIntro to Rings, Fields, Polynomials: Hardware Modeling by Modulo Arithmetic
Intro to Rings, Fields, Polynomials: Hardware Modeling by Modulo Arithmetic Priyank Kalla Associate Professor Electrical and Computer Engineering, University of Utah kalla@ece.utah.edu http://www.ece.utah.edu/~kalla
More informationHomework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More informationInteger roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationA SOFTWARE COMPARISON OF RSA AND ECC
International Journal Of Computer Science And Applications Vol. 2, No. 1, April / May 29 ISSN: 97413 A SOFTWARE COMPARISON OF RSA AND ECC Vivek B. Kute Lecturer. CSE Department, SVPCET, Nagpur 9975549138
More informationElliptic Curve Cryptography
Elliptic Curve Cryptography Elaine Brow, December 2010 Math 189A: Algebraic Geometry 1. Introduction to Public Key Cryptography To understand the motivation for elliptic curve cryptography, we must first
More informationCRYPTOGRAPHY AND NETWORK SECURITY Principles and Practice
CRYPTOGRAPHY AND NETWORK SECURITY Principles and Practice THIRD EDITION William Stallings Prentice Hall Pearson Education International CONTENTS CHAPTER 1 OVERVIEW 1 1.1 1.2 1.3 1.4 1.5 1.6 PART ONE CHAPTER
More informationCryptography and Network Security Chapter 8
Cryptography and Network Security Chapter 8 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 8 Introduction to Number Theory The Devil said to Daniel Webster:
More informationCryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Karagpur
Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Karagpur Lecture No. #06 Cryptanalysis of Classical Ciphers (Refer
More informationCHAPTER 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.
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationz 0 and y even had the form
Gaussian Integers The concepts of divisibility, primality and factoring are actually more general than the discussion so far. For the moment, we have been working in the integers, which we denote by Z
More informationAppendix A. Appendix. A.1 Algebra. Fields and Rings
Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.
More informationNetwork Security. Computer Networking Lecture 08. March 19, 2012. HKU SPACE Community College. HKU SPACE CC CN Lecture 08 1/23
Network Security Computer Networking Lecture 08 HKU SPACE Community College March 19, 2012 HKU SPACE CC CN Lecture 08 1/23 Outline Introduction Cryptography Algorithms Secret Key Algorithm Message Digest
More informationMathematics of Cryptography Part I
CHAPTER 2 Mathematics of Cryptography Part I (Solution to OddNumbered Problems) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinity to positive infinity.
More informationLecture 3: OneWay Encryption, RSA Example
ICS 180: Introduction to Cryptography April 13, 2004 Lecturer: Stanislaw Jarecki Lecture 3: OneWay Encryption, RSA Example 1 LECTURE SUMMARY We look at a different security property one might require
More informationFAREY FRACTION BASED VECTOR PROCESSING FOR SECURE DATA TRANSMISSION
FAREY FRACTION BASED VECTOR PROCESSING FOR SECURE DATA TRANSMISSION INTRODUCTION GANESH ESWAR KUMAR. P Dr. M.G.R University, Maduravoyal, Chennai. Email: geswarkumar@gmail.com Every day, millions of people
More informationTable of Contents. Bibliografische Informationen http://dnb.info/996514864. digitalisiert durch
1 Introduction to Cryptography and Data Security 1 1.1 Overview of Cryptology (and This Book) 2 1.2 Symmetric Cryptography 4 1.2.1 Basics 4 1.2.2 Simple Symmetric Encryption: The Substitution Cipher...
More informationAn Introduction to RSA PublicKey Cryptography
An Introduction to RSA PublicKey Cryptography David Boyhan August 5, 2008 According to the U.S. Census Bureau, in the 1st quarter of 2008, approximately $33 billion worth of retail sales were conducted
More informationCONTINUED FRACTIONS AND FACTORING. Niels Lauritzen
CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents
More informationLecture Note 5 PUBLICKEY CRYPTOGRAPHY. Sourav Mukhopadhyay
Lecture Note 5 PUBLICKEY CRYPTOGRAPHY Sourav Mukhopadhyay Cryptography and Network Security  MA61027 Modern/Publickey cryptography started in 1976 with the publication of the following paper. W. Diffie
More informationAn 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
More informationLecture 6: Finite Fields (PART 3) PART 3: Polynomial Arithmetic. Theoretical Underpinnings of Modern Cryptography
Lecture 6: Finite Fields (PART 3) PART 3: Polynomial Arithmetic Theoretical Underpinnings of Modern Cryptography Lecture Notes on Computer and Network Security by Avi Kak (kak@purdue.edu) January 29, 2015
More informationCryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur
Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #10 Symmetric Key Ciphers (Refer
More informationSoftware Tool for Implementing RSA Algorithm
Software Tool for Implementing RSA Algorithm Adriana Borodzhieva, Plamen Manoilov Rousse University Angel Kanchev, Rousse, Bulgaria Abstract: RSA is one of the mostcommon used algorithms for publickey
More informationFactoring 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
More informationMATH 289 PROBLEM SET 4: NUMBER THEORY
MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides
More informationFactorization in Polynomial Rings
Factorization in Polynomial Rings These notes are a summary of some of the important points on divisibility in polynomial rings from 17 and 18 of Gallian s Contemporary Abstract Algebra. Most of the important
More informationNotes 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
More informationRoots of Polynomials
Roots of Polynomials (Com S 477/577 Notes) YanBin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x
More informationThe RSA Algorithm. Evgeny Milanov. 3 June 2009
The RSA Algorithm Evgeny Milanov 3 June 2009 In 1978, Ron Rivest, Adi Shamir, and Leonard Adleman introduced a cryptographic algorithm, which was essentially to replace the less secure National Bureau
More information