Modular Arithmetic Some Basic Number Theory: CS70 Summer 2016 - Lecture 7A Midterm 2 Scores Out
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Number Theory, Dover Publications, Inc., New York, 1994
Theory of Numbers Divisibility Theory in the Integers, The Theory of Congruences, Number-Theoretic Functions, Primitive Roots, Quadratic Residues Yotsanan Meemark Informal style based on the course 2301331 Theory of Numbers, offered at Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University Second version August 2016 Any comment or suggestion, please write to [email protected] Contents 1 Divisibility Theory in the Integers 1 1.1 The Division Algorithm and GCD . 1 1.2 The Fundamental Theorem of Arithmetic . 5 1.3 The Euclidean Algorithm and Linear Diophantine Equations . 8 2 The Theory of Congruences 13 2.1 Basic Properties of Congruence . 13 2.2 Linear Congruences . 16 2.3 Reduced Residue Systems . 18 2.4 Polynomial Congruences . 21 3 Number-Theoretic Functions 25 3.1 Multiplicative Functions . 25 3.2 The Mobius¨ Inversion Formula . 28 3.3 The Greatest Integer Function . 30 4 Primitive Roots 33 4.1 The Order of an Integer Modulo n ............................ 33 4.2 Integers Having Primitive Roots . 35 4.3 nth power residues . 39 4.4 Hensel’s Lemma . 41 5 Quadratic Residues 45 5.1 The Legendre Symbol . 45 5.2 Quadratic Reciprocity . 48 Bibliography 53 Index 54 Chapter 1 Divisibility Theory in the Integers Let N denote the set of positive integers and let Z be the set of integers. 1.1 The Division Algorithm and GCD Theorem 1.1.1. [Well-Ordering Principle] Every nonempty set S of nonnegative integers contains a least element; that is, there is some integer a in S such that a ≤ b for all b 2 S. -
Modular Arithmetic
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range f0;1;:::;N ¡ 1g, and wraps around whenever you try to leave this range — like the hand of a clock (where N = 12) or the days of the week (where N = 7). Example: Calculating the day of the week. Suppose that you have mapped the sequence of days of the week (Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday) to the sequence of numbers (0;1;2;3;4;5;6) so that Sunday is 0, Monday is 1, etc. Suppose that today is Thursday (=4), and you want to calculate what day of the week will be 10 days from now. Intuitively, the answer is the remainder of 4 + 10 = 14 when divided by 7, that is, 0 —Sunday. In fact, it makes little sense to add a number like 10 in this context, you should probably find its remainder modulo 7, namely 3, and then add this to 4, to find 7, which is 0. What if we want to continue this in 10 day jumps? After 5 such jumps, we would have day 4 + 3 ¢ 5 = 19; which gives 5 modulo 7 (Friday). This example shows that in certain circumstances it makes sense to do arithmetic within the confines of a particular number (7 in this example), that is, to do arithmetic by always finding the remainder of each number modulo 7, say, and repeating this for the results, and so on. -
Discrete Mathematics
Slides for Part IA CST 2016/17 Discrete Mathematics <www.cl.cam.ac.uk/teaching/1617/DiscMath> Prof Marcelo Fiore [email protected] — 0 — What are we up to ? ◮ Learn to read and write, and also work with, mathematical arguments. ◮ Doing some basic discrete mathematics. ◮ Getting a taste of computer science applications. — 2 — What is Discrete Mathematics ? from Discrete Mathematics (second edition) by N. Biggs Discrete Mathematics is the branch of Mathematics in which we deal with questions involving finite or countably infinite sets. In particular this means that the numbers involved are either integers, or numbers closely related to them, such as fractions or ‘modular’ numbers. — 3 — What is it that we do ? In general: Build mathematical models and apply methods to analyse problems that arise in computer science. In particular: Make and study mathematical constructions by means of definitions and theorems. We aim at understanding their properties and limitations. — 4 — Lecture plan I. Proofs. II. Numbers. III. Sets. IV. Regular languages and finite automata. — 6 — Proofs Objectives ◮ To develop techniques for analysing and understanding mathematical statements. ◮ To be able to present logical arguments that establish mathematical statements in the form of clear proofs. ◮ To prove Fermat’s Little Theorem, a basic result in the theory of numbers that has many applications in computer science. — 16 — Proofs in practice We are interested in examining the following statement: The product of two odd integers is odd. This seems innocuous enough, but it is in fact full of baggage. — 18 — Proofs in practice We are interested in examining the following statement: The product of two odd integers is odd. -
EE 595 (PMP) Advanced Topics in Communication Theory Handout #1
EE 595 (PMP) Advanced Topics in Communication Theory Handout #1 Introduction to Cryptography. Symmetric Encryption.1 Wednesday, January 13, 2016 Tamara Bonaci Department of Electrical Engineering University of Washington, Seattle Outline: 1. Review - Security goals 2. Terminology 3. Secure communication { Symmetric vs. asymmetric setting 4. Background: Modular arithmetic 5. Classical cryptosystems { The shift cipher { The substitution cipher 6. Background: The Euclidean Algorithm 7. More classical cryptosystems { The affine cipher { The Vigen´erecipher { The Hill cipher { The permutation cipher 8. Cryptanalysis { The Kerchoff Principle { Types of cryptographic attacks { Cryptanalysis of the shift cipher { Cryptanalysis of the affine cipher { Cryptanalysis of the Vigen´erecipher { Cryptanalysis of the Hill cipher Review - Security goals Last lecture, we introduced the following security goals: 1. Confidentiality - ability to keep information secret from all but authorized users. 2. Data integrity - property ensuring that messages to and from a user have not been corrupted by communication errors or unauthorized entities on their way to a destination. 3. Identity authentication - ability to confirm the unique identity of a user. 4. Message authentication - ability to undeniably confirm message origin. 5. Authorization - ability to check whether a user has permission to conduct some action. 6. Non-repudiation - ability to prevent the denial of previous commitments or actions (think of a con- tract). 7. Certification - endorsement of information by a trusted entity. In the rest of today's lecture, we will focus on three of these goals, namely confidentiality, integrity and authentication (CIA). In doing so, we begin by introducing the necessary terminology. 1 We thank Professors Radha Poovendran and Andrew Clark for the help in preparing this material. -
The Complete Splittings of Finite Abelian Groups
The complete splittings of finite abelian groups Kevin Zhaoa aDepartment of Mathematics, South China normal university, Guangzhou 510631, China Abstract Let G be a finite group. We will say that M and S form a complete split- ting (splitting) of G if every element (nonzero element) g of G has a unique representation of the form g = ms with m ∈ M and s ∈ S, and 0 has a such representation (while 0 has no such representation). In this paper, we determine the structures of complete splittings of finite abelian groups. In particular, for complete splittings of cyclic groups our de- scription is more specific. Furthermore, we show some results for existence and nonexistence of complete splittings of cyclic groups and find a relationship between complete splittings and splittings for finite groups. Keywords: complete splittings, splittings, cyclic groups, finite abelian groups, relationship. 1. Introduction The splittings of finite abelian groups are closely related to lattice tilings and are easy to be generated to lattice packings. Let K0 be a polytope composed of unit cubes and v + K0 be a translate of K0 for some vector v. A family of translations {v + K0 : v ∈ H} is called an integer lattice packing if H is an n-dimensional subgroup of Zn and, for any two vectors v and w in H, the interiors of v + K0 and w + K0 are disjoint; furthermore, if the n-dimensional Euclidean space Zn is contained in the family of these translations, then we call it an integer lattice tiling. The splitting problem can be traceable to a geometric problem posed by H. -
Quaternion Algebra and Calculus
Quaternion Algebra and Calculus David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Created: March 2, 1999 Last Modified: August 18, 2010 Contents 1 Quaternion Algebra 2 2 Relationship of Quaternions to Rotations3 3 Quaternion Calculus 5 4 Spherical Linear Interpolation6 5 Spherical Cubic Interpolation7 6 Spline Interpolation of Quaternions8 1 This document provides a mathematical summary of quaternion algebra and calculus and how they relate to rotations and interpolation of rotations. The ideas are based on the article [1]. 1 Quaternion Algebra A quaternion is given by q = w + xi + yj + zk where w, x, y, and z are real numbers. Define qn = wn + xni + ynj + znk (n = 0; 1). Addition and subtraction of quaternions is defined by q0 ± q1 = (w0 + x0i + y0j + z0k) ± (w1 + x1i + y1j + z1k) (1) = (w0 ± w1) + (x0 ± x1)i + (y0 ± y1)j + (z0 ± z1)k: Multiplication for the primitive elements i, j, and k is defined by i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, and ki = −ik = j. Multiplication of quaternions is defined by q0q1 = (w0 + x0i + y0j + z0k)(w1 + x1i + y1j + z1k) = (w0w1 − x0x1 − y0y1 − z0z1)+ (w0x1 + x0w1 + y0z1 − z0y1)i+ (2) (w0y1 − x0z1 + y0w1 + z0x1)j+ (w0z1 + x0y1 − y0x1 + z0w1)k: Multiplication is not commutative in that the products q0q1 and q1q0 are not necessarily equal. -
WORKSHEET # 7 QUATERNIONS the Set of Quaternions Are the Set Of
WORKSHEET # 7 QUATERNIONS The set of quaternions are the set of all formal R-linear combinations of \symbols" i; j; k a + bi + cj + dk for a; b; c; d 2 R. We give them the following addition rule: (a + bi + cj + dk) + (a0 + b0i + c0j + d0k) = (a + a0) + (b + b0)i + (c + c0)j + (d + d0)k Multiplication is induced by the following rules (λ 2 R) i2 = j2 = k2 = −1 ij = k; jk = i; ki = j ji = −k kj = −i ik = −j λi = iλ λj = jλ λk = kλ combined with the distributive rule. 1. Use the above rules to carefully write down the product (a + bi + cj + dk)(a0 + b0i + c0j + d0k) as an element of the quaternions. Solution: (a + bi + cj + dk)(a0 + b0i + c0j + d0k) = aa0 + ab0i + ac0j + ad0k + ba0i − bb0 + bc0k − bd0j ca0j − cb0k − cc0 + cd0i + da0k + db0j − dc0i − dd0 = (aa0 − bb0 − cc0 − dd0) + (ab0 + a0b + cd0 − dc0)i (ac0 − bd0 + ca0 + db0)j + (ad0 + bc0 − cb0 + da0)k 2. Prove that the quaternions form a ring. Solution: It is obvious that the quaternions form a group under addition. We just showed that they were closed under multiplication. We have defined them to satisfy the distributive property. The only thing left is to show the associative property. I will only prove this for the elements i; j; k (this is sufficient by the distributive property). i(ij) = i(k) = ik = −j = (ii)j i(ik) = i(−j) = −ij = −k = (ii)k i(ji) = i(−k) = −ik = j = ki = (ij)i i(jj) = −i = kj = (ij)j i(jk) = i(i) = (−1) = (k)k = (ij)k i(ki) = ij = k = −ji = (ik)i i(kk) = −i = −jk = (ik)k j(ii) = −j = −ki = (ji)i j(ij) = jk = i = −kj = (ji)j j(ik) = j(−j) = 1 = −kk = (ji)k j(ji) = j(−k) = −jk = −i = (jj)i j(jk) = ji = −k = (jj)k j(ki) = jj = −1 = ii = (jk)i j(kj) = j(−i) = −ji = k = ij = (jk)j j(kk) = −j = ik = (jk)k k(ii) = −k = ji = (ki)i k(ij) = kk = −1 = jj = (ki)j k(ik) = k(−j) = −kj = i = jk = (ki)k k(ji) = k(−k) = 1 = (−i)i = (kj)i k(jj) = k(−1) = −k = −ij = (kj)j k(jk) = ki = j = −ik = (kj)k k(ki) = kj = −i = (kk)i k(kj) = k(−i) = −ki = −j = (kk)j 1 WORKSHEET #7 2 3. -
Math 412. §3.2, 3.3: Examples of Rings and Homomorphisms Professors Jack Jeffries and Karen E. Smith
Math 412. x3.2, 3.3: Examples of Rings and Homomorphisms Professors Jack Jeffries and Karen E. Smith DEFINITION:A subring of a ring R (with identity) is a subset S which is itself a ring (with identity) under the operations + and × for R. DEFINITION: An integral domain (or just domain) is a commutative ring R (with identity) satisfying the additional axiom: if xy = 0, then x or y = 0 for all x; y 2 R. φ DEFINITION:A ring homomorphism is a mapping R −! S between two rings (with identity) which satisfies: (1) φ(x + y) = φ(x) + φ(y) for all x; y 2 R. (2) φ(x · y) = φ(x) · φ(y) for all x; y 2 R. (3) φ(1) = 1. DEFINITION:A ring isomorphism is a bijective ring homomorphism. We say that two rings R and S are isomorphic if there is an isomorphism R ! S between them. You should think of a ring isomorphism as a renaming of the elements of a ring, so that two isomorphic rings are “the same ring” if you just change the names of the elements. A. WARM-UP: Which of the following rings is an integral domain? Which is a field? Which is commutative? In each case, be sure you understand what the implied ring structure is: why is each below a ring? What is the identity in each? (1) Z; Q. (2) Zn for n 2 Z. (3) R[x], the ring of polynomials with R-coefficients. (4) M2(Z), the ring of 2 × 2 matrices with Z coefficients. -
Vector Spaces
Chapter 1 Vector Spaces Linear algebra is the study of linear maps on finite-dimensional vec- tor spaces. Eventually we will learn what all these terms mean. In this chapter we will define vector spaces and discuss their elementary prop- erties. In some areas of mathematics, including linear algebra, better the- orems and more insight emerge if complex numbers are investigated along with real numbers. Thus we begin by introducing the complex numbers and their basic✽ properties. 1 2 Chapter 1. Vector Spaces Complex Numbers You should already be familiar with the basic properties of the set R of real numbers. Complex numbers were invented so that we can take square roots of negative numbers. The key idea is to assume we have i − i The symbol was√ first a square root of 1, denoted , and manipulate it using the usual rules used to denote −1 by of arithmetic. Formally, a complex number is an ordered pair (a, b), the Swiss where a, b ∈ R, but we will write this as a + bi. The set of all complex mathematician numbers is denoted by C: Leonhard Euler in 1777. C ={a + bi : a, b ∈ R}. If a ∈ R, we identify a + 0i with the real number a. Thus we can think of R as a subset of C. Addition and multiplication on C are defined by (a + bi) + (c + di) = (a + c) + (b + d)i, (a + bi)(c + di) = (ac − bd) + (ad + bc)i; here a, b, c, d ∈ R. Using multiplication as defined above, you should verify that i2 =−1. -
Primes in Arithmetical Progression
Colby College Digital Commons @ Colby Honors Theses Student Research 2019 Primes in Arithmetical Progression Edward C. Wessel Colby College Follow this and additional works at: https://digitalcommons.colby.edu/honorstheses Part of the Analysis Commons, and the Number Theory Commons Colby College theses are protected by copyright. They may be viewed or downloaded from this site for the purposes of research and scholarship. Reproduction or distribution for commercial purposes is prohibited without written permission of the author. Recommended Citation Wessel, Edward C., "Primes in Arithmetical Progression" (2019). Honors Theses. Paper 935. https://digitalcommons.colby.edu/honorstheses/935 This Honors Thesis (Open Access) is brought to you for free and open access by the Student Research at Digital Commons @ Colby. It has been accepted for inclusion in Honors Theses by an authorized administrator of Digital Commons @ Colby. Primes in Arithmetical Progression Edward (Teddy) Wessel April 2019 Contents 1 Message to the Reader 3 2 Introduction 3 2.1 Primes . 3 2.2 Euclid . 3 2.3 Euler and Dirichlet . 4 2.4 Shapiro . 4 2.5 Formal Statement . 5 3 Arithmetical Functions 6 3.1 Euler’s Totient Function . 6 3.2 The Mobius Function . 6 3.3 A Relationship Between j and m ................ 7 3.4 The Mangoldt Function . 8 4 Dirichlet Convolution 9 4.1 Definition . 10 4.2 Some Basic Properties of Dirichlet Multiplication . 10 4.3 Identity and Inverses within Dirichlet Multiplication . 11 4.4 Multiplicative Functions and their Relationship with Dirich- let Convolution . 13 4.5 Generalized Convolutions . 15 4.6 Partial Sums of Dirichlet Convolution . -
Number Theory
CS 5002: Discrete Structures Fall 2018 Lecture 5: October 4, 2018 1 Instructors: Tamara Bonaci, Adrienne Slaugther Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. Number Theory Readings for this week: Rosen, Chapter 4.1, 4.2, 4.3, 4.4 5.1 Overview 1. Review: set theory 2. Review: matrices and arrays 3. Number theory: divisibility and modular arithmetic 4. Number theory: prime numbers and greatest common divisor (gcd) 5. Number theory: solving congruences 6. Number theory: modular exponentiation and Fermat's little theorem 5.2 Introduction In today's lecture, we will dive into the branch of mathematics, studying the set of integers and their properties, known as number theory. Number theory has very important practical implications in computer science, but also in our every day life. For example, secure online communication, as we know it today, would not be possible without number theory because many of the encryption algorithms used to enable secure communication rely heavily of some famous (and in some cases, very old) results from number theory. We will first introduce the notion of divisibility of integers. From there, we will introduce modular arithmetic, and explore and prove some important results about modular arithmetic. We will then discuss prime numbers, and show that there are infinitely many primes. Finaly, we will explain how to solve linear congruences, and systems of linear congruences. 5-1 5-2 Lecture 5: October 4, 2018 5.3 Review 5.3.1 Set Theory In the last lecture, we talked about sets, and some of their properties. -
Number Theory & Asymmetric Cryptography
Number Theory & Asymmetric Cryptography Modular Arithmetic Notations Ζ={−∞,⋯−2,−1,0,1,2,⋯,∞} Ζm={0,1,2,⋯,m−2,m−1} a≡b mod m⇒ a=b+km, integer k Addition mod m Given a≡b mod m and c≡d mod m (a+c)≡(b+d) mod m a−b=km ,c−d=lm, integers k ,l (a+c)=b+d+(k+l)m=(b+d)+ jm Examples (25+15)≡(1+3)≡0 mod 4 (25+15)≡(4+1)≡5 mod 7 Multiplication mod m a≡b mod m,c≡d mod ac≡bd mod m ac=(b+km)(d+lm)=bd+(bl+kd+klm)m Example 26≡2 mod 4,11≡3 mod 4 26∗11=286≡2 mod 4 2∗3=6≡2 mod 4 What about division? Is division possible in Zm? (A long detour necessary at this point) Group, Abelian Group, Ring and Field Abelian Group Field Ring Group Group, Ring and Field Group: Addition is closed, and associative, additive identity and additive inverse exist Abelian group: Addition is also commutative Ring: Multiplication is closed, associative, commutative, and distributive; multiplicative identity exists Field: Every element except “additive identity” has multiplicative inverse In Zm additive identity is 0, multiplicative identity is 1. Its a Ring. Is Zm a Field? Multiplicative Inverse Consider m = 5, or {0,1,2,3,4} – 2*3 = 3*2 = 1 mod 5. So 2↔3 ● (3 and 2 are multiplicative inverses in Z5) – 4*4 = 1 mod 5, or 4 ↔ 4 – obviously, 1 ↔ 1 Z5 is clearly a field (all elements except additive identity have a multiplicative inverse) Consider m = 6, or {0,1,2,3,4,5} – 5↔5 , 1 ↔ 1 – No inverses for 2,3 and 4? (why?) – Z6 is not a Field (it is a Ring) – (As we shall see soon) the reason for this is that 6 is not a prime mumber Basic Theorems of Arithmetic Any number can be represented as a product th of powers of primes.