Rings of Residues. S. F. Ellermeyer. September 18, ; [1] m
|
|
- Ginger Short
- 6 years ago
- Views:
Transcription
1 Rings of Residues S F Ellermeyer September 18, 2006 If m is a positive integer, then we obtain the partition C = f[0] m ; [1] m ; : : : ; [m 1] m g of Z into m congruence classes (This is discussed in detail in the course notes that accompany Sections 11 and 21) Our goal is to de ne binary operations of addition and multiplication on the members of C De nition 1 A binary operation on a set, A, is a function mapping AA into A For example, the normal addition of integers is a binary operation on the set of integers because when we add two integers, we obtain another integer For example = 63 We can think of addition of integers as being a function called + mapping ZZ into Z where, for example, + (7; 56) = 63 Clearly, normal multiplication is also a binary operation on the integers Our operations of addition and multiplication on members of C will be de ned as follows: De nition 2 For any two members, [a] m and [b] m, of C, we de ne [a] m + [b] m = [a + b] m (where the + on the right hand side of this equation is the normal operation of addition of integers) and [a] m [b] m = [a b] m (where the on the right hand side of this equation is the normal operation of multiplication of integers) 1
2 Example 3 Compute [3] 6 + and [3] 6 Solution 4 [3] 6 + = [3 + 5] 6 = [8] 6 and [3] 6 = [3 5] 6 = [15] 6 Since [a] m and [b] m are sets, the operations de ned in De nition 2 make sense only if the results of the addition and multiplication do not depend on the particular choice of representative from [a] m and [b] m In other words, we need to assure ourselves that the operations of addition and multiplication that we have de ned are well de ned Some things that we have learned previously (in Section 21) allow us to conclude that these operations are indeed well de ned In particular, we know that x y mod m if and only if [x] m = [y] m and z w mod m if and only if [z] m = [w] m We also know that if a c mod m and b d mod m, then and (a + b) (c + d) mod m a b c d mod m We want to make sure that if [a] m = [c] m and [b] m = [d] m, then [a] m + [b] m = [c] m + [d] m Taking the above considerations into account, we see that this is in fact true because [a] m + [b] m = [a + b] m (by de nition) = [c + d] m (because (a + b) (c + d) mod m) = [c] m + [d] m (by de nition) By similar reasoning, we can also rest assured that multiplication, as de ned in De nition 2, is well de ned Example 5 In Example 3, we computed [3] 6 + = [3 + 5] 6 = [8] 6 and [3] 6 = [3 5] 6 = [15] 6 We will show that the same results are obtained when di erent representatives of [3] 6 and [4] 6 are used in doing these computations Note that 33 3 mod 6 and 17 5 mod 6 Thus [33] 6 = [3] 6 and [17] 6 = When we 33 and 17 as the representatives in our computations, we obtain [33] 6 + [17] 6 = [50] 6 However, both [8] 6 and [50] 6 are equal to [2] 6 Thus [33] 6 + [17] 6 = [3] 6 + 2
3 For the multiplication, we obtain [33] 6 [17] 6 = [561] 6 Since [15] 6 and [561] 6 are both equal to [3] 6, we see that [33] 6 [17] 6 = [3] 6 Recalling that every congruence class modulo m has a principal representative, a mod m (where 0 a mod m < m), let us agree to simplify things by always using the principal representative of any congruence class, [a] m, to represent [a] m Having agreed to this, we can simplify notation by omitting the brackets and the subscript m in doing addition and multiplication on elements of C (as long as the value of m is understood) For example, as long as it is understood that we are operating modulo 6, instead of writing [3] 6 + = [2] 6, we can simply write = 2, and instead of writing [3] 6 = [3] 6, we can just write 3 5 = 3 De nition 6 For a given positive integer m, the ring of residues modulo m is de ned to be the set A = f0; 1; : : : ; m 1g with binary operations of addition and multiplication as de ned above (That is, addition and multiplication are performed modulo m) The ring of residues modulo m is denoted by Z m Remark 7 The ring of residues modulo m is not a set of numbers! It actually consists of three things: 1 the set of numbers A = f0; 1; : : : ; m 1g 2 a binary operation called addition and denoted by + 3 a binary operation called multiplication and denoted by Thus Z m = fa; +; g Nonetheless, we will often abuse terminology and refer to Z m as though it is the set A For example, we will say things like Let x be a member of Z m and write x 2 Z m when what we really mean to say is that x is a member of the set A which is the underlying set of numbers that comprises Z m (but this is too long to say) 3
4 Example 8 Let us write down the addition and multiplication tables for Z 6 A Useful Diagram of the Ring Z As can be seen from the tables constructed in Example 8, there are some algebraic properties that hold in Z that do not hold in Z 6 For example, in Z, if we have ab = 0, then it must be true that either a = 0 or b = 0 This is not true in Z 6 though because, for example, 3 2 = 0 Also, in Z, if a b = a 4
5 and a 6= 0, then it must be true that b = 1 However this is not true in Z 6 because, for example, 3 3 = 3 Another interesting feature that contrasts Z and Z 6 is that the only numbers in Z that have multiplicative inverses in Z are 1 and 1; whereas there is a number not equal to 1 in Z 6 that has a multiplicative inverse In particular, since 5 5 = 1 in Z 6, we see that 5 has a multiplicative inverse (which is itself) Example 9 Let us write down the addition and multiplication tables for Z There are some important di erences between Z 5 and Z 6 that should be pointed out: 1 Notice that the only way to have a b = 0 in Z 5 is if a = 0 or b = 0 This is not true in Z 6 2 Notice that every non zero member of Z 5 has a multiplicative inverse For example, in Z 5, the multiplicative inverse of 2 is 3 because 23 = 1 However, not every non zero member of Z 6 has a multiplicative inverse (Only 5 has has a multiplicative inverse in Z 6 ) The following proposition lists some of the basic properties that are common to all rings of residues Proposition 10 In any ring of residues (modulo m), the following properties hold for any elements a, b, and c: 1 (associative properties of addition and multiplication) a + (b + c) = (a + b) + c and a (b c) = (a b) c 2 (commutative properties of addition and multiplication) a + b = b + a and a b = b a 5
6 3 (existence of an additive and multiplicative identity elements) a+0 = a and a 1 = a 4 (existence of additive inverses) For each element a, there exists an element a, called the additive inverse of a, such that a + ( a) = 0 5 (distributive property) a (b + c) = a b + a c Proof We will prove parts 1 and 5 of the proposition and leave the proofs of the other parts as homework 1 Let a, b, and c be elements of Z m Then a + (b + c) = [a] m + [b + c] m (converting from simpli ed notation) = [a + (b + c)] m (by de nition of addition in Z m ) = [(a + b) + c] m (by the associative property of addition for Z) = [a + b] m + [c] m (by de nition of addition in Z m ) = (a + b) + c (converting back to simpli ed notation) 5 Let a, b, and c be elements of Z m Then a (b + c) = [a] m [b + c] m = [a (b + c)] m = [a b + a c] m = [a b] m + [a c] m = a b + a c We will now de ne two terms that are obviously relevant to our study of the rings Z m De nition 11 An element, a, of Z m is called a zero divisor if a 6= 0 and there exists an element, x, of Z m such that a x = 0 De nition 12 An element, a, of Z m is called a unit if a has a multiplicative inverse in Z m In other words, a is a unit if there exists an element x in Z m such that a x = 1 6
7 Example 13 By referring to the multiplication tables in Examples 8 and 9, we see that the zero divisors in Z 6 are 2, 3, and 4 and that the units in Z 6 are 1 and 5 In Z 5, there are no zero divisors and every non zero member of Z 5 is a unit We now discuss how to determine if a given element in Z m is a zero divisor or a unit To determine if a given non zero element a 2 Z m is a zero divisor, we must determine whether or not there any non zero solutions to the equation a x = 0 Since this equation can be written as [ax] m = [0] m, we observe that we are looking for non zero solutions of the congruence equation ax 0 mod m The solution set of this equation is [0] m= gcd(a;m) If gcd (a; m) = 1, then the solution set is simply [0] m which means that x = 0 is the only solution of a x = 0 and hence that a is not a zero divisor On the other hand, if gcd (a; m) > 1, then [0] m= gcd(a;m) is the union of gcd (a; m) congruence classes modulo m All but one of the congruence classes in this union does not contain 0 Therefore, in this case, there are non zero solutions to the equation a x = 0, and this means that a is a zero divisor In summary, a is a zero divisor if and only if gcd (a; m) > 1 To determine if a given non zero element a 2 Z m is a unit, we must determine whether or not there any non zero solutions to the equation ax = 1 Since this equation can be written as [ax] m = [1] m, we observe that we are looking for solutions of the congruence equation ax 1 mod m If gcd (a; m) = 1, then this equation does have solutions (because, in this case, gcd (a; m) divides 1) and hence a x = 1 has a solution, meaning that a is a unit On the other hand, if gcd (a; m) > 1, then gcd (a; m) does not divide 1 and the congruence equation ax 1 mod m does not have any solutions This means that the equation a x = 1 does not have a solution and hence a is not a unit In summary, a is a unit if and only if gcd (a; m) = 1 We now see how to tell whether or not a given member of Z m is a unit or a zero divisor 1 If a 6= 0 and gcd (a; m) = 1, then a is a unit and not a zero divisor 2 If a 6= 0 and gcd (a; m) > 1, then a is a zero divisor and not a unit Note that any non zero element of Z m must be either a zero divisor or a unit and cannot be both Also note that if p is a prime number, then gcd (a; p) = 1 for all a such that 1 a < p This means that every non zero 7
8 element of Z p is a unit and that Z p has no zero divisors (We have already seen this illustrated in our study of Z 5 ) Example 14 In Z 8, since gcd (1; 8) = 1 gcd (2; 8) = 2 gcd (3; 8) = 1 gcd (4; 8) = 4 gcd (5; 8) = 1 gcd (6; 8) = 2 gcd (7; 8) = 1, we see that the zero divisors in Z 8 are 2; 4; and 6 and that the units in Z 8 are 1; 3; 5; and 7 Note, for example, that in Z 8 we have 2 4 = 0 and 3 3 = 1 (so 3 is its own multiplicative inverse in Z 8 ) Exercise 15 Make addition and multiplication tables for Z 1, Z 2, Z 3, Z 4, Z 7, and Z 8 Identify the zero divisors and the units of each of these rings Exercise 16 In the textbook, Section 24 (pages 80 and 81), do problems 2, 3, 5, 6, 11, 12, and 14 8
Rings, Integral Domains, and Fields
Rings, Integral Domains, and Fields S. F. Ellermeyer September 26, 2006 Suppose that A is a set of objects endowed with two binary operations called addition (and denoted by + ) and multiplication (denoted
More informationCongruences. September 16, 2006
Congruences September 16, 2006 1 Congruences If m is a given positive integer, then we can de ne an equivalence relation on Z (the set of all integers) by requiring that an integer a is related to an integer
More informationPart V. Chapter 19. Congruence of integers
Part V. Chapter 19. Congruence of integers Congruence modulo m Let m be a positive integer. Definition. Integers a and b are congruent modulo m if and only if a b is divisible by m. For example, 1. 277
More informationIntroduction to Groups
Introduction to Groups S F Ellermeyer November 2, 2006 A group, G, is a set, A, endowed with a single binary operation,, such that: The operation is associative, meaning that a (b c) = (a b) c for all
More informationSolutions to Section 2.1 Homework Problems S. F. Ellermeyer
Solutions to Section 21 Homework Problems S F Ellermeyer 1 [13] 9 = f13; 22; 31; 40; : : :g [ f4; 5; 14; : : :g [3] 10 = f3; 13; 23; 33; : : :g [ f 7; 17; 27; : : :g [4] 11 = f4; 15; 26; : : :g [ f 7;
More information3+4=2 5+6=3 7 4=4. a + b =(a + b) mod m
Rings and fields The ring Z m -part2(z 5 and Z 8 examples) Suppose we are working in the ring Z 5, consisting of the set of congruence classes Z 5 := {[0] 5, [1] 5, [2] 5, [3] 5, [4] 5 } with the operations
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationSOLUTIONS Math 345 Homework 6 10/11/2017. Exercise 23. (a) Solve the following congruences: (i) x (mod 12) Answer. We have
Exercise 23. (a) Solve the following congruences: (i) x 101 7 (mod 12) Answer. We have φ(12) = #{1, 5, 7, 11}. Since gcd(7, 12) = 1, we must have gcd(x, 12) = 1. So 1 12 x φ(12) = x 4. Therefore 7 12 x
More information1 The Well Ordering Principle, Induction, and Equivalence Relations
1 The Well Ordering Principle, Induction, and Equivalence Relations The set of natural numbers is the set N = f1; 2; 3; : : :g. (Some authors also include the number 0 in the natural numbers, but number
More informationSolution Sheet (i) q = 5, r = 15 (ii) q = 58, r = 15 (iii) q = 3, r = 7 (iv) q = 6, r = (i) gcd (97, 157) = 1 = ,
Solution Sheet 2 1. (i) q = 5, r = 15 (ii) q = 58, r = 15 (iii) q = 3, r = 7 (iv) q = 6, r = 3. 2. (i) gcd (97, 157) = 1 = 34 97 21 157, (ii) gcd (527, 697) = 17 = 4 527 3 697, (iii) gcd (2323, 1679) =
More information2x 1 7. A linear congruence in modular arithmetic is an equation of the form. Why is the solution a set of integers rather than a unique integer?
Chapter 3: Theory of Modular Arithmetic 25 SECTION C Solving Linear Congruences By the end of this section you will be able to solve congruence equations determine the number of solutions find the multiplicative
More informationAnswers and Solutions to Selected Homework Problems From Section 2.5 S. F. Ellermeyer. and B =. 0 2
Answers and Solutions to Selected Homework Problems From Section 2.5 S. F. Ellermeyer 5. Since gcd (2; 4) 6, then 2 is a zero divisor (and not a unit) in Z 4. In fact, we see that 2 2 0 in Z 4. Thus 2x
More informationCOMP239: Mathematics for Computer Science II. Prof. Chadi Assi EV7.635
COMP239: Mathematics for Computer Science II Prof. Chadi Assi assi@ciise.concordia.ca EV7.635 The Euclidean Algorithm The Euclidean Algorithm Finding the GCD of two numbers using prime factorization is
More informationCoordinate Systems. S. F. Ellermeyer. July 10, 2009
Coordinate Systems S F Ellermeyer July 10, 009 These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (rd edition) These notes are
More informationMATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences.
MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. Congruences Let n be a postive integer. The integers a and b are called congruent modulo n if they have the same
More informationHomework #2 solutions Due: June 15, 2012
All of the following exercises are based on the material in the handout on integers found on the class website. 1. Find d = gcd(475, 385) and express it as a linear combination of 475 and 385. That is
More informationMath 131 notes. Jason Riedy. 6 October, Linear Diophantine equations : Likely delayed 6
Math 131 notes Jason Riedy 6 October, 2008 Contents 1 Modular arithmetic 2 2 Divisibility rules 3 3 Greatest common divisor 4 4 Least common multiple 4 5 Euclidean GCD algorithm 5 6 Linear Diophantine
More informationMATH 115, SUMMER 2012 LECTURE 4 THURSDAY, JUNE 21ST
MATH 115, SUMMER 2012 LECTURE 4 THURSDAY, JUNE 21ST JAMES MCIVOR Today we enter Chapter 2, which is the heart of this subject. Before starting, recall that last time we saw the integers have unique factorization
More information44.(ii) In this case we have that (12, 38) = 2 which does not divide 5 and so there are no solutions.
Solutions to Assignment 3 5E More Properties of Congruence 40. We can factor 729 = 7 3 9 so it is enough to show that 3 728 (mod 7), 3 728 (mod 3) and 3 728 (mod 9). 3 728 =(3 3 ) 576 = (27) 576 ( ) 576
More informationALGEBRA. 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers
ALGEBRA CHRISTIAN REMLING 1. Some elementary number theory 1.1. Primes and divisibility. We denote the collection of integers by Z = {..., 2, 1, 0, 1,...}. Given a, b Z, we write a b if b = ac for some
More information4. Congruence Classes
4 Congruence Classes Definition (p21) The congruence class mod m of a Z is Example With m = 3 we have Theorem For a b Z Proof p22 = {b Z : b a mod m} [0] 3 = { 6 3 0 3 6 } [1] 3 = { 2 1 4 7 } [2] 3 = {
More information1 2 3 style total. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points.
1 2 3 style total Math 415 Examination 3 Please print your name: Answer Key 1 True/false Circle the correct answer; no explanation is required. Each problem in this section counts 5 points. 1. The rings
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationMATH 25 CLASS 21 NOTES, NOV Contents. 2. Subgroups 2 3. Isomorphisms 4
MATH 25 CLASS 21 NOTES, NOV 7 2011 Contents 1. Groups: definition 1 2. Subgroups 2 3. Isomorphisms 4 1. Groups: definition Even though we have been learning number theory without using any other parts
More informationWORKSHEET MATH 215, FALL 15, WHYTE. We begin our course with the natural numbers:
WORKSHEET MATH 215, FALL 15, WHYTE We begin our course with the natural numbers: N = {1, 2, 3,...} which are a subset of the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } We will assume familiarity with their
More informationGroups, Rings, and Finite Fields. Andreas Klappenecker. September 12, 2002
Background on Groups, Rings, and Finite Fields Andreas Klappenecker September 12, 2002 A thorough understanding of the Agrawal, Kayal, and Saxena primality test requires some tools from algebra and elementary
More informationTomáš Madaras Congruence classes
Congruence classes For given integer m 2, the congruence relation modulo m at the set Z is the equivalence relation, thus, it provides a corresponding partition of Z into mutually disjoint sets. Definition
More informationSection 19 Integral domains
Section 19 Integral domains Instructor: Yifan Yang Spring 2007 Observation and motivation There are rings in which ab = 0 implies a = 0 or b = 0 For examples, Z, Q, R, C, and Z[x] are all such rings There
More information2 Arithmetic. 2.1 Greatest common divisors. This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}.
2 Arithmetic This chapter is about properties of the integers Z = {..., 2, 1, 0, 1, 2,...}. (See [Houston, Chapters 27 & 28]) 2.1 Greatest common divisors Definition 2.16. If a, b are integers, we say
More informationMathematical Foundations of Cryptography
Mathematical Foundations of Cryptography Cryptography is based on mathematics In this chapter we study finite fields, the basis of the Advanced Encryption Standard (AES) and elliptical curve cryptography
More informationSignature: (In Ink) UNIVERSITY OF MANITOBA TEST 1 SOLUTIONS COURSE: MATH 2170 DATE & TIME: February 11, 2019, 16:30 17:15
PAGE: 1 of 7 I understand that cheating is a serious offence: Signature: (In Ink) PAGE: 2 of 7 1. Let a, b, m, be integers, m > 1. [1] (a) Define a b. Solution: a b iff for some d, ad = b. [1] (b) Define
More informationFor your quiz in recitation this week, refer to these exercise generators:
Monday, Oct 29 Today we will talk about inverses in modular arithmetic, and the use of inverses to solve linear congruences. For your quiz in recitation this week, refer to these exercise generators: GCD
More informationEquivalence Relations and Partitions, Normal Subgroups, Quotient Groups, and Homomorphisms
Equivalence Relations and Partitions, Normal Subgroups, Quotient Groups, and Homomorphisms Math 356 Abstract We sum up the main features of our last three class sessions, which list of topics are given
More informationEconomics 241B Review of Limit Theorems for Sequences of Random Variables
Economics 241B Review of Limit Theorems for Sequences of Random Variables Convergence in Distribution The previous de nitions of convergence focus on the outcome sequences of a random variable. Convergence
More informationMath 312/ AMS 351 (Fall 17) Sample Questions for Final
Math 312/ AMS 351 (Fall 17) Sample Questions for Final 1. Solve the system of equations 2x 1 mod 3 x 2 mod 7 x 7 mod 8 First note that the inverse of 2 is 2 mod 3. Thus, the first equation becomes (multiply
More informationPrime and irreducible elements of the ring of integers modulo n
Prime and irreducible elements of the ring of integers modulo n M. H. Jafari and A. R. Madadi Department of Pure Mathematics, Faculty of Mathematical Sciences University of Tabriz, Tabriz, Iran Abstract
More informationIntroduction to Proofs
Introduction to Proofs Many times in economics we will need to prove theorems to show that our theories can be supported by speci c assumptions. While economics is an observational science, we use mathematics
More information14 Equivalence Relations
14 Equivalence Relations Tom Lewis Fall Term 2010 Tom Lewis () 14 Equivalence Relations Fall Term 2010 1 / 10 Outline 1 The definition 2 Congruence modulo n 3 Has-the-same-size-as 4 Equivalence classes
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 57 5. p-adic Numbers 5.1. Motivating examples. We all know that 2 is irrational, so that 2 is not a square in the rational field Q, but that we can
More informationWORKSHEET ON NUMBERS, MATH 215 FALL. We start our study of numbers with the integers: N = {1, 2, 3,...}
WORKSHEET ON NUMBERS, MATH 215 FALL 18(WHYTE) We start our study of numbers with the integers: Z = {..., 2, 1, 0, 1, 2, 3,... } and their subset of natural numbers: N = {1, 2, 3,...} For now we will not
More informationNumber Theory Notes Spring 2011
PRELIMINARIES The counting numbers or natural numbers are 1, 2, 3, 4, 5, 6.... The whole numbers are the counting numbers with zero 0, 1, 2, 3, 4, 5, 6.... The integers are the counting numbers and zero
More informationNumber Theory Math 420 Silverman Exam #1 February 27, 2018
Name: Number Theory Math 420 Silverman Exam #1 February 27, 2018 INSTRUCTIONS Read Carefully Time: 50 minutes There are 5 problems. Write your name neatly at the top of this page. Write your final answer
More informationChapter 4 Finite Fields
Chapter 4 Finite Fields Introduction will now introduce finite fields of increasing importance in cryptography AES, Elliptic Curve, IDEA, Public Key concern operations on numbers what constitutes a number
More informationMATH 4400 SOLUTIONS TO SOME EXERCISES. 1. Chapter 1
MATH 4400 SOLUTIONS TO SOME EXERCISES 1.1.3. If a b and b c show that a c. 1. Chapter 1 Solution: a b means that b = na and b c that c = mb. Substituting b = na gives c = (mn)a, that is, a c. 1.2.1. Find
More informationLecture Notes. Advanced Discrete Structures COT S
Lecture Notes Advanced Discrete Structures COT 4115.001 S15 2015-01-13 Recap Divisibility Prime Number Theorem Euclid s Lemma Fundamental Theorem of Arithmetic Euclidean Algorithm Basic Notions - Section
More informationQuadratic Congruences, the Quadratic Formula, and Euler s Criterion
Quadratic Congruences, the Quadratic Formula, and Euler s Criterion R. C. Trinity University Number Theory Introduction Let R be a (commutative) ring in which 2 = 1 R + 1 R R. Consider a quadratic equation
More information7.2 Applications of Euler s and Fermat s Theorem.
7.2 Applications of Euler s and Fermat s Theorem. i) Finding and using inverses. From Fermat s Little Theorem we see that if p is prime and p a then a p 1 1 mod p, or equivalently a p 2 a 1 mod p. This
More informationMATH 361: NUMBER THEORY FOURTH LECTURE
MATH 361: NUMBER THEORY FOURTH LECTURE 1. Introduction Everybody knows that three hours after 10:00, the time is 1:00. That is, everybody is familiar with modular arithmetic, the usual arithmetic of the
More informationMATH 3240Q Introduction to Number Theory Homework 4
If the Sun refused to shine I don t mind I don t mind If the mountains fell in the sea Let it be it ain t me Now if six turned out to be nine Oh I don t mind I don t mind Jimi Hendrix If Six Was Nine from
More informationChapter 5. Modular arithmetic. 5.1 The modular ring
Chapter 5 Modular arithmetic 5.1 The modular ring Definition 5.1. Suppose n N and x, y Z. Then we say that x, y are equivalent modulo n, and we write x y mod n if n x y. It is evident that equivalence
More informationBasic elements of number theory
Cryptography Basic elements of number theory Marius Zimand By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a k for some integer k. Notation
More informationBasic elements of number theory
Cryptography Basic elements of number theory Marius Zimand 1 Divisibility, prime numbers By default all the variables, such as a, b, k, etc., denote integer numbers. Divisibility a 0 divides b if b = a
More information1 Structure of Finite Fields
T-79.5501 Cryptology Additional material September 27, 2005 1 Structure of Finite Fields This section contains complementary material to Section 5.2.3 of the text-book. It is not entirely self-contained
More informationAN ALGEBRAIC PROOF OF RSA ENCRYPTION AND DECRYPTION
AN ALGEBRAIC PROOF OF RSA ENCRYPTION AND DECRYPTION Recall that RSA works as follows. A wants B to communicate with A, but without E understanding the transmitted message. To do so: A broadcasts RSA method,
More informationCPSC 467: Cryptography and Computer Security
CPSC 467: Cryptography and Computer Security Michael J. Fischer Lecture 9 September 30, 2015 CPSC 467, Lecture 9 1/47 Fast Exponentiation Algorithms Number Theory Needed for RSA Elementary Number Theory
More informationKnow the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element.
The first exam will be on Monday, June 8, 202. The syllabus will be sections. and.2 in Lax, and the number theory handout found on the class web site, plus the handout on the method of successive squaring
More informationProof 1: Using only ch. 6 results. Since gcd(a, b) = 1, we have
Exercise 13. Consider positive integers a, b, and c. (a) Suppose gcd(a, b) = 1. (i) Show that if a divides the product bc, then a must divide c. I give two proofs here, to illustrate the different methods.
More informationThe Euclidean Algorithm and Multiplicative Inverses
1 The Euclidean Algorithm and Multiplicative Inverses Lecture notes for Access 2009 The Euclidean Algorithm is a set of instructions for finding the greatest common divisor of any two positive integers.
More informationThe Integers. Peter J. Kahn
Math 3040: Spring 2009 The Integers Peter J. Kahn Contents 1. The Basic Construction 1 2. Adding integers 6 3. Ordering integers 16 4. Multiplying integers 18 Before we begin the mathematics of this section,
More informationMTH 346: The Chinese Remainder Theorem
MTH 346: The Chinese Remainder Theorem March 3, 2014 1 Introduction In this lab we are studying the Chinese Remainder Theorem. We are going to study how to solve two congruences, find what conditions are
More informationMath 222A W03 D. Congruence relations
Math 222A W03 D. 1. The concept Congruence relations Let s start with a familiar case: congruence mod n on the ring Z of integers. Just to be specific, let s use n = 6. This congruence is an equivalence
More informationFoundations of Mathematics
Foundations of Mathematics L. Pedro Poitevin 1. Preliminaries 1.1. Sets We will naively think of a set as a collection of mathematical objects, called its elements or members. To indicate that an object
More informationABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH
ABSTRACT ALGEBRA 1, LECTURE NOTES 5: HOMOMORPHISMS, ISOMORPHISMS, SUBGROUPS, QUOTIENT ( FACTOR ) GROUPS. ANDREW SALCH 1. Homomorphisms and isomorphisms between groups. Definition 1.1. Let G, H be groups.
More informationMath 109 HW 9 Solutions
Math 109 HW 9 Solutions Problems IV 18. Solve the linear diophantine equation 6m + 10n + 15p = 1 Solution: Let y = 10n + 15p. Since (10, 15) is 5, we must have that y = 5x for some integer x, and (as we
More informationAlgebraic number theory Solutions to exercise sheet for chapter 4
Algebraic number theory Solutions to exercise sheet for chapter 4 Nicolas Mascot n.a.v.mascot@warwick.ac.uk), Aurel Page a.r.page@warwick.ac.uk) TAs: Chris Birkbeck c.d.birkbeck@warwick.ac.uk), George
More information1/30: Polynomials over Z/n.
1/30: Polynomials over Z/n. Last time to establish the existence of primitive roots we rely on the following key lemma: Lemma 6.1. Let s > 0 be an integer with s p 1, then we have #{α Z/pZ α s = 1} = s.
More information1. multiplication is commutative and associative;
Chapter 4 The Arithmetic of Z In this chapter, we start by introducing the concept of congruences; these are used in our proof (going back to Gauss 1 ) that every integer has a unique prime factorization.
More informationABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS
ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2006 2 This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair ISBN 1 57766
More informationFoundations of Mathematics Worksheet 2
Foundations of Mathematics Worksheet 2 L. Pedro Poitevin June 24, 2007 1. What are the atomic truth assignments on {a 1,..., a n } that satisfy: (a) The proposition p = ((a 1 a 2 ) (a 2 a 3 ) (a n 1 a
More information2.3 In modular arithmetic, all arithmetic operations are performed modulo some integer.
CHAPTER 2 INTRODUCTION TO NUMBER THEORY ANSWERS TO QUESTIONS 2.1 A nonzero b is a divisor of a if a = mb for some m, where a, b, and m are integers. That is, b is a divisor of a if there is no remainder
More informationAbstract Algebra, Second Edition, by John A. Beachy and William D. Blair. Corrections and clarifications
1 Abstract Algebra, Second Edition, by John A. Beachy and William D. Blair Corrections and clarifications Note: Some corrections were made after the first printing of the text. page 9, line 8 For of the
More informationLecture 7: Polynomial rings
Lecture 7: Polynomial rings Rajat Mittal IIT Kanpur You have seen polynomials many a times till now. The purpose of this lecture is to give a formal treatment to constructing polynomials and the rules
More informationThe Chinese Remainder Theorem
Chapter 4 The Chinese Remainder Theorem The Monkey-Sailor-Coconut Problem Three sailors pick up a number of coconuts, place them in a pile and retire for the night. During the night, the first sailor wanting
More informationThe number of ways to choose r elements (without replacement) from an n-element set is. = r r!(n r)!.
The first exam will be on Friday, September 23, 2011. The syllabus will be sections 0.1 through 0.4 and 0.6 in Nagpaul and Jain, and the corresponding parts of the number theory handout found on the class
More informationFROM GROUPS TO GALOIS Amin Witno
WON Series in Discrete Mathematics and Modern Algebra Volume 6 FROM GROUPS TO GALOIS Amin Witno These notes 1 have been prepared for the students at Philadelphia University (Jordan) who are taking the
More informationDiscrete Mathematics with Applications MATH236
Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet
More informationNumber Theory. Modular Arithmetic
Number Theory The branch of mathematics that is important in IT security especially in cryptography. Deals only in integer numbers and the process can be done in a very fast manner. Modular Arithmetic
More informationCoding Theory ( Mathematical Background I)
N.L.Manev, Lectures on Coding Theory (Maths I) p. 1/18 Coding Theory ( Mathematical Background I) Lector: Nikolai L. Manev Institute of Mathematics and Informatics, Sofia, Bulgaria N.L.Manev, Lectures
More informationDiscrete Mathematics and Probability Theory Fall 2018 Alistair Sinclair and Yun Song Note 6
CS 70 Discrete Mathematics and Probability Theory Fall 2018 Alistair Sinclair and Yun Song Note 6 1 Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes
More informationM381 Number Theory 2004 Page 1
M81 Number Theory 2004 Page 1 [[ Comments are written like this. Please send me (dave@wildd.freeserve.co.uk) details of any errors you find or suggestions for improvements. ]] Question 1 20 = 2 * 10 +
More informationExamples of Groups
Examples of Groups 8-23-2016 In this section, I ll look at some additional examples of groups. Some of these will be discussed in more detail later on. In many of these examples, I ll assume familiar things
More information2a 2 4ac), provided there is an element r in our
MTH 310002 Test II Review Spring 2012 Absractions versus examples The purpose of abstraction is to reduce ideas to their essentials, uncluttered by the details of a specific situation Our lectures built
More information1 Overview and revision
MTH6128 Number Theory Notes 1 Spring 2018 1 Overview and revision In this section we will meet some of the concerns of Number Theory, and have a brief revision of some of the relevant material from Introduction
More informationFinal Exam Review. 2. Let A = {, { }}. What is the cardinality of A? Is
1. Describe the elements of the set (Z Q) R N. Is this set countable or uncountable? Solution: The set is equal to {(x, y) x Z, y N} = Z N. Since the Cartesian product of two denumerable sets is denumerable,
More informationMATH 310: Homework 7
1 MATH 310: Homework 7 Due Thursday, 12/1 in class Reading: Davenport III.1, III.2, III.3, III.4, III.5 1. Show that x is a root of unity modulo m if and only if (x, m 1. (Hint: Use Euler s theorem and
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Math 0 Solutions to homework Due 9//6. An element [a] Z/nZ is idempotent if [a] 2 [a]. Find all idempotent elements in Z/0Z and in Z/Z. Solution. First note we clearly have [0] 2 [0] so [0] is idempotent
More informationNumber theory (Chapter 4)
EECS 203 Spring 2016 Lecture 10 Page 1 of 8 Number theory (Chapter 4) Review Questions: 1. Does 5 1? Does 1 5? 2. Does (129+63) mod 10 = (129 mod 10)+(63 mod 10)? 3. Does (129+63) mod 10 = ((129 mod 10)+(63
More informationINTRODUCTION TO THE GROUP THEORY
Lecture Notes on Structure of Algebra INTRODUCTION TO THE GROUP THEORY By : Drs. Antonius Cahya Prihandoko, M.App.Sc e-mail: antoniuscp.fkip@unej.ac.id Mathematics Education Study Program Faculty of Teacher
More informationPolynomials. Chapter 4
Chapter 4 Polynomials In this Chapter we shall see that everything we did with integers in the last Chapter we can also do with polynomials. Fix a field F (e.g. F = Q, R, C or Z/(p) for a prime p). Notation
More informationLECTURE NOTES IN CRYPTOGRAPHY
1 LECTURE NOTES IN CRYPTOGRAPHY Thomas Johansson 2005/2006 c Thomas Johansson 2006 2 Chapter 1 Abstract algebra and Number theory Before we start the treatment of cryptography we need to review some basic
More information1 Selected Homework Solutions
Selected Homework Solutions Mathematics 4600 A. Bathi Kasturiarachi September 2006. Selected Solutions to HW # HW #: (.) 5, 7, 8, 0; (.2):, 2 ; (.4): ; (.5): 3 (.): #0 For each of the following subsets
More informationNotes on Systems of Linear Congruences
MATH 324 Summer 2012 Elementary Number Theory Notes on Systems of Linear Congruences In this note we will discuss systems of linear congruences where the moduli are all different. Definition. Given the
More informationFunctions Some Basic Ideas and Some Examples
Functions Some Basic Ideas and Some Examples S. F. Ellermeyer May 30, 2001 1 Basic Ideas and Terminology Let D < with D 6= A real valued function, f, ond is an assignment of each number x D toasinglerealnumber,y.
More informationName: Solutions Final Exam
Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] All of
More information3.2 Solving linear congruences. v3
3.2 Solving linear congruences. v3 Solving equations of the form ax b (mod m), where x is an unknown integer. Example (i) Find an integer x for which 56x 1 mod 93. Solution We have already solved this
More informationGalois fields/1. (M3) There is an element 1 (not equal to 0) such that a 1 = a for all a.
Galois fields 1 Fields A field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except by zero) can be performed, and satisfy the usual rules. More
More informationNumber Theory Proof Portfolio
Number Theory Proof Portfolio Jordan Rock May 12, 2015 This portfolio is a collection of Number Theory proofs and problems done by Jordan Rock in the Spring of 2014. The problems are organized first by
More informationNOTES ON SIMPLE NUMBER THEORY
NOTES ON SIMPLE NUMBER THEORY DAMIEN PITMAN 1. Definitions & Theorems Definition: We say d divides m iff d is positive integer and m is an integer and there is an integer q such that m = dq. In this case,
More informationPart IX. Factorization
IX.45. Unique Factorization Domains 1 Part IX. Factorization Section IX.45. Unique Factorization Domains Note. In this section we return to integral domains and concern ourselves with factoring (with respect
More informationHomework #5 Solutions
Homework #5 Solutions p 83, #16. In order to find a chain a 1 a 2 a n of subgroups of Z 240 with n as large as possible, we start at the top with a n = 1 so that a n = Z 240. In general, given a i we will
More information