Foundations of Cryptography


 Allison Bell
 5 years ago
 Views:
Transcription
1 Foundations of Cryptography Ville Junnila Department of Mathematics and Statistics University of Turku 2015 Ville Junnila Lecture 7 1 of 18
2 Cosets Definition 2.12 Let G be a group, H G and a G. Then we have the following definitions: ah = {ah G h H} is the left coset of H in G defined by a and Ha = {ha G h H} is the right coset of H in G defined by a. Theorem 2.9 Let H be a subgroup of G. Then for any a, b G the following statements are equivalent: 1 ah = bh, 2 a bh, i.e., a = bh for some h H and 3 b 1 a H. Ville Junnila viljun@utu.fi Lecture 7 2 of 18
3 Cosets Definition 2.13 If H G, then the set of all cosets of H is denoted by G/H. Theorem 2.10 Let H G. 1 Each element of G belongs to exactly on left coset of H. 2 If G is finite, then each left coset has the same number of elements. 3 The subgroup H itself is a coset; H = 1 H. Ville Junnila viljun@utu.fi Lecture 7 3 of 18
4 Cosets Theorem 2.11 (Lagrange s theorem) Let G be a finite group and H G. If the number of left cosets of H is i, then we have G = i H. In particular, G is divided by H. Example Let G be a finite group with 21 elements. Consider the possible orders of subgroups of G. Ville Junnila viljun@utu.fi Lecture 7 4 of 18
5 Order of group element Definition 2.14 Let G be a group. If a G is such that a k 1 G for all k Z \ {0}, then the order of a is infinite. Otherwise, the order of a G is the smallest positive integer n such that a n = 1 G. Theorem Let G be a finite cyclic group, G = c. If n is the smallest positive integer such that c n = 1 G, then G = n and G = {1, c, c 2,..., c n 1 }. Theorem 2.12 Let G be a group and a G. The order of a is equal to the order of a = {a k k Z}. Ville Junnila viljun@utu.fi Lecture 7 5 of 18
6 Order of group element Theorem 2.13 Let G be a finite group and a G. Then the order of a G divides G since a divides G by Lagrange s theorem. Therefore, Example a G = 1 G. Consider the group (Z 25, ). Determine the order of 2 Z 25. Ville Junnila viljun@utu.fi Lecture 7 6 of 18
7 Order of group element Theorem 2.14 (Euler s theorem) Considering the group (Z n, ), we have a ϕ(n) = 1 for all a Z n. In other words, for all a Z such that gcd(a, n) = 1, we have a ϕ(n) 1 (mod n). Theorem 2.15 (Fermat s little theorem) If p P and a Z is not divisible by p, then a p 1 1 (mod p). Ville Junnila viljun@utu.fi Lecture 7 7 of 18
8 Rings Definition 2.15 A triplet (R, +, ) is called a ring, if + and are binary operations defined over R and the following conditions hold: Re1 (R, +) is an abelian group (the additive group of the ring) Re2 a(bc) = (ab)c multiplication) a, b, c R (the associativity of Re3 there exists 1 R such that 1 a = a 1 = a element or identity element of the ring) Re4 a(b + c) = ab + ac; (a + b)c = ac + bc (distributivity). a R (unit a, b, c R If multiplication is also commutative, i.e., ab = ba a, b R, we call R a commutative ring. Remark The unit element 1 of the ring is unique. Ville Junnila viljun@utu.fi Lecture 7 8 of 18
9 Rings Example 2.11 Each of the sets Z, Q and R form a commutative ring under the usual + and. Example 2.12 (Polynomial ring) The set of polynomials R[x] = {a o + a 1 x + + a n x n n 0, a k R (k = 0, 1,..., n)} with the operations (f, g R[x]) (f + g)(x) = f (x) + g(x) and (fg)(x) = f (x)g(x) for all x R is a commutative ring. Similarly, Z[x] and Q[x] are commutative rings. Ville Junnila viljun@utu.fi Lecture 7 9 of 18
10 Rings Example 2.13 (Quotient ring or residue class ring) The set Z m is a commutative ring under the following + and : a + b = a + b and a b = ab. The zero element is 0 and the unit element 1. The ring is finite and commutative. Definition 2.16 Let R be a ring. A subset I R is an ideal in R if I1 (I, +) is a subgroup of (R, +), I2 ra I for all r R and a I, and I3 ar I for all r R and a I. Ville Junnila viljun@utu.fi Lecture 7 10 of 18
11 Ideals Example Consider the ring (Z, +, ). Let us show that m = mz is an ideal of Z. Example 2.14(a) Consider the polynomial ring (R[x], +, ). Let us show that is an ideal in R[x]. I = {p(x) R[x] p(0) = 0} Ville Junnila viljun@utu.fi Lecture 7 11 of 18
12 Ideals Example 2.14(c) Let us show that I = {a m x m + a m+1 x m a n x n R[x] n m} is an ideal in R[x]. Definition If S 1, S 2,..., S k are subsets of a ring R, then S 1 + S S k = {r 1 + r r k r i S i }. Ville Junnila viljun@utu.fi Lecture 7 12 of 18
13 Ideals Theorem 2.16 Let R be a ring. 1 If I and J are ideals in R, then I + J is an ideal. Generally, if I 1, I 2,..., I n are ideals in R, then I 1 + I I n is an ideal. 2 If I and J are ideals in R, then I J is an ideal. Generally, if I i (i I) are ideals in R, then the intersection i I I i is an ideal. Definition (Generating an ideal) Let R be a ring. A subset S R generates an ideal S = I, S I where I goes through all such ideals in R. Indeed, by Theorem 2.16, S is an ideal. Ville Junnila viljun@utu.fi Lecture 7 13 of 18
14 Ideals Remark The ideal S is the smallest one including S, i.e., if J is an ideal such that S J, then S J. Definition If S is a finite set, say S = {a 1, a 2,..., a k }, then we denote S = a 1, a 2,..., a k and say that the ideal S is finitely generated. An ideal generated by one element, say a, is called a principal ideal. Example 2.15 The trivial ideals R and {0} are principal ideal since R = 1 and {0} = 0. Ville Junnila viljun@utu.fi Lecture 7 14 of 18
15 Ideals Example 2.16 The ideals of the ring Z are principal ideals m = mz (m 0) (by Theorem 2.6). Example 2.17 Consider the principal ideal x m in the polynomial ring R[x]. By the definition of ideal, p(x)x m x m for any p(x) R[x]. Therefore, the ideal (of Example 2.14(c)) I = {a m x m + a m+1 x m a n x n R[x] n m} is such that I x m. Since x m I, then by the minimality of x m, we have x m I. Thus, I = x m. Ville Junnila viljun@utu.fi Lecture 7 15 of 18
16 Ideals Theorem 2.17 If R is a commutative ring, then for any a 1, a 2,..., a k R we have a 1, a 2,..., a k = {r 1 a 1 + r 2 a r k a k r i R}. Ville Junnila viljun@utu.fi Lecture 7 16 of 18
17 Quotient ring Remark Let (R, +, ) be a ring and I an ideal in R. Recall that I R under the addition +. Hence, we may consider the cosets a + I in the group (R, +). Recall that R/I denoted the set of all cosets. Theorem 2.18 Let (R, +, ) be a ring and I an ideal in R. The equations (a + I ) + (b + I ) = (a + b) + I and (a + I )(b + I ) = ab + I give welldefined binary operations from R/I R/I to R/I that form a ring (R/I, +, ). Ville Junnila viljun@utu.fi Lecture 7 17 of 18
18 Quotient ring Remark Recall that if I is an ideal in a ring R, then (I, +) is a subgroup of (R, +). Theorem 2.9 Let H be a subgroup of G. Then for any a, b G the following statements are equivalent: 1 ah = bh, 2 a bh, i.e., a = bh for some h H and 3 ab 1 H. Example Consider the ring (Z/3Z, +, ). Ville Junnila viljun@utu.fi Lecture 7 18 of 18
CHAPTER 14. Ideals and Factor Rings
CHAPTER 14 Ideals and Factor Rings Ideals Definition (Ideal). A subring A of a ring R is called a (twosided) ideal of R if for every r 2 R and every a 2 A, ra 2 A and ar 2 A. Note. (1) A absorbs elements
More informationAlgebra Review. Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor. June 15, 2001
Algebra Review Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor June 15, 2001 1 Groups Definition 1.1 A semigroup (G, ) is a set G with a binary operation such that: Axiom 1 ( a,
More informationMany of the groups with which we are familiar are arithmetical in nature, and they tend to share key structures that combine more than one operation.
12. Rings 1 Rings Many of the groups with which we are familiar are arithmetical in nature, and they tend to share key structures that combine more than one operation. Example: Z, Q, R, and C are an Abelian
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 informationSection 18 Rings and fields
Section 18 Rings and fields Instructor: Yifan Yang Spring 2007 Motivation Many sets in mathematics have two binary operations (and thus two algebraic structures) For example, the sets Z, Q, R, M n (R)
More information5 Group theory. 5.1 Binary operations
5 Group theory This section is an introduction to abstract algebra. This is a very useful and important subject for those of you who will continue to study pure mathematics. 5.1 Binary operations 5.1.1
More informationMathematics for Cryptography
Mathematics for Cryptography Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, N2L 3G1, Canada March 15, 2016 1 Groups and Modular Arithmetic 1.1
More informationRings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.
Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary
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 informationSection 10: Counting the Elements of a Finite Group
Section 10: Counting the Elements of a Finite Group Let G be a group and H a subgroup. Because the right cosets are the family of equivalence classes with respect to an equivalence relation on G, it follows
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 informationMATH 101: ALGEBRA I WORKSHEET, DAY #3. Fill in the blanks as we finish our first pass on prerequisites of group theory.
MATH 101: ALGEBRA I WORKSHEET, DAY #3 Fill in the blanks as we finish our first pass on prerequisites of group theory 1 Subgroups, cosets Let G be a group Recall that a subgroup H G is a subset that is
More informationIdeals, congruence modulo ideal, factor rings
Ideals, congruence modulo ideal, factor rings Sergei Silvestrov Spring term 2011, Lecture 6 Contents of the lecture Homomorphisms of rings Ideals Factor rings Typeset by FoilTEX Congruence in F[x] and
More informationSupplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.
Glossary 1 Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.23 Abelian Group. A group G, (or just G for short) is
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 informationCHAPTER I. Rings. Definition A ring R is a set with two binary operations, addition + and
CHAPTER I Rings 1.1 Definitions and Examples Definition 1.1.1. A ring R is a set with two binary operations, addition + and multiplication satisfying the following conditions for all a, b, c in R : (i)
More information( ) 3 = ab 3 a!1. ( ) 3 = aba!1 a ( ) = 4 " 5 3 " 4 = ( )! 2 3 ( ) =! 5 4. Math 546 Problem Set 15
Math 546 Problem Set 15 1. Let G be a finite group. (a). Suppose that H is a subgroup of G and o(h) = 4. Suppose that K is a subgroup of G and o(k) = 5. What is H! K (and why)? Solution: H! K = {e} since
More informationRings and Fields Theorems
Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a nonempty set. Let + and (multiplication)
More informationGroups Subgroups Normal subgroups Quotient groups Homomorphisms Cyclic groups Permutation groups Cayley s theorem Class equations Sylow theorems
Group Theory Groups Subgroups Normal subgroups Quotient groups Homomorphisms Cyclic groups Permutation groups Cayley s theorem Class equations Sylow theorems Groups Definition : A nonempty set ( G,*)
More informationMA441: Algebraic Structures I. Lecture 18
MA441: Algebraic Structures I Lecture 18 5 November 2003 1 Review from Lecture 17: Theorem 6.5: Aut(Z/nZ) U(n) For every positive integer n, Aut(Z/nZ) is isomorphic to U(n). The proof used the map T :
More informationGroups in Cryptography. Çetin Kaya Koç Winter / 13
http://koclab.org Çetin Kaya Koç Winter 2017 1 / 13 A set S and a binary operation A group G = (S, ) if S and satisfy: Closure: If a, b S then a b S Associativity: For a, b, c S, (a b) c = a (b c) A neutral
More informationElementary Algebra Chinese Remainder Theorem Euclidean Algorithm
Elementary Algebra Chinese Remainder Theorem Euclidean Algorithm April 11, 2010 1 Algebra We start by discussing algebraic structures and their properties. This is presented in more depth than what we
More informationExample 2: Let R be any commutative ring with 1, fix a R, and let. I = ar = {ar : r R},
25. Ideals and quotient rings We continue our study of rings by making analogies with groups. The next concept we introduce is that of an ideal of a ring. Ideals are ringtheoretic counterparts of normal
More informationKevin James. MTHSC 412 Section 3.1 Definition and Examples of Rings
MTHSC 412 Section 3.1 Definition and Examples of Rings A ring R is a nonempty set R together with two binary operations (usually written as addition and multiplication) that satisfy the following axioms.
More informationALGEBRA I (LECTURE NOTES 2017/2018) LECTURE 9  CYCLIC GROUPS AND EULER S FUNCTION
ALGEBRA I (LECTURE NOTES 2017/2018) LECTURE 9  CYCLIC GROUPS AND EULER S FUNCTION PAVEL RŮŽIČKA 9.1. Congruence modulo n. Let us have a closer look at a particular example of a congruence relation on
More informationTotal 100
Math 542 Midterm Exam, Spring 2016 Prof: Paul Terwilliger Your Name (please print) SOLUTIONS NO CALCULATORS/ELECTRONIC DEVICES ALLOWED. MAKE SURE YOUR CELL PHONE IS OFF. Problem Value 1 10 2 10 3 10 4
More informationFinite Fields. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay
1 / 25 Finite Fields Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay September 25, 2014 2 / 25 Fields Definition A set F together
More informationCSIR  Algebra Problems
CSIR  Algebra Problems N. Annamalai DST  INSPIRE Fellow (SRF) Department of Mathematics Bharathidasan University Tiruchirappalli 620024 Email: algebra.annamalai@gmail.com Website: https://annamalaimaths.wordpress.com
More informationAlgebraic Structures Exam File Fall 2013 Exam #1
Algebraic Structures Exam File Fall 2013 Exam #1 1.) Find all four solutions to the equation x 4 + 16 = 0. Give your answers as complex numbers in standard form, a + bi. 2.) Do the following. a.) Write
More informationQuizzes for Math 401
Quizzes for Math 401 QUIZ 1. a) Let a,b be integers such that λa+µb = 1 for some inetegrs λ,µ. Prove that gcd(a,b) = 1. b) Use Euclid s algorithm to compute gcd(803, 154) and find integers λ,µ such that
More informationAlgebraic structures I
MTH5100 Assignment 110 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one
More information(Rgs) Rings Math 683L (Summer 2003)
(Rgs) Rings Math 683L (Summer 2003) We will first summarise the general results that we will need from the theory of rings. A unital ring, R, is a set equipped with two binary operations + and such that
More informationDiscrete Mathematics with Applications MATH236
Discrete Mathematics with Applications MATH236 Dr. Hung P. TongViet School of Mathematics, Statistics and Computer Science University of KwaZuluNatal Pietermaritzburg Campus Semester 1, 2013 TongViet
More informationChapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples
Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter
More informationMath Introduction to Modern Algebra
Math 343  Introduction to Modern Algebra Notes Rings and Special Kinds of Rings Let R be a (nonempty) set. R is a ring if there are two binary operations + and such that (A) (R, +) is an abelian group.
More informationFinite Fields. Sophie Huczynska. Semester 2, Academic Year
Finite Fields Sophie Huczynska Semester 2, Academic Year 200506 2 Chapter 1. Introduction Finite fields is a branch of mathematics which has come to the fore in the last 50 years due to its numerous applications,
More informationMath 2070BC Term 2 Weeks 1 13 Lecture Notes
Math 2070BC 2017 18 Term 2 Weeks 1 13 Lecture Notes Keywords: group operation multiplication associative identity element inverse commutative abelian group Special Linear Group order infinite order cyclic
More informationThe number of ways to choose r elements (without replacement) from an nelement 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 informationPublickey Cryptography: Theory and Practice
Publickey Cryptography Theory and Practice Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Chapter 2: Mathematical Concepts Divisibility Congruence Quadratic Residues
More informationECEN 5022 Cryptography
Elementary Algebra and Number Theory University of Colorado Spring 2008 Divisibility, Primes Definition. N denotes the set {1, 2, 3,...} of natural numbers and Z denotes the set of integers {..., 2, 1,
More informationModern Algebra I. Circle the correct answer; no explanation is required. Each problem in this section counts 5 points.
1 2 3 style total Math 415 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. Every group of order 6
More informationTo hand in: (a) Prove that a group G is abelian (= commutative) if and only if (xy) 2 = x 2 y 2 for all x, y G.
Homework #6. Due Thursday, October 14th Reading: For this homework assignment: Sections 3.3 and 3.4 (up to page 167) Before the class next Thursday: Sections 3.5 and 3.4 (pp. 168171). Also review the
More informationSUMMARY ALGEBRA I LOUISPHILIPPE THIBAULT
SUMMARY ALGEBRA I LOUISPHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.
More informationSome practice problems for midterm 2
Some practice problems for midterm 2 Kiumars Kaveh November 14, 2011 Problem: Let Z = {a G ax = xa, x G} be the center of a group G. Prove that Z is a normal subgroup of G. Solution: First we prove Z is
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 information2 ALGEBRA II. Contents
ALGEBRA II 1 2 ALGEBRA II Contents 1. Results from elementary number theory 3 2. Groups 4 2.1. Denition, Subgroup, Order of an element 4 2.2. Equivalence relation, Lagrange's theorem, Cyclic group 9 2.3.
More informationMath 546, Exam 2 Information.
Math 546, Exam 2 Information. 10/21/09, LC 303B, 10:1011:00. Exam 2 will be based on: Sections 3.2, 3.3, 3.4, 3.5; The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/546fa09/546.html)
More informationMATH 433 Applied Algebra Lecture 19: Subgroups (continued). Errordetecting and errorcorrecting codes.
MATH 433 Applied Algebra Lecture 19: Subgroups (continued). Errordetecting and errorcorrecting codes. Subgroups Definition. A group H is a called a subgroup of a group G if H is a subset of G and the
More informationLecture 7 Cyclic groups and subgroups
Lecture 7 Cyclic groups and subgroups Review Types of groups we know Numbers: Z, Q, R, C, Q, R, C Matrices: (M n (F ), +), GL n (F ), where F = Q, R, or C. Modular groups: Z/nZ and (Z/nZ) Dihedral groups:
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 informationMath 4400, Spring 08, Sample problems Final Exam.
Math 4400, Spring 08, Sample problems Final Exam. 1. Groups (1) (a) Let a be an element of a group G. Define the notions of exponent of a and period of a. (b) Suppose a has a finite period. Prove that
More informationModern Algebra (MA 521) Synopsis of lectures JulyNov 2015 semester, IIT Guwahati
Modern Algebra (MA 521) Synopsis of lectures JulyNov 2015 semester, IIT Guwahati Shyamashree Upadhyay Contents 1 Lecture 1 4 1.1 Properties of Integers....................... 4 1.2 Sets, relations and
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
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 email: antoniuscp.fkip@unej.ac.id Mathematics Education Study Program Faculty of Teacher
More informationDefinition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson
Definition List Modern Algebra, Fall 2011 Anders O.F. Hendrickson On almost every Friday of the semester, we will have a brief quiz to make sure you have memorized the definitions encountered in our studies.
More information1. Group Theory Permutations.
1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7
More informationSchool of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information
MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout 0: Course Information Lecturer: Martyn Quick, Room 326. Prerequisite: MT3505 (or MT4517) Rings & Fields Lectures: Tutorials: Mon
More informationSUMMARY OF GROUPS AND RINGS GROUPS AND RINGS III Week 1 Lecture 1 Tuesday 3 March.
SUMMARY OF GROUPS AND RINGS GROUPS AND RINGS III 2009 Week 1 Lecture 1 Tuesday 3 March. 1. Introduction (Background from Algebra II) 1.1. Groups and Subgroups. Definition 1.1. A binary operation on a set
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 informationTheorems and Definitions in Group Theory
Theorems and Definitions in Group Theory Shunan Zhao Contents 1 Basics of a group 3 1.1 Basic Properties of Groups.......................... 3 1.2 Properties of Inverses............................. 3
More informationFirst Semester Abstract Algebra for Undergraduates
First Semester Abstract Algebra for Undergraduates Lecture notes by: Khim R Shrestha, Ph. D. Assistant Professor of Mathematics University of Great Falls Great Falls, Montana Contents 1 Introduction to
More informationPrime Rational Functions and Integral Polynomials. Jesse Larone, Bachelor of Science. Mathematics and Statistics
Prime Rational Functions and Integral Polynomials Jesse Larone, Bachelor of Science Mathematics and Statistics Submitted in partial fulfillment of the requirements for the degree of Master of Science Faculty
More informationMT5836 Galois Theory MRQ
MT5836 Galois Theory MRQ May 3, 2017 Contents Introduction 3 Structure of the lecture course............................... 4 Recommended texts..................................... 4 1 Rings, Fields and
More informationI216e Discrete Math (for Review)
I216e Discrete Math (for Review) Nov 22nd, 2017 To check your understanding. Proofs of do not appear in the exam. 1 Monoid Let (G, ) be a monoid. Proposition 1 Uniquness of Identity An idenity e is unique,
More informationA. (Groups of order 8.) (a) Which of the five groups G (as specified in the question) have the following property: G has a normal subgroup N such that
MATH 402A  Solutions for the suggested problems. A. (Groups of order 8. (a Which of the five groups G (as specified in the question have the following property: G has a normal subgroup N such that N =
More informationNote that a unit is unique: 1 = 11 = 1. Examples: Nonnegative integers under addition; all integers under multiplication.
Algebra fact sheet An algebraic structure (such as group, ring, field, etc.) is a set with some operations and distinguished elements (such as 0, 1) satisfying some axioms. This is a fact sheet with definitions
More informationMay 6, Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem. Show all of your work.
Math 236H May 6, 2008 Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem. Show all of your work. 1. (15 points) Prove that the symmetric group S 4 is generated
More informationPractice problems for first midterm, Spring 98
Practice problems for first midterm, Spring 98 midterm to be held Wednesday, February 25, 1998, in class Dave Bayer, Modern Algebra All rings are assumed to be commutative with identity, as in our text.
More informationBasic Definitions: Group, subgroup, order of a group, order of an element, Abelian, center, centralizer, identity, inverse, closed.
Math 546 Review Exam 2 NOTE: An (*) at the end of a line indicates that you will not be asked for the proof of that specific item on the exam But you should still understand the idea and be able to apply
More informationEuler s, Fermat s and Wilson s Theorems
Euler s, Fermat s and Wilson s Theorems R. C. Daileda February 17, 2018 1 Euler s Theorem Consider the following example. Example 1. Find the remainder when 3 103 is divided by 14. We begin by computing
More informationRINGS: SUMMARY OF MATERIAL
RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 1113 of Artin. Definitions not included here may be considered
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
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 informationModular Arithmetic and Elementary Algebra
18.310 lecture notes September 2, 2013 Modular Arithmetic and Elementary Algebra Lecturer: Michel Goemans These notes cover basic notions in algebra which will be needed for discussing several topics of
More informationApplied Cryptography and Computer Security CSE 664 Spring 2018
Applied Cryptography and Computer Security Lecture 12: Introduction to Number Theory II Department of Computer Science and Engineering University at Buffalo 1 Lecture Outline This time we ll finish the
More informationGroup Theory. 1. Show that Φ maps a conjugacy class of G into a conjugacy class of G.
Group Theory Jan 2012 #6 Prove that if G is a nonabelian group, then G/Z(G) is not cyclic. Aug 2011 #9 (Jan 2010 #5) Prove that any group of order p 2 is an abelian group. Jan 2012 #7 G is nonabelian nite
More informationCongruences and Residue Class Rings
Congruences and Residue Class Rings (Chapter 2 of J. A. Buchmann, Introduction to Cryptography, 2nd Ed., 2004) Shoichi Hirose Faculty of Engineering, University of Fukui S. Hirose (U. Fukui) Congruences
More informationIdeals: Definitions & Examples
Ideals: Definitions & Examples Defn: An ideal I of a commutative ring R is a subset of R such that for a, b I and r R we have a + b, a b, ra I Examples: All ideals of Z have form nz = (n) = {..., n, 0,
More informationDownloaded from
Question 1: Exercise 2.1 The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (i) (ii) (iii) Page 1 of 24 (iv) (v) (v) Page
More informationALGEBRA QUALIFYING EXAM SPRING 2012
ALGEBRA QUALIFYING EXAM SPRING 2012 Work all of the problems. Justify the statements in your solutions by reference to specific results, as appropriate. Partial credit is awarded for partial solutions.
More informationKevin James. MTHSC 412 Section 3.4 Cyclic Groups
MTHSC 412 Section 3.4 Cyclic Groups Definition If G is a cyclic group and G =< a > then a is a generator of G. Definition If G is a cyclic group and G =< a > then a is a generator of G. Example 1 Z is
More informationSection 15 Factorgroup computation and simple groups
Section 15 Factorgroup computation and simple groups Instructor: Yifan Yang Fall 2006 Outline Factorgroup computation Simple groups The problem Problem Given a factor group G/H, find an isomorphic group
More informationENTRY GROUP THEORY. [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld.
ENTRY GROUP THEORY [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld Group theory [Group theory] is studies algebraic objects called groups.
More informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
More informationPage Points Possible Points. Total 200
Instructions: 1. The point value of each exercise occurs adjacent to the problem. 2. No books or notes or calculators are allowed. Page Points Possible Points 2 20 3 20 4 18 5 18 6 24 7 18 8 24 9 20 10
More informationPRACTICE FINAL MATH , MIT, SPRING 13. You have three hours. This test is closed book, closed notes, no calculators.
PRACTICE FINAL MATH 18.703, MIT, SPRING 13 You have three hours. This test is closed book, closed notes, no calculators. There are 11 problems, and the total number of points is 180. Show all your work.
More informationWe begin with some definitions which apply to sets in general, not just groups.
Chapter 8 Cosets In this chapter, we develop new tools which will allow us to extend to every finite group some of the results we already know for cyclic groups. More specifically, we will be able to generalize
More informationAbstract Algebra: Chapters 16 and 17
Study polynomials, their factorization, and the construction of fields. Chapter 16 Polynomial Rings Notation Let R be a commutative ring. The ring of polynomials over R in the indeterminate x is the set
More informationPh.D. Qualifying Examination in Algebra Department of Mathematics University of Louisville January 2018
Ph.D. Qualifying Examination in Algebra Department of Mathematics University of Louisville January 2018 Do 6 problems with at least 2 in each section. Group theory problems: (1) Suppose G is a group. The
More informationPart II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
More informationMath 4400 First Midterm Examination September 21, 2012 ANSWER KEY. Please indicate your reasoning and show all work on this exam paper.
Name: Math 4400 First Midterm Examination September 21, 2012 ANSWER KEY Please indicate your reasoning and show all work on this exam paper. Relax and good luck! Problem Points Score 1 20 20 2 20 20 3
More informationφ(a + b) = φ(a) + φ(b) φ(a b) = φ(a) φ(b),
16. Ring Homomorphisms and Ideals efinition 16.1. Let φ: R S be a function between two rings. We say that φ is a ring homomorphism if for every a and b R, and in addition φ(1) = 1. φ(a + b) = φ(a) + φ(b)
More informationMATH HL OPTION  REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis
MATH HL OPTION  REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis PART B: GROUPS GROUPS 1. ab The binary operation a * b is defined by a * b = a+ b +. (a) Prove that * is associative.
More informationDiscrete Logarithms. Let s begin by recalling the definitions and a theorem. Let m be a given modulus. Then the finite set
Discrete Logarithms Let s begin by recalling the definitions and a theorem. Let m be a given modulus. Then the finite set Z/mZ = {[0], [1],..., [m 1]} = {0, 1,..., m 1} of residue classes modulo m is called
More informationELG 5372 Error Control Coding. Lecture 12: Ideals in Rings and Algebraic Description of Cyclic Codes
ELG 5372 Error Control Coding Lecture 12: Ideals in Rings and Algebraic Description of Cyclic Codes Quotient Ring Example + Quotient Ring Example Quotient Ring Recall the quotient ring R={,,, }, where
More informationCHAPTEER  TWO SUBGROUPS. ( Z, + ) is subgroup of ( R, + ). 1) Find all subgroups of the group ( Z 8, + 8 ).
CHAPTEER  TWO SUBGROUPS Definition 21. Let (G, ) be a group and H G be a nonempty subset of G. The pair ( H, ) is said to be a SUBGROUP of (G, ) if ( H, ) is group. Example. ( Z, + ) is subgroup of (
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationMATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:0010:00 PM
MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:0010:00 PM Basic Questions 1. Compute the factor group Z 3 Z 9 / (1, 6). The subgroup generated by (1, 6) is
More informationIntroduction to Information Security
Introduction to Information Security Lecture 5: Number Theory 007. 6. Prof. Byoungcheon Lee sultan (at) joongbu. ac. kr Information and Communications University Contents 1. Number Theory Divisibility
More information