DOCSLIB.ORG

On the Algebraic Structure of Quasi Group Codes 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Algebraic Structure of Quasi Group Codes 1 ON THE ALGEBRAIC STRUCTURE OF QUASI GROUP CODES MARTINO BORELLO AND WOLFGANG WILLEMS Abstract. In this note, an intrinsic description of some families of linear codes with symmetries is given, showing that they can be described more generally as quasi group codes, that is as linear codes with a free group of permutation automorphisms. An algebraic description, including the concatenated structure, of such codes is presented. Finally, self-duality of quasi group codes is investigated. 1. Introduction In the theory of error correcting codes, the linear ones play a central role for their algebraic properties which allow, for example, their easy description and storage. Quite early in their study, it appeared convenient to add additional algebraic structure in order to get more information about the parameters and to facilitate the decoding process. In 1957, E. Prange introduced the now well- known class of cyclic codes [18], which are the forefathers of many other families of codes with symmetries introduced later. In particular, abelian codes [2], group codes [15], quasi-cyclic codes [9] and quasi-abelian codes [19] are distinguished descendants of cyclic codes: they have a nice algebraic structure and many optimal codes belong to these families. The aim of the current note is to give an intrinsic description of these classes in terms of their permutation automorphism groups: we will show that all of them have a free group of permutation automorphisms (which is not necessarily the full permutation automorphism group of the code). We will define a code with this property as a quasi group code. So all families cited above can be described in this way. Note that quasi group codes (over Frobenius rings) has been introduced also in a recent paper by S. Dougherty et al. [10]. As it turns out very easily from our description, the families of quasi group codes, quasi-abelian codes and quasi-cyclic codes coincide. Anyway, we give the most general algebraic description (Theorem 4.1) for quasi group codes describing their concatenated structure (Theorem 4.2) without any restriction on the group acting on. Also no assumption on semi-simplicity (unlike in previous papers on the subject) is needed. In the last part of the paper, we deal with self-duality of quasi group codes, proving a necessary and sufficient condition on their underlying field and index for their existence. 2. Background We collect in this section preliminaries which are crucial in the subsequent sections. Let K be a finite field of cardinality q. A linear code C of length n is a subspace of Kn. An element c = (c1,...,cn)∈C is called a codeword and its (Hamming) weight is given by wt(c) ∶= S{i ∈ {1,...,n} S ci ≠ 0}S The minimum distance of C is defined by d(C) ∶= minc∈C∖{0} wt(c). An n,k,d q code is a linear code of length n, dimension k and minimum distance d over a field of[ cardinality] q. These are usually called the parameters of the code. M. Borello is with LAGA, UMR 7539, CNRS, Universit´eParis 13 - Sorbonne Paris Cit´e, Universit´eParis 8, F-93526, Saint-Denis, France. W. Willems is with Otto-von-Guericke Universit¨at, Magdeburg, Germany and Universidad del Norte, Barranquilla, Colombia. 1 2 MARTINOBORELLOANDWOLFGANGWILLEMS There is a standard way of combining codes to obtain a code of larger length. To describe this process, let K ⊆ L be a field extension, m be an integer greater than or equal to L ∶ K and π ∶ L → Km be a K-linear injection. In the concatenation process, L-linear codes in L[n are called] outer codes and the K-linear code I ∶= π L is called inner code. The K-linear code ( ) n⋅m I ◻π C ∶= π c1 ,...,π cn c1,...,cn ∈C ⊆ K {( ( ) ( )) S ( ) } is the concatenation of C with I by π, or simply the concatenated code. This construction obviously depends on the choice of π, but there are some properties that do not depend on π. For example, the length of I ◻π C is n ⋅ m, the dimension over K of I ◻π C is the product dimK I ⋅dimL C and we have the bound d I ◻π C ≥ d I ⋅d C (see for example [11]). ( ) ( ) ( ) ( ) ( ) There are some natural group actions associated to linear codes. The symmetric group Sn acts on the coordinates 1,...,n by definition. This action induces an action on the elements of Kn, n { } namely for v ∈ K and σ ∈ Sn we have σ v ∶= vσ−1 1 ,vσ−1 2 ,...,vσ−1 n . ( ( ) ( ) ( )) This action induces an action on subsets (and in particular on linear codes). For C ⊆ Kn we put Cσ ∶= cσ c ∈C . { S } σ An element σ ∈ Sn is an automorphism of C if C =C. The stabilizer σ PAut C ∶= σ ∈ Sn C =C ( ) { S } is called the permutation automorphism group of C. Moreover, a linear code C1 is equivalent or ( ) better permutation equivalent to a linear code C2 (and we denote that by C1 C2) if there exists σ ∼ σ Sn such that C1 =C2. This is not the most general definition of equivalence but it is sufficient for∈ the purpose of this note. It is easy to see that PAut Cσ = PAut C σ (the conjugate of PAut C ( ) ( ) ( ) in Sn by σ). From group theory we recall that a (right) action of a group G on a set X is called ● transitive if X ≠ ∅ and for all x,y ∈ X there exists g ∈ G such that xg = y; ● free if xg = x for some x ∈ X and g ∈ G, then g is the unit in G; ● regular if it is both transitive and free. In the case that the group G is finite and acts freely, it immediately follows from the definition that all orbits of G have cardinality G . In particular, G divides X . Moreover, if G is regular, then G = X . S S S S S S S S S S A subgroup H of the symmetric group Sn is called transitive (free, resp. regular) if the restriction to H of the action of Sn on the coordinates 1,...,n is transitive (free, resp. regular). { } Finally, given a group G and a field K, the group algebra KG is the set of formal sums ⎧ R ⎫ KG ∶= ⎪a = Q agg R ag ∈ K⎪ , ⎨ R ⎬ ⎪ g∈G R ⎪ ⎩⎪ R ⎭⎪ which is a K-vector space in a natural way and which becomes a K-algebra via the multiplication ab ∶= Q Q ahbh−1g g, g∈G h∈G for a = Q agg and b = Q bgg. g∈G g∈G ONTHEALGEBRAICSTRUCTUREOFQUASIGROUPCODES 3 3. Codes with a free group of symmetries Let G be a finite group of cardinality n and let K be a finite field. Recall that the group algebra KG is isomorphic to Kn as a K-vector space, where n = G . There is a standard way of constructing such an isomorphism, which allows us to transfer manyS codingS theoretical properties from Kn to KG. Once an ordering g1,...,gn of the elements of G is chosen, we may define ϕ ∶ gi ↦ ei, where n e1,...,en is the standard basis of K . Then we extend this map K-linearly so that { } n (1) ϕ ∶ Q aigi ↦ a1,...,an . i=1 ( ) The isomorphism ϕ obtained is not canonical, since it depends on the ordering of the group, but different orderings lead only to a permutation of the coordinates. Via the isomorphism ϕ, we can transfer the Hamming metric from Kn to KG. For a ∈ KG, we define wt a ∶= wt ϕ a . So, from a coding theoretical point of view, we can consider linear codes either in (KG) or in( K( n))without any difference. However, the algebraic structure of KG allows us to consider codes with more structure than linearity. Definition 3.1 ([2, 15]). A G-code (or a group code) is a right ideal C in the group algebra KG. If the group G is abelian, then the code C is called an abelian code. A particular and well-known class of abelian codes is the family of cyclic codes. If G is cyclic, i hence generated by a certain g ∈ G, and the isomorphism ϕ sends g → ei+1 for all i ∈ 0,...,n − 1 , then ϕ C is cyclic. In fact, in this case KG ≅ K x xn − 1 via the isomorphism {g ↦ x. Thus,} a cyclic( code) turns out to be an ideal in the factor[ ]~( algebra K) x xn − 1 , which is the classical definition. [ ]~( ) Now let C be a G-code. Observe that on G the right multiplication by one of its elements, say g, induces a permutation σg ∈ Sn defined by (2) σg i = j iff gig = gj. ( ) Note that g ↦ σg is a faithful permutation representation of G, which again depends on the chosen ordering of G. If this last map coincides with that chosen for ϕ, then σ G ∶= σg g ∈ G is a subgroup of PAut ϕ C . This is due to the fact that a right ideal is stable( ) by multipli{ S cation} on the right. Since the( action( )) of right multiplication is regular, the group σ G is regular. ( ) Suppose that C is a linear code in Kn admitting a regular subgroup G of PAut C . Since G is a group of automorphisms, C becomes a right KG-module via the action ( ) g (3) c ⋅ ⎛ Q agg⎞ ∶= Q agc ∈ ∈ ⎝g G ⎠ g G for c ∈C and ag ∈ K. Moreover, as every regular action of G is isomorphic to the action of G on itself given by right multiplication, there is an ordering of G such that ϕ−1 C is a G-code in KG.
Load more

Recommended publications
  • SOME ALGEBRAIC DEFINITIONS and CONSTRUCTIONS Definition
    SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS Definition 1. A monoid is a set M with an element e and an associative multipli- cation M M M for which e is a two-sided identity element: em = m = me for all m M×. A−→group is a monoid in which each element m has an inverse element m−1, so∈ that mm−1 = e = m−1m. A homomorphism f : M N of monoids is a function f such that f(mn) = −→ f(m)f(n) and f(eM )= eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n M, we often write the product as +, the identity element as 0, and the inverse of∈m as m. As a convention, it is convenient to say that a commutative monoid is “Abelian”− when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1. Definition 2. The Grothendieck construction on an Abelian monoid is an Abelian group G(M) together with a homomorphism of Abelian monoids i : M G(M) such that, for any Abelian group A and homomorphism of Abelian monoids−→ f : M A, there exists a unique homomorphism of Abelian groups f˜ : G(M) A −→ −→ such that f˜ i = f. ◦ We construct G(M) explicitly by taking equivalence classes of ordered pairs (m,n) of elements of M, thought of as “m n”, under the equivalence relation generated by (m,n) (m′,n′) if m + n′ = −n + m′.
    [Show full text]
  • A Review of Commutative Ring Theory Mathematics Undergraduate Seminar: Toric Varieties
    A REVIEW OF COMMUTATIVE RING THEORY MATHEMATICS UNDERGRADUATE SEMINAR: TORIC VARIETIES ADRIANO FERNANDES Contents 1. Basic Definitions and Examples 1 2. Ideals and Quotient Rings 3 3. Properties and Types of Ideals 5 4. C-algebras 7 References 7 1. Basic Definitions and Examples In this first section, I define a ring and give some relevant examples of rings we have encountered before (and might have not thought of as abstract algebraic structures.) I will not cover many of the intermediate structures arising between rings and fields (e.g. integral domains, unique factorization domains, etc.) The interested reader is referred to Dummit and Foote. Definition 1.1 (Rings). The algebraic structure “ring” R is a set with two binary opera- tions + and , respectively named addition and multiplication, satisfying · (R, +) is an abelian group (i.e. a group with commutative addition), • is associative (i.e. a, b, c R, (a b) c = a (b c)) , • and the distributive8 law holds2 (i.e.· a,· b, c ·R, (·a + b) c = a c + b c, a (b + c)= • a b + a c.) 8 2 · · · · · · Moreover, the ring is commutative if multiplication is commutative. The ring has an identity, conventionally denoted 1, if there exists an element 1 R s.t. a R, 1 a = a 1=a. 2 8 2 · ·From now on, all rings considered will be commutative rings (after all, this is a review of commutative ring theory...) Since we will be talking substantially about the complex field C, let us recall the definition of such structure. Definition 1.2 (Fields).
    [Show full text]
  • Algebraic Structures Lecture 18 Thursday, April 4, 2019 1 Type
    Harvard School of Engineering and Applied Sciences — CS 152: Programming Languages Algebraic structures Lecture 18 Thursday, April 4, 2019 In abstract algebra, algebraic structures are defined by a set of elements and operations on those ele- ments that satisfy certain laws. Some of these algebraic structures have interesting and useful computa- tional interpretations. In this lecture we will consider several algebraic structures (monoids, functors, and monads), and consider the computational patterns that these algebraic structures capture. We will look at Haskell, a functional programming language named after Haskell Curry, which provides support for defin- ing and using such algebraic structures. Indeed, monads are central to practical programming in Haskell. First, however, we consider type constructors, and see two new type constructors. 1 Type constructors A type constructor allows us to create new types from existing types. We have already seen several different type constructors, including product types, sum types, reference types, and parametric types. The product type constructor × takes existing types τ1 and τ2 and constructs the product type τ1 × τ2 from them. Similarly, the sum type constructor + takes existing types τ1 and τ2 and constructs the product type τ1 + τ2 from them. We will briefly introduce list types and option types as more examples of type constructors. 1.1 Lists A list type τ list is the type of lists with elements of type τ. We write [] for the empty list, and v1 :: v2 for the list that contains value v1 as the first element, and v2 is the rest of the list. We also provide a way to check whether a list is empty (isempty? e) and to get the head and the tail of a list (head e and tail e).
    [Show full text]
  • Semilattice Sums of Algebras and Mal'tsev Products of Varieties
    Mathematics Publications Mathematics 5-20-2020 Semilattice sums of algebras and Mal’tsev products of varieties Clifford Bergman Iowa State University, [email protected] T. Penza Warsaw University of Technology A. B. Romanowska Warsaw University of Technology Follow this and additional works at: https://lib.dr.iastate.edu/math_pubs Part of the Algebra Commons The complete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ math_pubs/215. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Mathematics at Iowa State University Digital Repository. It has been accepted for inclusion in Mathematics Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Semilattice sums of algebras and Mal’tsev products of varieties Abstract The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider when this quasivariety is a variety. The main result shows that if V is a strongly irregular variety with no nullary operations, and S is a variety, of the same type as V, equivalent to the variety of semilattices, then the Mal’tsev product V ◦ S is a variety. It consists precisely of semilattice sums of algebras in V. We derive an equational basis for the product from an equational basis for V. However, if V is a regular variety, then the Mal’tsev product may not be a variety. We discuss examples of various applications of the main result, and examine some detailed representations of algebras in V ◦ S.
    [Show full text]
  • Problems and Comments on Boolean Algebras Rosen, Fifth Edition: Chapter 10; Sixth Edition: Chapter 11 Boolean Functions
    Problems and Comments on Boolean Algebras Rosen, Fifth Edition: Chapter 10; Sixth Edition: Chapter 11 Boolean Functions Section 10. 1, Problems: 1, 2, 3, 4, 10, 11, 29, 36, 37 (fifth edition); Section 11.1, Problems: 1, 2, 5, 6, 12, 13, 31, 40, 41 (sixth edition) The notation ""forOR is bad and misleading. Just think that in the context of boolean functions, the author uses instead of ∨.The integers modulo 2, that is ℤ2 0,1, have an addition where 1 1 0 while 1 ∨ 1 1. AsetA is partially ordered by a binary relation ≤, if this relation is reflexive, that is a ≤ a holds for every element a ∈ S,it is transitive, that is if a ≤ b and b ≤ c hold for elements a,b,c ∈ S, then one also has that a ≤ c, and ≤ is anti-symmetric, that is a ≤ b and b ≤ a can hold for elements a,b ∈ S only if a b. The subsets of any set S are partially ordered by set inclusion. that is the power set PS,⊆ is a partially ordered set. A partial ordering on S is a total ordering if for any two elements a,b of S one has that a ≤ b or b ≤ a. The natural numbers ℕ,≤ with their ordinary ordering are totally ordered. A bounded lattice L is a partially ordered set where every finite subset has a least upper bound and a greatest lower bound.The least upper bound of the empty subset is defined as 0, it is the smallest element of L.
    [Show full text]
  • N-Algebraic Structures and S-N-Algebraic Structures
    N-ALGEBRAIC STRUCTURES AND S-N-ALGEBRAIC STRUCTURES W. B. Vasantha Kandasamy Florentin Smarandache 2005 1 N-ALGEBRAIC STRUCTURES AND S-N-ALGEBRAIC STRUCTURES W. B. Vasantha Kandasamy e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache e-mail: [email protected] 2005 2 CONTENTS Preface 5 Chapter One INTRODUCTORY CONCEPTS 1.1 Group, Smarandache semigroup and its basic properties 7 1.2 Loops, Smarandache Loops and their basic properties 13 1.3 Groupoids and Smarandache Groupoids 23 Chapter Two N-GROUPS AND SMARANDACHE N-GROUPS 2.1 Basic Definition of N-groups and their properties 31 2.2 Smarandache N-groups and some of their properties 50 Chapter Three N-LOOPS AND SMARANDACHE N-LOOPS 3.1 Definition of N-loops and their properties 63 3.2 Smarandache N-loops and their properties 74 Chapter Four N-GROUPOIDS AND SMARANDACHE N-GROUPOIDS 4.1 Introduction to bigroupoids and Smarandache bigroupoids 83 3 4.2 N-groupoids and their properties 90 4.3 Smarandache N-groupoid 99 4.4 Application of N-groupoids and S-N-groupoids 104 Chapter Five MIXED N-ALGEBRAIC STRUCTURES 5.1 N-group semigroup algebraic structure 107 5.2 N-loop-groupoids and their properties 134 5.3 N-group loop semigroup groupoid (glsg) algebraic structures 163 Chapter Six PROBLEMS 185 FURTHER READING 191 INDEX 195 ABOUT THE AUTHORS 209 4 PREFACE In this book, for the first time we introduce the notions of N- groups, N-semigroups, N-loops and N-groupoids. We also define a mixed N-algebraic structure.
    [Show full text]
  • An Introduction to the Theory of Lattice Ý Jinfang Wang £
    An Introduction to the Theory of Lattice Ý Jinfang Wang £ Graduate School of Science and Technology, Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan May 11, 2006 £ Fax: 81-43-290-3663. E-Mail addresses: [email protected] Ý Partially supported by Grand-in-Aid 15500179 of the Japanese Ministry of Education, Science, Sports and Cul- ture. 1 1 Introduction A lattice1 is a partially ordered set (or poset), in which all nonempty finite subsets have both a supremum (join) and an infimum (meet). Lattices can also be characterized as algebraic structures that satisfy certain identities. Since both views can be used interchangeably, lattice theory can draw upon applications and methods both from order theory and from universal algebra. Lattices constitute one of the most prominent representatives of a series of “lattice-like” structures which admit order-theoretic as well as algebraic descriptions, such as semilattices, Heyting algebras, and Boolean algebras. 2 Semilattice A semilattice is a partially ordered set within which either all binary sets have a supremum (join) or all binary sets have an infimum (meet). Consequently, one speaks of either a join-semilattice or a meet-semilattice. Semilattices may be regarded as a generalization of the more prominent concept of a lattice. In the literature, join-semilattices sometimes are sometimes additionally required to have a least element (the join of the empty set). Dually, meet-semilattices may include a greatest element. 2.1 Definitions Semilattices as posets Ë µ Ü DEFINITION 2.1 (MEET-SEMILATTICE). A poset ´ is a meet-semilattice if for all elements Ë Ü Ý and Ý of , the greatest lower bound (meet) of the set exists.
    [Show full text]
  • How Structure Sense for Algebraic Expressions Or Equations Is Related to Structure Sense for Abstract Algebra
    Mathematics Education Research Journal 2008, Vol. 20, No. 2, 93-104 How Structure Sense for Algebraic Expressions or Equations is Related to Structure Sense for Abstract Algebra Jarmila Novotná Maureen Hoch Charles University, Czech Republic Tel Aviv University, Israel Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense components, and that several components of university algebra structure sense are analogies of high school algebra structure sense components. We present a theoretical argument for these hypotheses, with some examples. We recommend emphasizing structure sense in high school algebra in the hope of easing students’ paths in university algebra. The cooperation of the authors in the domain of structure sense originated at a scientific conference where they each presented the results of their research in their own countries: Israel and the Czech Republic. Their findings clearly show that they are dealing with similar situations, concepts, obstacles, and so on, at two different levels—high school and university. In their classrooms, high school teachers are dismayed by students’ inability to apply basic algebraic techniques in contexts different from those they have experienced. Many students who arrive in high school with excellent grades in mathematics from the junior-high school prove to be poor at algebraic manipulations. Even students who succeed well in 10th grade algebra show disappointing results later on, because of the algebra.
    [Show full text]
  • Arxiv:2001.00808V2
    LATTICES, SPECTRAL SPACES, AND CLOSURE OPERATIONS ON IDEMPOTENT SEMIRINGS JAIUNG JUN, SAMARPITA RAY, AND JEFFREY TOLLIVER Abstract. Spectral spaces, introduced by Hochster, are topological spaces homeomorphic to the prime spectra of commutative rings. In this paper we study spectral spaces in perspective of idempo- tent semirings which are algebraic structures receiving a lot of attention due to its several applications to tropical geometry. We first prove that a space is spectral if and only if it is the saturated prime spec- trum of an idempotent semiring. In fact, we enrich Hochster’s theorem by constructing a subcategory of idempotent semirings which is equivalent to the category of spectral spaces. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the space of valuations and the space of prime congruences on an idempotent semiring are spectral, and there is a natural bijection of sets between two; this shows a stark difference between rings and idempotent semirings. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of closure operations for semirings, and provide several examples. In particular, we introduce an integral closure operation and a Frobenius closure operation for idempotent semirings. 1. Introduction A semiring is an algebraic structure which assumes the same axioms as a ring except that one does not necessarily require additive inverses to exist. Typical examples include the semiring N of natural numbers, the Boolean semiring B, or the tropical semiring T. The theory of semirings has its own charms, but also recently people have found several applications of semirings, making the theory of semirings even more interesting.
    [Show full text]
  • Algebraic Structure
    Algebraic Structure Groups: A group (G) is a set of elements with binary operations “•” denoted as G = < {}, • > that satisfies four following properties: Closure If a and b are elements of G then c = a • b is also an element of G. this means that the result of applying the operation on any two elements is the set is another element in the set. Associativity If a, b and c are the elements of G, then (a•b)•c = a•(b•c). In other words, it does not matter in which order we apply the operation on more than two elements. Existence of Identity For all a in G, there exists an element e, called the identity element such that e•a = a•e = a. Existence of Inverse For each element a on G, there exists an element a’, called the inverse of a, such that a•a’ = a’•a = e. Abelian Group (Commutative Group): A group is called Abelian Group or Commutative Group if the operator satisfies the four properties for groups plus an extra property, commutative. Commutativity For all a and b in G, we have a•b=b•a. note that this property needs to be satisfied only for Abelian group. As an example The set of residue integers with the addition operator, G = <Zn, + > is a commutative group. Let’s check the properties. Santaseel Chakraborty Page 1 Closure: The result of adding two integers in Zn is another integer in Zn. Associativity: The result of (3+2)+4 = 3+(2+4). Commutativity: The result of 3+5 = 5+3.
    [Show full text]
  • Notes on Algebraic Structures
    Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the second-year course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester 2005–2006. After a short introductory chapter consisting mainly of reminders about such topics as functions, equivalence relations, matrices, polynomials and permuta- tions, the notes fall into two chapters, dealing with rings and groups respec- tively. I have chosen this order because everybody is familiar with the ring of integers and can appreciate what we are trying to do when we generalise its prop- erties; there is no well-known group to play the same role. Fairly large parts of the two chapters (subrings/subgroups, homomorphisms, ideals/normal subgroups, Isomorphism Theorems) run parallel to each other, so the results on groups serve as revision for the results on rings. Towards the end, the two topics diverge. In ring theory, we study factorisation in integral domains, and apply it to the con- struction of fields; in group theory we prove Cayley’s Theorem and look at some small groups. The set text for the course is my own book Introduction to Algebra, Ox- ford University Press. I have refrained from reading the book while teaching the course, preferring to have another go at writing out this material. According to the learning outcomes for the course, a studing passing the course is expected to be able to do the following: • Give the following. Definitions of binary operations, associative, commuta- tive, identity element, inverses, cancellation. Proofs of uniqueness of iden- tity element, and of inverse.
    [Show full text]
  • Mathematics of Cryptography Algebraic Structures ALGEBRAIC STRUCTURES
    Mathematics of Cryptography Algebraic Structures ALGEBRAIC STRUCTURES Cryptography requires sets of integers and specific operations that are defined for those sets. The combination of the set and the operations that are applied to the elements of the set is called an algebraic structure. We will define three common algebraic structures: groups, rings, and fields. 1 Groups 2 Rings 3 Fields Groups A group (G) is a set of elements with a binary operation (•) that satisfies four properties (or axioms). A commutative group satisfies an extra property, commutativity: ❏ Closure: If a and b are elements of G, then c=a•b is also an element of G. ❏ Associativity: a•(b•c)=(a•b)•c ❏ Commutativity: a•b=b•a ❏ Existence of identity: For all element a in G, there exists and element e, identity element, s. t. e•a=a•e=a ❏ Existence of inverse: for each a there exists an a’, inverse of a, s.t. a•a’=a’•a=e. Figure Group Application Although a group involves a single operation, the properties imposed on the operation allow the use of a pair of operations as long as they are inverses of each other. Example The set of residue integers with the addition operator, G = < Zn , +>, is a commutative group. We can perform addition and subtraction on the elements of this set without moving out of the set. Example The set Zn* with the multiplication operator, G = <Zn*, ×>, is also an abelian group. Identity element is 1. Example Let us define a set G = < {a, b, c, d}, •> and the operation as shown below.
    [Show full text]