(Ebook) Elements of quasigroup theory with applications in coding and cryptology 1st Edition by Victor Shcherbacov ISBN 1498721559 9781498721554

(Ebook) Elements of quasigroup theory with applications in coding and cryptology 1st Edition by Victor Shcherbacov ISBN 1498721559 9781498721554

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ISBN 10: 1498721559 
ISBN 13: 9781498721554
Author: Victor Shcherbacov

Understanding Interaction is a book that explores the interaction between people and technology, in the broader context of the relations between the human made and the natural environments. It is not just about digital technologies – our computers, smart phones, the Internet – but all our technologies such as mechanical, electrical and electronic. Our ancestors started creating mechanical tools and shaping their environments millions of years ago, developing cultures and languages, which in turn influenced our evolution. Volume 1 of Understanding Interaction looks into this deep history – starting from the tool creating period (the longest and most influential on our physical and mental capacities), to the settlement period (agriculture, domestication, villages and cities, written language), the industrial period (science, engineering, reformation and renaissance), and finally the communication period (mass media, digital technologies, global networks). Volume 2 looks into humans in interaction – our physiology, anatomy, neurology, psychology, how we experience and influence the world, and how we (think we) think. From this transdisciplinary understanding, design approaches and frameworks are presented, to potentially guide future developments and innovations. The aim of the book is to be guide and inspiration for designers, artists, engineers, psychologists, media producers, social scientists etc., and as such be useful for both novices and more experienced practitioners.

(Ebook) Elements of quasigroup theory with applications in coding and cryptology 1st Table of contents:

I: Foundations
1: Elements of quasigroup theory
1.1 Introduction
1.1.1 The role of definitions
1.1.2 Sets
1.1.3 Products and partitions
1.1.4 Maps
1.2 Objects
1.2.1 Groupoids and quasigroups
1.2.2 Parastrophy: Quasigroup as an algebra
1.2.2.1 Parastrophy
1.2.2.2 Middle translations
1.2.2.3 Some groupoids
1.2.2.4 Substitutions in groupoid identities
1.2.2.5 Equational definitions
1.2.3 Some other definitions of e-quasigroups
1.2.4 Quasigroup-based cryptosystem
1.2.5 Identity elements
1.2.5.1 Local identity elements
1.2.5.2 Left and right identity elements
1.2.5.3 Loops
1.2.5.4 Identity elements of quasigroup parastrophes
1.2.5.5 The equivalence of loop definitions
1.2.5.6 Identity elements in some quasigroups
1.2.5.7 Inverse elements in loops
1.2.6 Multiplication groups of quasigroups
1.2.7 Transversals: “Come back way”
1.2.8 Generators of inner multiplication groups
1.3 Morphisms
1.3.1 Isotopism
1.3.2 Group action
1.3.3 Isotopism: Another point of view
1.3.4 Autotopisms of binary quasigroups
1.3.5 Automorphisms of quasigroups
1.3.6 Pseudo-automorphisms and G-loops
1.3.7 Parastrophisms as operators
1.3.8 Isostrophism
1.3.9 Autostrophisms
1.3.9.1 Coincidence of quasigroup parastrophes
1.3.10 Inverse loops to a fixed loop
1.3.11 Anti-autotopy
1.3.12 Translations of isotopic quasigroups
1.4 Sub-objects
1.4.1 Subquasigroups: Nuclei and center
1.4.1.1 Sub-objects
1.4.1.2 Nuclei
1.4.1.3 Center
1.4.2 Bol and Moufang nuclei
1.4.3 The coincidence of loop nuclei
1.4.3.1 Nuclei coincidence and identities
1.4.4 Quasigroup nuclei and center
1.4.4.1 Historical notes
1.4.4.2 Quasigroup nuclei
1.4.4.3 Quasigroup center
1.4.5 Regular permutations
1.4.6 A-nuclei of quasigroups
1.4.7 A-pseudo-automorphisms by isostrophy
1.4.8 Commutators and associators
1.5 Congruences
1.5.1 Congruences of quasigroups
1.5.1.1 Congruences in universal algebra
1.5.1.2 Normal congruences
1.5.2 Quasigroup homomorphisms
1.5.3 Normal subquasigroups
1.5.4 Normal subloops
1.5.5 Antihomomorphisms and endomorphisms
1.5.6 Homotopism
1.5.7 Congruences and isotopism
1.5.8 Congruence permutability
1.6 Constructions
1.6.1 Direct product
1.6.2 Semidirect product
1.6.3 Crossed (quasi-direct) product
1.6.4 n-Ary crossed product
1.6.5 Generalized crossed product
1.6.6 Generalized singular direct product
1.6.7 Sabinin’s product
1.7 Quasigroups and combinatorics
1.7.1 Orthogonality
1.7.1.1 Orthogonality of binary operations
1.7.1.2 Orthogonality of n-ary operations
1.7.1.3 Easy way to construct n-ary orthogonal operations
1.7.2 Partial Latin squares: Latin trades
1.7.3 Critical sets of Latin squares, Sudoku
1.7.4 Transversals in Latin squares
1.7.5 Quasigroup prolongations: Combinatorial aspect
1.7.5.1 Bruck-Belousov prolongation
1.7.5.2 Belyavskaya prolongation
1.7.5.3 Algebraic approach
1.7.5.4 Prolongation using quasicomplete mappings
1.7.5.5 Two-step mixed procedure
1.7.5.6 Brualdi problem
1.7.5.7 Contractions of quasigroups
1.7.6 Orthomorphisms
1.7.7 Neo-fields and left neo-fields
1.7.8 Sign of translations
1.7.9 The number of quasigroups
1.7.10 Latin squares and graphs
1.7.11 Orthogonal arrays
2: Some quasigroup classes
2.1 Definitions of loop and quasigroup classes
2.1.1 Moufang loops, Bol loops, and generalizations
2.1.2 Some linear quasigroups
2.2 Classical inverse quasigroups
2.2.1 Definitions and properties
2.2.2 Autotopies of LIP- and IP-loops
2.2.3 Moufang and Bol elements in LIP-loops
2.2.4 Loops with the property Il = Ir
2.3 Medial quasigroups
2.3.1 Linear forms: Toyoda theorem
2.3.2 Direct decompositions: Murdoch theorem
2.3.3 Simple quasigroups
2.3.4 Examples
2.4 Paramedial quasigroups
2.4.1 Kepka-Nemec theorem
2.4.2 Antiendomorphisms
2.4.3 Direct decomposition
2.4.4 Simple paramedial quasigroups
2.4.5 Quasigroups of order 4
2.5 CMLs and their isotopes
2.5.1 CMLs
2.5.2 Distributive quasigroups
2.6 Left distributive quasigroups
2.6.1 Examples, constructions, orders
2.6.2 Properties, simple quasigroups, loop isotopes
2.7 TS-quasigroups
2.7.1 Constructions, loop isotopes
2.7.2 2-nilpotent TS-loops
2.7.3 Some properties of TS-quasigroups
2.8 Schröder quasigroups
2.9 Incidence systems and block designs
2.9.1 Introduction
2.9.2 3-nets and binary quasigroups
2.9.3 On orders of finite projective planes
2.9.4 Steiner systems
2.9.5 Mendelsohn design
2.9.6 Spectra of quasigroups with 2-variable identities
2.10 Linear quasigroups
2.10.1 Introduction
2.10.2 Definitions
2.10.3 Group isotopes and identities
2.10.4 Nuclei, identities
2.10.5 Parastrophes of linear quasigroups
2.10.6 On the forms of n-T-quasigroups
2.10.7 (m,n)-Linear quasigroups
2.11 Miscellaneous
2.11.1 Groups with triality
2.11.2 Universal properties of quasigroups
2.11.3 Alternative and various conjugate closed quasigroups and loops
3: Binary inverse quasigroups
3.1 Definitions
3.1.1 Definitions of “general” inverse quasigroups
3.2 (r,s,t)-Inverse quasigroups
3.2.1 Elementary properties and examples
3.2.2 Left-linear quasigroups which are (r,s,t)-inverse
3.2.3 Main theorems
3.2.4 Direct product of (r,s,t)-quasigroups
3.2.5 The existence of (r,s,t)-inverse quasigroups
3.2.6 WIP-quasigroups
3.2.7 Examples of WIP-quasigroups
3.2.8 Generalized balanced parastrophic identities
3.2.9 Historical notes
4: A-nuclei of quasigroups
4.1 Preliminaries
4.1.1 Isotopism
4.1.2 Quasigroup derivatives
4.1.2.1 G-quasigroups
4.1.2.2 Garrison’s nuclei in quasigroups
4.1.2.3 Mixed derivatives
4.1.3 Set of maps
4.2 Garrison’s nuclei and A-nuclei
4.2.1 Definitions of nuclei and A-nuclei
4.2.2 Components of A-nuclei and identity elements
4.2.3 A-nuclei of loops by isostrophy
4.2.4 Isomorphisms of A-nuclei
4.2.5 A-nuclei by some isotopisms
4.2.6 Quasigroup bundle and nuclei
4.2.7 A-nuclei actions
4.2.8 A-nuclear quasigroups
4.2.9 Identities with permutation and group isotopes
4.3 A-centers of a quasigroup
4.3.1 Normality of A-nuclei and autotopy group
4.3.2 A-centers of a loop
4.3.3 A-centers of a quasigroup
4.4 A-nuclei and quasigroup congruences
4.4.1 Normality of equivalences in quasigroups
4.4.2 Additional conditions of normality of equivalences
4.4.3 A-nuclei and quasigroup congruences
4.4.4 A-nuclei and loop congruences
4.4.5 On loops with nucleus of index two
4.5 Coincidence of A-nuclei in inverse quasigroups
4.5.1 (α, β,ϒ )-inverse quasigroups
4.5.2 λ-, ρ- and µ-inverse quasigroups
4.6 Relations between a loop and its inverses
4.6.1 Nuclei of inverse loops in Belousov sense
4.6.2 LIP- and AAIP-loops
4.6.3 Invariants of reciprocally inverse loops
4.6.3.1 Middle Bol loops
4.6.3.2 Some invariants
4.6.3.3 Term-equivalent loops
4.6.4 Nuclei of loops that are inverse to a fixed loop
II: Theory
5: On two Belousov problems
5.1 The existence of identity elements in quasigroups
5.1.1 On quasigroups with Moufang identities
5.1.2 Identities that define a CML
5.2 Bruck-Belousov problem
5.2.1 Introduction
5.2.2 Congruences of quasigroups
5.2.3 Congruences of inverse quasigroups
5.2.4 Behavior of congruences by an isotopy
5.2.5 Regularity of quasigroup congruences
6: Quasigroups which have an endomorphism
6.1 Introduction
6.1.1 Parastrophe invariants and isostrophisms
6.2 Left and right F-, E-, SM-quasigroups
6.2.1 Direct decompositions
6.2.2 F-quasigroups
6.2.3 E-quasigroups
6.2.4 SM-quasigroups
6.2.5 Finite simple quasigroups
6.2.6 Left FESM-quasigroups
6.2.7 CML as an SM-quasigroup
6.3 Loop isotopes
6.3.1 Left F-quasigroups
6.3.2 F-quasigroups
6.3.3 Left SM-quasigroups
6.3.4 Left E-quasigroups
7: Structure of n-ary medial quasigroups
7.1 On n-ary medial quasigroups
7.1.1 n-ary quasigroups: Isotopy and translations
7.1.2 Linear n-ary quasigroups
7.1.3 n-Ary medial quasigroups
7.1.4 Homomorphisms of n-ary quasigroups
7.1.5 Direct product of n-ary quasigroups
7.1.6 Multiplication group of n-ary T -quasigroup
7.1.7 Homomorphisms of n-ary linear quasigroups
7.1.8 n-Ary analog of Murdoch theorem
7.2 Properties of n-ary simple T -quasigroups
7.2.1 Simple n-ary quasigroups
7.2.2 Congruences of linear n-ary quasigroups
7.2.3 Simple n-T -quasigroups
7.2.4 Simple n-ary medial quasigroups
7.3 Solvability of finite n-ary medial quasigroups
8: Automorphisms of some quasigroups
8.1 On autotopies of n-ary linear quasigroups
8.1.1 Autotopies of derivative groups
8.1.2 Automorphisms of n-T -quasigroups
8.1.3 Automorphisms of some quasigroup isotopes
8.1.4 Automorphisms of medial n-quasigroups
8.1.5 Examples
8.2 Automorphism groups of some binary quasigroups
8.2.1 Isomorphisms of IP-loop isotopes
8.2.2 Automorphisms of loop isotopes
8.2.3 Automorphisms of LD-quasigroups
8.2.4 Automorphisms of isotopes of LD-quasigroups
8.2.5 Quasigroups with transitive automorphism group
8.3 Non-isomorphic isotopic quasigroups
9: Orthogonality of quasigroups
9.1 Orthogonality: Introduction
9.1.1 Squares and Latin squares
9.1.2 m-Tuples of maps and its product
9.1.3 m-Tuples of maps and groupoids
9.1.4 T-Property
9.1.5 Definitions of orthogonality
9.1.6 Orthogonality in works of V.D. Belousov
9.1.7 Product of squares
9.2 Orthogonality and parastroph orthogonality
9.2.1 Orthogonality of left quasigroups
9.2.2 Orthogonality of quasigroup parastrophes
9.2.3 Orthogonality in the language of quasi-identities
9.2.4 Orthogonality of parastrophes in the language of identities
9.2.5 Spectra of some parastroph orthogonal quasigroups
9.3 Orthogonality of linear and alinear quasigroups
9.3.1 Orthogonality of one-sided linear quasigroups
9.3.2 Orthogonality of linear and alinear quasigroups
9.3.3 Orthogonality of parastrophes
9.3.4 Parastrophe orthogonality of T -quasigroups
9.3.5 (12)-parastrophe orthogonality
9.3.6 totCO-quasigroups
9.4 Nets and orthogonality of the systems of quasigroups
9.4.1 k-nets and systems of orthogonal binary quasigroups
9.4.2 Algebraic (k,n)-nets and systems of orthogonal n-ary quasigroups
9.4.3 Orthogonality of n-ary quasigroups and identities
9.5 Transformations which preserve orthogonality
9.5.1 Isotopy and (12)-isostrophy
9.5.2 Generalized isotopy
9.5.3 Gisotopy and orthogonality
9.5.4 Mann’s operations
III: Applications
10: Quasigroups and codes
10.1 One check symbol codes and quasigroups
10.1.1 Introduction
10.1.2 On possibilities of quasigroup codes
10.1.3 TAC-quasigroups and n-quasigroup codes
10.1.4 5-n-quasigroup codes
10.1.5 Phonetic errors
10.1.6 Examples of codes
10.2 Recursive MDS-codes
10.2.1 Some definitions
10.2.2 Singleton bound
10.2.3 MDS-codes
10.2.4 Recursive codes
10.2.5 Gonsales-Couselo-Markov-Nechaev construction
10.2.6 Orthogonal quasigroups of order ten
10.2.7 Additional information
10.3 On signs of Bol loop translations
11: Quasigroups in cryptology
11.1 Introduction
11.1.1 Quasigroups in “classical” cryptology
11.2 Quasigroup-based stream ciphers
11.2.1 Introduction
11.2.2 Modifications and generalizations
11.2.3 Further development
11.2.4 Some applications
11.2.5 Additional modifications of Algorithm 1.69
11.2.6 n-Ary analogs of binary algorithms
11.2.7 Further development of Algorithm 11.10
11.3 Cryptanalysis of some stream ciphers
11.3.1 Chosen ciphertext attack
11.3.2 Chosen plaintext attack
11.4 Combined algorithms
11.4.1 Ciphers based on the systems of orthogonal n-ary operation
11.4.2 Modifications of Algorithm 11.14
11.4.3 Stream cipher based on orthogonal system of quasigroups
11.4.4 T-quasigroup-based stream cipher
11.4.5 Generalization of functions of Algorithm 11.16
11.4.6 On quasigroup-based cryptcode
11.4.6.1 Code part
11.4.6.2 Cryptographical part
11.4.6.3 Decoding
11.4.6.4 Resistance
11.4.6.5 A code-crypt algorithm
11.4.7 Comparison of the power of the proposed algorithms
11.5 One-way and hash functions
11.5.1 One-way function
11.5.2 Hash function
11.6 Secret-sharing schemes
11.6.1 Critical sets
11.6.2 Youden squares
11.6.3 Reed-Solomon codes
11.6.4 Orthogonality and secret-sharing schemes
11.7 Some algebraic systems in cryptology
11.7.1 Inverse quasigroups in cryptology
11.7.2 Some groups in cryptology
11.7.2.1 El Gamal cryptosystem
11.7.2.2 De-symmetrization of Algorithm 1.69
11.7.2.3 RSA and GM cryptosystems
11.7.2.4 Homomorphic encryption
11.7.2.5 MOR cryptosystem
11.7.3 El Gamal signature scheme
11.7.4 Polynomially complete quasigroups in cryptology
11.7.5 Cryptosystems which are based on row-Latin squares
11.7.6 Non-binary pseudo-random sequences over Galois fields
11.7.7 Authentication of a message
11.7.8 Zero-knowledge protocol
11.7.9 Hamming distance between quasigroups
11.7.10 Generation of quasigroups for cryptographical needs

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Tags: Victor Shcherbacov, Elements, quasigroup