DOCSLIB.ORG

Fully Homomorphic Encryption Over Exterior Product Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fully Homomorphic Encryption Over Exterior Product Spaces FULLY HOMOMORPHIC ENCRYPTION OVER EXTERIOR PRODUCT SPACES by DAVID WILLIAM HONORIO ARAUJO DA SILVA B.S.B.A., Universidade Potiguar (Brazil), 2012 A thesis submitted to the Graduate Faculty of the University of Colorado Colorado Springs in partial fulfillment of the requirements for the degree of Master of Science Department of Computer Science 2017 © Copyright by David William Honorio Araujo da Silva 2017 All Rights Reserved This thesis for the Master of Science degree by David William Honorio Araujo da Silva has been approved for the Department of Computer Science by C. Edward Chow, Chair Carlos Paz de Araujo Jonathan Ventura 9 December 2017 Date ii Honorio Araujo da Silva, David William (M.S., Computer Science) Fully Homomorphic Encryption Over Exterior Product Spaces Thesis directed by Professor C. Edward Chow ABSTRACT In this work I propose a new symmetric fully homomorphic encryption powered by Exterior Algebra and Product Spaces, more specifically by Geometric Algebra as a mathematical language for creating cryptographic solutions, which is organized and presented as the En- hanced Data-Centric Homomorphic Encryption - EDCHE, invented by Dr. Carlos Paz de Araujo, Professor and Associate Dean at the University of Colorado Colorado Springs, in the Electrical Engineering department. Given GA as mathematical language, EDCHE is the framework for developing solutions for cryptology, such as encryption primitives and sub-primitives. In 1978 Rivest et al introduced the idea of an encryption scheme able to provide security and the manipulation of encrypted data, without decrypting it. With such encryption scheme, it would be possible to process encrypted data in a meaningful way. In 2009, Craig Gentry proposed the first fully homomorphic encryption scheme based on ideal lattices. Although his original solution has faced some important improvements, Gentry’s scheme is currently impracticable. Geometric Algebra is a field of mathematics already explored in many areas of Com- puter Science, Physics, Electrical Engineering, to cite a few. However, this is the first time that Geometric Algebra is proposed as the main building block of cryptographic solutions. Numbers are represented as multivectors, the special object for data representation, and through the GA operations (and some of its extensions), a series of transformations are performed, providing the mathematical under-determinacy required for encryption. Since the mathematics that empowers EDCHE is naturally homomorphic, EDCHE is illustrated as an intrinsically fully homomorphic encryption, allowing the required ho- momorphic additive and multiplicative properties and adding special capabilities such as comparison, sorting and searching. iii DEDICATION Dedicated to my parents, Janildo and Elisabete, my wife Cimaria, and my son and daughters, Johnathan, Samara and Sarah. Not even a thousand theses would be enough to properly express how much I love you all. iv ACKNOWLEDGEMENTS I would like to thank God, my Lord and Savior, for the gift of life and for so much love, for what I am constantly astonished and amazed. I also want to thank Dr. Carlos A. Paz de Araujo for believing in my potential and for investing in my academic and professional career in so many ways. Thank you for being such an inspiration, for mentoring me with absolute excellence and for teaching me that learning and creating are achievable goals, not matter the size or the complexity of the challenges in front of us. I want to thank Greg Jones for supporting me in my personal and professional develop- ment with the type of guidance that makes the difference in the life of any successful person. Thank you for being always optimistic and and willing to serve. I want to thank Marcelo Araujo Xavier for being a great incentive for my academic growth, helping me whenever I needed it. My gratitude also goes to my advisor Dr. Edward Chow for believing in me and in my ideas, for giving me practicable orientation during the entire research process and teaching me how to think as a Computer Scientist. I learned a lot during this whole process. v Table of Contents CHAPTER 1Introduction 1 1.1 Fundamental concepts and definitions . 1 1.2 Types and properties of secrecy system . 3 1.3 Statisticalmethods................................ 4 1.4 Homomorphism................................... 6 1.5 Homomorphicencryption ............................. 7 1.6 Fullyhomomorphicencryption . 8 1.7 Exterior product spaces and Geometric Algebra . 9 1.8 Enhanced Data-Centric Homomorphic Encryption . ... 9 1.9 EDCHESpecialApplications . 11 1.10MyContribution .................................. 13 2 FHE Using Ideal Lattices 15 2.1 Introduction..................................... 15 2.2 FullyHomomorphicEncryption . 16 2.3 Gentry’sFullyHomomorphicEncryption. .... 16 2.4 LimitationsofGentry’sscheme . 18 2.5 Intrinsic and Extrinsic Homomorphism . ....... 20 2.6 Conclusion...................................... 21 3ProductspacesandGeometricAlgebra 22 3.1 Introduction..................................... 22 3.2 Exterior Algebra and Product Spaces . 23 3.3 GeometricAlgebra ................................. 26 vi 3.4 Conclusion...................................... 35 4EncryptionusingMultivectors 37 4.1 Introduction..................................... 37 4.2 BasicMultivectorPackingScheme . 41 4.3 Sylvester’sEquation ............................... 43 4.4 Tripleproduct.................................... 44 4.5 The Underdeterminacy of the EDCHE Primitives . 45 4.6 EDCHEasaFramework.............................. 47 4.7 EDCHESub-primitives .............................. 48 4.8 SecretKeyExchange................................ 50 4.9 Sendingwithoutsending.............................. 55 4.10 Hierarchy Identity-Based Encryption . .... 59 4.11 ContinuousAuthentication . 62 4.12 EDCHE With Real Secret Keys . 64 4.13 Generating Keys from Existing Keys . 66 4.14 AutomaticKeyUpdate .............................. 68 4.15 Multivectortranslation. .... 68 4.16EDCHEwithXOR................................. 71 4.17 EDCHEWithRationalNumbers . 73 4.18Cryptanalysis ................................... 76 4.19 TimeandSpaceComplexity. 82 4.20Conclusion...................................... 83 5 FHE with EDCHE 85 5.1 Introduction..................................... 85 5.2 Selected Encryption Primitive . 86 5.3 DynamicPackingScheme ............................. 86 5.4 Additive Homomorphism . 90 5.5 ScalarMultiplicativeHomomorphism . ..... 92 5.6 Multiplicative Properties of the Rationalize . ....... 93 vii 5.7 Multiplicative Homomorphism with Cartesian product . ...... 95 5.8 Multiplicative Homomorphism With the Edge Product . ....... 102 5.9 Conclusion...................................... 105 6 EDCHE Applications 107 6.1 Introduction..................................... 107 6.2 Additive Homomorphism . 108 6.3 ScalarMultiplicativeHomomorphism . ..... 109 6.4 MultiplicativeHomomorphism. .... 111 6.5 HomomorphicSearch................................ 114 6.6 HomomorphicNumericSorting . 118 6.7 Homomorphic Alphanumeric Sort . 122 6.8 Conclusions ..................................... 126 7 AES Overall Comparison with EDCHE 127 7.1 Introduction..................................... 127 7.2 BlockCiphers.................................... 128 7.3 OverviewofAESDesign.............................. 130 7.4 Mathematical Concepts and Terminology Used in AES . 131 7.5 BooleanFunctions ................................. 136 7.6 Bundle Partitions, Transpositions and Bricklayer Functions .......... 136 7.7 Overall Differences Between AES and EDCHE . 139 7.8 PerformanceAnalysis............................... 141 7.9 Conclusions ..................................... 145 8 Conclusions and Future Work 147 8.1 Futurework.....................................153 Bibliography 155 APPENDIX viii AAdditionalExamples 159 A.1 ScaleProperties................................... 159 A.2 RatioProperties .................................. 161 A.3 SortingProperties ................................. 162 B Ruby Codes 165 B.1 MultivectorClass.................................. 165 B.2 ToolsClass ..................................... 171 B.3 EDCHEModule .................................. 172 B.4 Loader........................................ 174 B.5 ExampleWithPre-definedKeys . 174 B.6 ExampleWithCustomKeys ........................... 175 ix List of Tables TABLE 4.1 Diffie-HellmanKeyExchange . 51 7.1 Main Differences Between AES and EDCHE . 140 7.2 EDCHEPerformanceSpecs ............................ 142 7.3 EncryptionTime .................................. 142 7.4 CiphtertextSize................................... 143 7.5 EncryptionTime .................................. 143 7.6 AESPerformanceSpecs .............................. 144 7.7 AESPerformanceResults ............................. 144 x List of Figures FIGURE 4.1 EncryptionwiththeSylvester’sEquation . .... 44 4.2 EncryptionwiththeTripleProduct. 45 4.3 Multi-Ciphertexts Multivector . 60 4.4 NestedCiphtertexts ................................ 61 4.5 ContinuousAuthentication . 63 4.6 Key Generation via Rationalize . 68 4.7 Rationalnumbers.................................. 74 5.1 EDCHEOverview ................................. 85 7.1 BlockCipherDiagram ............................... 129 7.2 AESStateDiagram ................................130 7.3 ExampleofBundleTransposition . .... 137 7.4 ExampleofBricklayerTransformation . ..... 138 7.5 IterativeBooleanTransformation . .... 139 xi CHAPTER 1 Introduction After years of technological evolution, data is literally everywhere.
Load more

Recommended publications
  • Geometric Algebra for Vector Fields Analysis and Visualization: Mathematical Settings, Overview and Applications Chantal Oberson Ausoni, Pascal Frey
    Geometric algebra for vector fields analysis and visualization: mathematical settings, overview and applications Chantal Oberson Ausoni, Pascal Frey To cite this version: Chantal Oberson Ausoni, Pascal Frey. Geometric algebra for vector fields analysis and visualization: mathematical settings, overview and applications. 2014. hal-00920544v2 HAL Id: hal-00920544 https://hal.sorbonne-universite.fr/hal-00920544v2 Preprint submitted on 18 Sep 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Geometric algebra for vector field analysis and visualization: mathematical settings, overview and applications Chantal Oberson Ausoni and Pascal Frey Abstract The formal language of Clifford’s algebras is attracting an increasingly large community of mathematicians, physicists and software developers seduced by the conciseness and the efficiency of this compelling system of mathematics. This contribution will suggest how these concepts can be used to serve the purpose of scientific visualization and more specifically to reveal the general structure of complex vector fields. We will emphasize the elegance and the ubiquitous nature of the geometric algebra approach, as well as point out the computational issues at stake. 1 Introduction Nowadays, complex numerical simulations (e.g. in climate modelling, weather fore- cast, aeronautics, genomics, etc.) produce very large data sets, often several ter- abytes, that become almost impossible to process in a reasonable amount of time.
    [Show full text]
  • Exploring Physics with Geometric Algebra, Book II., , C December 2016 COPYRIGHT
    peeter joot [email protected] EXPLORINGPHYSICSWITHGEOMETRICALGEBRA,BOOKII. EXPLORINGPHYSICSWITHGEOMETRICALGEBRA,BOOKII. peeter joot [email protected] December 2016 – version v.1.3 Peeter Joot [email protected]: Exploring physics with Geometric Algebra, Book II., , c December 2016 COPYRIGHT Copyright c 2016 Peeter Joot All Rights Reserved This book may be reproduced and distributed in whole or in part, without fee, subject to the following conditions: • The copyright notice above and this permission notice must be preserved complete on all complete or partial copies. • Any translation or derived work must be approved by the author in writing before distri- bution. • If you distribute this work in part, instructions for obtaining the complete version of this document must be included, and a means for obtaining a complete version provided. • Small portions may be reproduced as illustrations for reviews or quotes in other works without this permission notice if proper citation is given. Exceptions to these rules may be granted for academic purposes: Write to the author and ask. Disclaimer: I confess to violating somebody’s copyright when I copied this copyright state- ment. v DOCUMENTVERSION Version 0.6465 Sources for this notes compilation can be found in the github repository https://github.com/peeterjoot/physicsplay The last commit (Dec/5/2016), associated with this pdf was 595cc0ba1748328b765c9dea0767b85311a26b3d vii Dedicated to: Aurora and Lance, my awesome kids, and Sofia, who not only tolerates and encourages my studies, but is also awesome enough to think that math is sexy. PREFACE This is an exploratory collection of notes containing worked examples of more advanced appli- cations of Geometric Algebra (GA), also known as Clifford Algebra.
    [Show full text]
  • DERIVATIONS and PROJECTIONS on JORDAN TRIPLES an Introduction to Nonassociative Algebra, Continuous Cohomology, and Quantum Functional Analysis
    DERIVATIONS AND PROJECTIONS ON JORDAN TRIPLES An introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis Bernard Russo July 29, 2014 This paper is an elaborated version of the material presented by the author in a three hour minicourse at V International Course of Mathematical Analysis in Andalusia, at Almeria, Spain September 12-16, 2011. The author wishes to thank the scientific committee for the opportunity to present the course and to the organizing committee for their hospitality. The author also personally thanks Antonio Peralta for his collegiality and encouragement. The minicourse on which this paper is based had its genesis in a series of talks the author had given to undergraduates at Fullerton College in California. I thank my former student Dana Clahane for his initiative in running the remarkable undergraduate research program at Fullerton College of which the seminar series is a part. With their knowledge only of the product rule for differentiation as a starting point, these enthusiastic students were introduced to some aspects of the esoteric subject of non associative algebra, including triple systems as well as algebras. Slides of these talks and of the minicourse lectures, as well as other related material, can be found at the author's website (www.math.uci.edu/∼brusso). Conversely, these undergraduate talks were motivated by the author's past and recent joint works on derivations of Jordan triples ([116],[117],[200]), which are among the many results discussed here. Part I (Derivations) is devoted to an exposition of the properties of derivations on various algebras and triple systems in finite and infinite dimensions, the primary questions addressed being whether the derivation is automatically continuous and to what extent it is an inner derivation.
    [Show full text]
  • Triple Product Formula and the Subconvexity Bound of Triple Product L-Function in Level Aspect
    TRIPLE PRODUCT FORMULA AND THE SUBCONVEXITY BOUND OF TRIPLE PRODUCT L-FUNCTION IN LEVEL ASPECT YUEKE HU Abstract. In this paper we derived a nice general formula for the local integrals of triple product formula whenever one of the representations has sufficiently higher level than the other two. As an application we generalized Venkatesh and Woodbury’s work on the subconvexity bound of triple product L-function in level aspect, allowing joint ramifications, higher ramifications, general unitary central characters and general special values of local epsilon factors. 1. introduction 1.1. Triple product formula. Let F be a number field. Let ⇡i, i = 1, 2, 3 be three irreducible unitary cuspidal automorphic representations, such that the product of their central characters is trivial: (1.1) w⇡i = 1. Yi Let ⇧=⇡ ⇡ ⇡ . Then one can define the triple product L-function L(⇧, s) associated to them. 1 ⌦ 2 ⌦ 3 It was first studied in [6] by Garrett in classical languages, where explicit integral representation was given. In particular the triple product L-function has analytic continuation and functional equation. Later on Shapiro and Rallis in [19] reformulated his work in adelic languages. In this paper we shall study the following integral representing the special value of triple product L function (see Section 2.2 for more details): − ⇣2(2)L(⇧, 1/2) (1.2) f (g) f (g) f (g)dg 2 = F I0( f , f , f ), | 1 2 3 | ⇧, , v 1,v 2,v 3,v 8L( Ad 1) v ZAD (FZ) D (A) ⇤ \ ⇤ Y D D Here fi ⇡i for a specific quaternion algebra D, and ⇡i is the image of ⇡i under Jacquet-Langlands 2 0 correspondence.
    [Show full text]
  • Vectors, Matrices and Coordinate Transformations
    S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties. Vectors For our purposes we will think of a vector as a mathematical representation of a physical entity which has both magnitude and direction in a 3D space. Examples of physical vectors are forces, moments, and velocities. Geometrically, a vector can be represented as arrows. The length of the arrow represents its magnitude. Unless indicated otherwise, we shall assume that parallel translation does not change a vector, and we shall call the vectors satisfying this property, free vectors. Thus, two vectors are equal if and only if they are parallel, point in the same direction, and have equal length. Vectors are usually typed in boldface and scalar quantities appear in lightface italic type, e.g. the vector quantity A has magnitude, or modulus, A = |A|. In handwritten text, vectors are often expressed using the −→ arrow, or underbar notation, e.g. A , A. Vector Algebra Here, we introduce a few useful operations which are defined for free vectors. Multiplication by a scalar If we multiply a vector A by a scalar α, the result is a vector B = αA, which has magnitude B = |α|A. The vector B, is parallel to A and points in the same direction if α > 0.
    [Show full text]
  • Matrices and Tensors
    APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM..
    [Show full text]
  • Finite Projective Geometries 243
    FINITE PROJECTÎVEGEOMETRIES* BY OSWALD VEBLEN and W. H. BUSSEY By means of such a generalized conception of geometry as is inevitably suggested by the recent and wide-spread researches in the foundations of that science, there is given in § 1 a definition of a class of tactical configurations which includes many well known configurations as well as many new ones. In § 2 there is developed a method for the construction of these configurations which is proved to furnish all configurations that satisfy the definition. In §§ 4-8 the configurations are shown to have a geometrical theory identical in most of its general theorems with ordinary projective geometry and thus to afford a treatment of finite linear group theory analogous to the ordinary theory of collineations. In § 9 reference is made to other definitions of some of the configurations included in the class defined in § 1. § 1. Synthetic definition. By a finite projective geometry is meant a set of elements which, for sugges- tiveness, are called points, subject to the following five conditions : I. The set contains a finite number ( > 2 ) of points. It contains subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains A and B. HI. If A, B, C are non-collinear points and if a line I contains a point D of the line AB and a point E of the line BC, but does not contain A, B, or C, then the line I contains a point F of the line CA (Fig.
    [Show full text]
  • New Foundations for Geometric Algebra1
    Text published in the electronic journal Clifford Analysis, Clifford Algebras and their Applications vol. 2, No. 3 (2013) pp. 193-211 New foundations for geometric algebra1 Ramon González Calvet Institut Pere Calders, Campus Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain E-mail : [email protected] Abstract. New foundations for geometric algebra are proposed based upon the existing isomorphisms between geometric and matrix algebras. Each geometric algebra always has a faithful real matrix representation with a periodicity of 8. On the other hand, each matrix algebra is always embedded in a geometric algebra of a convenient dimension. The geometric product is also isomorphic to the matrix product, and many vector transformations such as rotations, axial symmetries and Lorentz transformations can be written in a form isomorphic to a similarity transformation of matrices. We collect the idea Dirac applied to develop the relativistic electron equation when he took a basis of matrices for the geometric algebra instead of a basis of geometric vectors. Of course, this way of understanding the geometric algebra requires new definitions: the geometric vector space is defined as the algebraic subspace that generates the rest of the matrix algebra by addition and multiplication; isometries are simply defined as the similarity transformations of matrices as shown above, and finally the norm of any element of the geometric algebra is defined as the nth root of the determinant of its representative matrix of order n. The main idea of this proposal is an arithmetic point of view consisting of reversing the roles of matrix and geometric algebras in the sense that geometric algebra is a way of accessing, working and understanding the most fundamental conception of matrix algebra as the algebra of transformations of multiple quantities.
    [Show full text]
  • Notices of the American Mathematical Society 35 Monticello Place, Pawtucket, RI 02861 USA American Mathematical Society Distribution Center
    ISSN 0002-9920 Notices of the American Mathematical Society 35 Monticello Place, Pawtucket, RI 02861 USA Society Distribution Center American Mathematical of the American Mathematical Society February 2012 Volume 59, Number 2 Conformal Mappings in Geometric Algebra Page 264 Infl uential Mathematicians: Birth, Education, and Affi liation Page 274 Supporting the Next Generation of “Stewards” in Mathematics Education Page 288 Lawrence Meeting Page 352 Volume 59, Number 2, Pages 257–360, February 2012 About the Cover: MathJax (see page 346) Trim: 8.25" x 10.75" 104 pages on 40 lb Velocity • Spine: 1/8" • Print Cover on 9pt Carolina AMS-Simons Travel Grants TS AN GR EL The AMS is accepting applications for the second RAV T year of the AMS-Simons Travel Grants program. Each grant provides an early career mathematician with $2,000 per year for two years to reimburse travel expenses related to research. Sixty new awards will be made in 2012. Individuals who are not more than four years past the comple- tion of the PhD are eligible. The department of the awardee will also receive a small amount of funding to help enhance its research atmosphere. The deadline for 2012 applications is March 30, 2012. Applicants must be located in the United States or be U.S. citizens. For complete details of eligibility and application instructions, visit: www.ams.org/programs/travel-grants/AMS-SimonsTG Solve the differential equation. t ln t dr + r = 7tet dt 7et + C r = ln t WHO HAS THE #1 HOMEWORK SYSTEM FOR CALCULUS? THE ANSWER IS IN THE QUESTIONS.
    [Show full text]
  • Essential Concepts of Projective Geomtry
    Essential Concepts of Projective Geomtry Course Notes, MA 561 Purdue University August, 1973 Corrected and supplemented, August, 1978 Reprinted and revised, 2007 Department of Mathematics University of California, Riverside 2007 Table of Contents Preface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : i Prerequisites: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :iv Suggestions for using these notes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :v I. Synthetic and analytic geometry: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :1 1. Axioms for Euclidean geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. Cartesian coordinate interpretations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 3 3. Lines and planes in R and R : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 II. Affine geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1. Synthetic affine geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2. Affine subspaces of vector spaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 3. Affine bases: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :19 4. Properties of coordinate
    [Show full text]
  • Jordan Triple Systems with Completely Reducible Derivation Or Structure Algebras
    Pacific Journal of Mathematics JORDAN TRIPLE SYSTEMS WITH COMPLETELY REDUCIBLE DERIVATION OR STRUCTURE ALGEBRAS E. NEHER Vol. 113, No. 1 March 1984 PACIFIC JOURNAL OF MATHEMATICS Vol. 113, No. 1,1984 JORDAN TRIPLE SYSTEMS WITH COMPLETELY REDUCIBLE DERIVATION OR STRUCTURE ALGEBRAS ERHARD NEHER We prove that a finite-dimensional Jordan triple system over a field k of characteristic zero has a completely reducible structure algebra iff it is a direct sum of a trivial and a semisimple ideal. This theorem depends on a classification of Jordan triple systems with completely reducible derivation algebra in the case where k is algebraically closed. As another application we characterize real Jordan triple systems with compact automorphism group. The main topic of this paper is finite-dimensional Jordan triple systems over a field of characteristic zero which have a completely reducible derivation algebra. The history of the subject begins with [7] where G. Hochschild proved, among other results, that for an associative algebra & the deriva- tion algebra is semisimple iff & itself is semisimple. Later on R. D. Schafer considered in [18] the case of a Jordan algebra £. His result was that Der f is semisimple if and only if $ is semisimple with each simple component of dimension not equal to 3 over its center. This theorem was extended by K.-H. Helwig, who proved in [6]: Let f be a Jordan algebra which is finite-dimensional over a field of characteristic zero. Then the following are equivlent: (1) Der % is completely reducible and every derivation of % has trace zero, (2) £ is semisimple, (3) the bilinear form on Der f> given by (Dl9 D2) -> trace(Z>!Z>2) is non-degenerate and every derivation of % is inner.
    [Show full text]
  • Geometric Reasoning with Invariant Algebras
    Geometric Reasoning with Invariant Algebras Hongbo Li [email protected] Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080, P. R. China Abstract: Geometric reasoning is a common task in Mathematics Education, Computer-Aided Design, Com- puter Vision and Robot Navigation. Traditional geometric reasoning follows either a logical approach in Arti¯cial Intelligence, or a coordinate approach in Computer Algebra, or an approach of basic geometric invariants such as areas, volumes and distances. In algebraic approaches to geometric reasoning, geometric interpretation is needed for the result after algebraic manipulation, but in general this is a di±cult task. It is hoped that more advanced geometric invariants can make some contribution to the problem. In this paper, a hierarchical framework of invariant algebras is introduced for algebraic manipulation of geometric problems. The bottom level is the algebra of basic geometric invariants, and the higher levels are more complicated invariants. High level invariants can keep more geometric nature within algebraic structure. They are bene¯cial to geometric explanation but more di±cult to handle with than low level ones. This paper focuses on geometric computing at high invariant levels, for both automated theorem proving and new theorem discovering. Using the new techniques developed in recent years for hierarchical invariant algebras, better geometric results can be obtained from algebraic computation, in addition to more e±cient algebraic manipulation. 1 Background: Geometric Invariants What is geometric computing? Computing is an algebraic thing. If the objects involved are geometric, then in some suitable algebraic framework, the geometric problem is translated into an algebraic one, and an algebraic result is obtained by computation [18].
    [Show full text]