Authentification codes and combinatorial designs

Researchers and practitioners of cryptography and information security are constantly challenged to respond to new attacks and threats to information systems. Authentication Codes and Combinatorial Designs presents new findings and original work on perfect authentication codes characterized in terms of combinatorial designs, namely strong partially balanced designs (SPBD). Beginning with examples illustrating the concepts of authentication schemes and combinatorial designs, the book considers the probability of successful deceptions followed by schemes involving three and four participants, respectively. From this point, the author constructs the perfect authentication schemes and explores encoding rules for such schemes in some special cases. Using rational normal curves in projective spaces over finite fields, the author constructs a new family of SPBD. He then presents some established combinatorial designs that can be used to construct perfect schemes, such as t-designs, orthogonal arrays of index unity, and designs constructed by finite geometry. The book concludes by studying definitions of perfect secrecy, properties of perfectly secure schemes, and constructions of perfect secrecy schemes with and without authentication. Supplying an appendix of construction schemes for authentication and secrecy schemes, Authentication Codes and Combinatorial Designs points to new applications of combinatorial designs in cryptography. FM Discrete Mathematics And Its Applications 1 Continued Titles 2 Contents 5 Preface 7 Notations 9 Chapter 1 Introduction 14 1.1 Authentication Problem 14 1.2 Authentication Schemes 16 1.3 Combinatorial Designs 18 Chapter 2 Authentication Schemes 20 2.1 Model with Three Participants (A-Codes) 20 2.2 Model with Four Participants (A2-Codes) 23 2.3 Comments 26 Chapter 3 Authentication Schemes with Three Participants 27 3.1 Entropy 27 3.2 Information- Theoretic Bound 31 3.3 Perfect Authentication Schemes 33 3.4 Perfect Cartesian Codes 38 3.5 Combinatorial Bound 48 3.6 Comments 50 3.7 Exercises 51 Chapter 4 Authentication Schemes with Arbitration 53 4.1 Lower Bounds 53 4.2 Perfect Schemes with Arbitration 60 4.3 Perfect Cartesian A2-Codes 68 4.4 Combinatorial Bounds of A2-Codes 78 4.5 Comments 84 4.6 Exercises 85 Chapter 5 A-Codes Based on Rational Normal Curves 86 5.1 SPBD Based on RNC 86 5.2 A Family of Non- Cartesian Perfect A-Codes 91 5.3 Encoding Rules (n = 2, q Odd) 96 5.4 Encoding Rules (n = 2, q Even) 114 5.5 Comments 124 5.6 Exercises 124 Chapter 6 t-Designs 126 6.1 2 - (v, k, 1) Designs 127 6.2 Steiner Triple System 128 6.3 3 - (v, k, 1) Designs 133 6.4 Comments 135 6.5 Exercises 135 Chapter 7 Orthogonal Arrays of Index Unity 137 7.1 OA with Strength t = 2 and Orthogonal Latin Squares 137 7.2 Transversal Designs 145 7.3 Existence of OA(n2, 4, n, 2) 150 7.4 Bush's Construction 152 7.5 OA and Error- Correcting Codes 155 7.6 MDS Codes 157 7.7 Comments 160 7.8 Exercises 161 Chapter 8 A-Codes from Finite Geometries 162 8.1 Symplectic Spaces over Finite Fields 162 8.2 A- Codes from Symplectic Spaces 178 8.3 A- Codes from Unitary Spaces 187 8.4 Comments 191 8.5 Exercises 191 Chapter 9 Authentication/Secrecy Schemes 193 9.1 Perfect Secrecy Schemes 194 9.2 Construction of Perfect Secrecy Schemes 202 9.3 Authentication Schemes with Perfect Secrecy 212 9.4 Construction of Perfect Authentication/ Secrecy Schemes 217 9.5 Comments 220 9.6 Exercises 221 Appendix A Survey of Constructions for A-Codes 222 A. 1 Key Grouping Technique 222 A. 2 Perpendicular Arrays 224 A. 3 Generalized Quadrangles 228 A. 4 Resolvable Block Design and A2-Codes 233 A. 5 Regular Bipartite Graphs 237 References 239 The concept of authentication schemes was first introduced in the early 1980s as a means of providing in unconditionally secure systems. Connecting the theory of authentication schemes to the mathematical theory of combinatorial designs, Pei (mathematics, Graduate School of the Chinese Academy of Science) presents some original contributions in the The authentication of codes is an important area of cryptography. This book ties together the notion of authentication codes & combinatorial designs & demonstrates how ideas from combinatorics can be used for cryptographic applications