Authentication Codes and Combinatorial Designs / Edition 1

by Dingyi Pei

Overview

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.

Product Details

ISBN-13:	9780367391249
Publisher:	Taylor & Francis
Publication date:	09/05/2019
Series:	Discrete Mathematics and Its Applications
Pages:	256
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Dingyi Pei is a Professor of Mathematics at the Graduate School of the Chinese Academy of Sciences, Beijing, and at the Guangzhou University, Guangzhou, China. His current research interests include number theory, algebra, and cryptology.

Table of Contents

Preface vii

1 Introduction 1

1.1 Authentication Problem 1

1.2 Authentication Schemes 3

1.3 Combinatorial Designs 5

2 Authentication Schemes 7

2.1 Model with Three Participants (A-Codes) 7

2.2 Model with Four Participants (A²-Codes) 10

2.3 Comments 13

3 Authentication Schemes with Three Participants 15

3.1 Entropy 15

3.2 Information-Theoretic Bound 19

3.3 Perfect Authentication Schemes 21

3.4 Perfect Cartesian Codes 26

3.5 Combinatorial Bound 36

3.6 Comments 38

3.7 Exercises 39

4 Authentication Schemes with Arbitration 41

4.1 Lower Bounds 41

4.2 Perfect Schemes with Arbitration 48

4.3 Perfect Cartesian A²-Codes 56

4.4 Combinatorial Bounds of A²-Codes 66

4.5 Comments 72

4.6 Exercises 73

5 A-Codes Based on Rational Normal Curves 75

5.1 SPBD Based on RNC 75

5.2 A Family of Non-Cartesian Perfect A-Codes 80

5.3 Encoding Rules (n = 2, q Odd) 85

5.4 Encoding Rules (n = 2, q Even) 103

5.5 Comments 113

5.6 Exercises 113

6 t-Designs 115

6.1 2- (v, k, 1) Designs 116

6.2 Steiner Triple System 117

6.3 3-(v, k, 1) Designs 122

6.4 Comments 124

6.5 Exercises 124

7 Orthogonal Arrays of Index Unity 127

7.1 OA with Strength t = 2 and Orthogonal Latin Squares 127

7.2 Transversal Designs 135

7.3 Existence of OA(n²,4, n, 2) 140

7.4 Bush's Construction 142

7.5 OA and Error-Correcting Codes 145

7.6 MDS Codes 147

7.7 Comments 150

7.8 Exercises 151

8 A-Codes from Finite Geometries 153

8.1 Symplectic Spaces over Finite Fields 153

8.2 A-Codes from Symplectic Spaces 169

8.3 A-Codes from Unitary Spaces 178

8.4 Comments 182

8.5 Exercises 182

9 Authentication/Secrecy Schemes 185

9.1 Perfect Secrecy Schemes 186

9.2 Construction of Perfect Secrecy Schemes 194

9.3 Authentication Schemes with Perfect Secrecy 204

9.4 Construction of Perfect Authentication/Secrecy Schemes 209

9.5 Comments 212

9.6 Exercises 213

Appendix: A Survey of Constructions for A-Codes 215

A.1 Key Grouping Technique 215

A.2 Perpendicular Arrays 217

A.3 Generalized Quadrangles 221

A.4 Resolvable Block Design and A²-Codes 226

A.5 Regular Bipartite Graphs 230

References 233

Notations 237

Index 243