Authentication Codes and Combinatorial Designs
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Summary
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.
Table of Contents
INTRODUCTION
Authentication Problems
Authentication Schemes
Combinatorial Designs
AUTHENTICATION SCHEMES
Model with Three Participants (A-Codes)
Model with Four Participants (A2-Codes)
Comments
AUTHENTICATION SCHEMES WITH THREE PARTICIPANTS
Entropy
Information-Theoretic Bound
Perfect Authentication Schemes
Perfect Cartesian Codes
Combinatorial Bound
Comments
Exercises
AUTHENTICATION SCHEMES WITH ARBITRATION
Lower Bounds
Perfect Schemes with Arbitration
Perfect Cartesian A2-Codes
Combinatorial Bounds of A2-Codes
Comments
Exercises
A-CODES BASED ON RATIONAL NORMAL CURVES
SPBD Based on RNC
A Family of Non-Cartesian Perfect A-Codes
Encoding Rules (n = 2, q Odd)
Encoding Rules (n = 2, q Even)
Comments
Exercises
t-DEISGNS
2 - (v, k, 1) Designs
Steiner Triple System
3 - (v, k, 1) Designs
Comments
Exercises
ORTHOGONAL ARRAYS OF INDEX UNITY
OA with Strength t = 2 and Orthogonal Latin Squares
Transversal Designs
Existence of OA (n2, 4, n, 2)
Bush's Construction
OA and Error-Correcting Codes
MDS Codes
Comments
Exercises
A-CODES FROM FINITE GEOMETRIES
Symplectic Spaces over Finite Fields
A-Codes from Symplectic Spaces
A-Codes from Unitary Spaces
Comments
Exercises
AUTHENTICATION/SECRECY SCHEMES
Perfect Secrecy Schemes
Construction of Perfect Secrecy Schemes
Authentication Schemes with Perfect Secrecy
Construction of Perfect Authentication/Secrecy Schemes
Comments
Exercises
APPENDIX A: A SURVEY OF CONSTRUCTIONS FOR A-CODES
Key Grouping Technique
Perpendicular Arrays
Generalized Quadrangles
Resolvable Block Design and A2-Codes
Regular Bipartite Graphs
REFERENCES
NOTATIONS
INDEX
Editorial Reviews
"This monograph is devoted to the study of authentication codes and their related combinatorial designs. It is well suited for readers who want to learn how to reduce a cryptographic problem to a combinatorial problem, and how to use combinatorial design theory to solve problems in the field of authentication codes. ...one of the most interesting references for the study of authentication codes and their related combinatorial designs."
- Mathematical Reviews
