Semirings as building blocks in cryptography

Contents......Page 6
Acknowledgements......Page 9
Preface......Page 10
Chapter One......Page 12
  2.1.1 Definitions......Page 15
  2.1.2 Public key Cryptosystems based on the Discrete Logarithm Problem......Page 16
  2.1.3 Generic algorithms for solving the Discrete Logarithm Problem......Page 19
  2.1.4 Index Calculus algorithms for finite fields......Page 21
  2.2.1 The Discrete Logarithm Problem in abelian groups......Page 23
  2.2.2 The Discrete Logarithm Problem in nonabelian groups......Page 24
  2.2.3 Groups used as platforms for cryptosystems......Page 28
  2.3.1 The Diffie-Hellman protocol in terms of formal algebra –a parallel between groups and rings......Page 33
  2.3.2 The Discrete Logarithm Problem in noncommutative rings......Page 35
  2.3.3 The Discrete Logarithm Problem and cryptography using semirings......Page 37
  2.3.4 Semirings and cryptography......Page 39
Bibliography to Chapters One and Two......Page 42
 3.1 Idempotent Algebra......Page 64
  3.1.1 Dioids as lattices......Page 67
  3.1.2 Residuated maps......Page 69
  3.1.3 Semimodules over idempotent semirings......Page 70
  3.1.4 Residuated lattices......Page 73
  3.1.5 Construction of complete dioid......Page 74
  3.1.6 Matrices defined over idempotent semifields......Page 76
  3.1.7 Residuated operations over matrix semimodules......Page 80
  3.1.8 Max-plus algebra......Page 81
  3.1.9 Min-plus algebra......Page 84
  3.1.10 Max-time algebra......Page 88
  3.1.11 Min-time algebra......Page 89
 3.2 Linear Equations......Page 91
 3.3 Applications of Idempotent Algebra in Cryptography......Page 95
Bibliography to Chapter Three......Page 98
 4.1 Definitions and Motivation to Study......Page 107
 4.2 Distributed Secure Multicast Protocol......Page 109
Bibliography to Chapter Four......Page 113
 5.1 Endomorphism Semirings......Page 115
 5.2 Simplices and Simplicial Complexes......Page 118
  5.3.1 Motivation, basic definitions and notations......Page 123
  5.3.2 Three key-exchange protocols based on endomorphism semirings......Page 126
  5.3.3 Four key exchange protocols using right identities of the simplex......Page 128
Bibliography to Chapter Five......Page 132