# Research on development of methods of graph theory and automata in steganography and searchable encryption

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 1 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . 4

1.1 Basic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Deterministic Finite Automata . . . . . . . . . . . . . . . . . . . . 6

1.1.4 The Galois Field GF(pm) . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Digital Image Steganography . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Exact Pattern Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Longest Common Subsequence . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Searchable Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

CHAPTER 2 DIGITAL IMAGE STEGANOGRAPHY BASED ON THE

GALOIS FIELD USING GRAPH THEORY AND AUTOMATA . . . . . 16

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 The Digital Image Steganography Problem . . . . . . . . . . . . . . . . . . 18

2.3 A New Digital Image Steganography Approach . . . . . . . . . . . . . . . . 19

2.3.1 Mathematical Basis based on The Galois Field . . . . . . . . . . . . 19

2.3.2 Digital Image Steganography Based on The Galois Field GF(pm)

Using Graph Theory and Automata . . . . . . . . . . . . . . . . . . 21

2.4 The Near Optimal and Optimal Data Hiding Schemes for Gray and Palette

Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

CHAPTER 3 AN AUTOMATA APPROACH TO EXACT PATTERN

MATCHING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 The New Algorithm - The MRc Algorithm . . . . . . . . . . . . . . . . . . 41

3.3 Analysis of The MRc Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

CHAPTER 4 AUTOMATA TECHNIQUE FOR THE LONGEST

COMMON SUBSEQUENCE PROBLEM . . . . . . . . . . . . . . . . . . 56

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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