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

MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY |||||||||| Nguyen Huy Truong RESEARCH ON DEVELOPMENT OF METHODS OF GRAPH THEORY AND AUTOMATA IN STEGANOGRAPHY AND SEARCHABLE ENCRYPTION DOCTORAL DISSERTATION IN MATHEMATICS AND INFORMATICS Hanoi - 2020 MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY |||||||||| Nguyen Huy Truong RESEARCH ON DEVELOPMENT OF METHODS OF GRAPH THEORY AND AUTOMATA IN STEGANOGRAPHY AND SEARCHABLE ENCRYPTION Major: Mathematics and Informatics Major code: 9460117 DOCTORAL DISSERTATION IN MATHEMATICS AND INFORMATICS SUPERVISORS: Assoc Prof Dr Sc Phan Thi Ha Duong Dr Vu Thanh Nam Hanoi - 2020 DECLARATION OF AUTHORSHIP I hereby certify that I am the author of this dissertation, and that I have completed it under the supervision of Assoc Prof Dr Sc Phan Thi Ha Duong and Dr Vu Thanh Nam I also certify that the dissertation’s results have not been published by other authors Hanoi, May 18, 2020 PhD Student Nguyen Huy Truong Supervisors Assoc Prof Dr Sc Phan Thi Ha Duong Dr Vu Thanh Nam ACKNOWLEDGMENTS I am extremely grateful to Assoc Prof Dr Sc Phan Thi Ha Duong I want to thank Dr Vu Thanh Nam I would also like to extend my deepest gratitude to Late Assoc Prof Dr Phan Trung Huy I would like to thank my co-workers from School of Applied Mathematics and Informatics, Hanoi University of Science and Technology for all their help I also wish to thank members of Seminar on Mathematical Foundations for Computer Science at Institute of Mathematics, Vietnam Academy of Science and Technology for their valuable comments and helpful advice I give thanks to PhD students of Late Assoc Prof Dr Phan Trung Huy for sharing and exchanging information in steganography and searchable encryption Finally, I must also thank my family for supporting all my work CONTENTS LISTOFSYMBOLS LISTOFABBREVIATIONS LISTOFFIGURES LISTOFTABLES INTRODUCTION CHAPTER1 PRELIMINARIES 1.1 Basic Structures 1.1.1 Strings 1.1.2 Graph 1.1.3 Deterministic Finite Automata 1.1.4 The Galois Field GF (pm) 1.2 Digital Image Steganography 1.3 Exact Pattern Matching 1.4 Longest Common Subsequence 1.5 Searchable Encryption CHAPTER DIGITAL IMAGE STEGANOGRAPHY BASED ON THE GALOIS FIELD USING GRAPH THEORY AND AUTOMATA 16 2.1 Introduction 2.2 The Digital Image Steganography Problem 2.3 A New Digital Image Steganography Approach 2.3.1 Mathematical Basis based on The Galois Field Page iii iv v vi 4 4 11 12 15 m 16 18 19 19 2.3.2 Digital Image Steganography Based on The Galois Field GF (p ) 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 38 CHAPTER AN AUTOMATA APPROACH TO EXACT PATTERN MATCHING 40 3.1 Introduction 40 3.2 The New Algorithm - The MRc Algorithm 42 3.3 Analysis of The MRc Algorithm 48 3.4 Experimental Results 51 3.5 Conclusions 56 CHAPTER AUTOMATA TECHNIQUE FOR THE LONGEST COMMON SUBSEQUENCE PROBLEM 57 4.1 Introduction 57 i 4.2 4.3 4.4 4.5 Mathematical Basis Automata Models for Solving The LCS Problem Experimental Results Conclusions CHAPTER CRYPTOGRAPHY BASED ON STEGANOGRAPHY AND AUTOMATA METHODS FOR SEARCHABLE ENCRYPTION 69 5.1 Introduction 5.2 A Novel Cryptosystem Based on The Data Hiding Scheme (2; 9; 8) 5.3 Automata Technique for Exact Pattern Matching on Encrypted Data 5.4 Automata Technique for Approximate Pattern Matching on Encrypted Data 5.5 Conclusions CONCLUSION LISTOFPUBLICATIONS BIBLIOGRAPHY ii 58 62 67 68 69 71 75 77 79 81 82 83 LIST OF SYMBOLS ? jSj juj m GF (p ) n m (GF (p ); +; ) LCS(p; x) lcs(p; x) LeftID(u) An alphabet The set of all strings on The empty set The empty string The number of elements of a set S The length of a string u The Galois eld is constructed from the polynomial ring Z p[x], where p is prime and m is a positive integer m A vector space over the eld GF (p ) A longest common subsequence of p and x The length of a LCS(p; x) The least element the leftmost location of u The last component of LeftID(u) in p Rmp(u) (I; M; K; Em; Ex) A data hiding scheme A set of all image blocks with the same size and image format I A nite set of secret elements M A nite set of secret keys K Em An embedding function embeds a secret element in an image block Ex An extracting function extracts an embedded secret element from an image block q The number of di erent ways to change the colour of each colour pixel in an arbitrary image block I An image block M A secret element K A secret key Adjacent(cp; a) An adjacent vertex of cp Posp(z) The last position of appearance of z in p c block Mp Con g(p) Wp(u) Wp(C) WCon g(p) i Wp (a) Wmp(a) W (a) A string of length c An automaton accepting the pattern p The set of all the gurations of p The weight of u in p The weight of C The set of the weights of all the gurations of p The weight of a at the location i in p The heaviest weight of a in p The weight of a in p iii LIST OF ABBREVIATIONS AOSO BF BFS BMH BNDM CTL EBOM ER FJS FOPA FSBNDM HASH HCIH LBNDM LCS LSB MSDR MSE NP OPA PA PCT PSNR RGB SA SAE SBNDM SE SSE TVSBS WF WL Average Optimal Shift Or Brute Force Breadth First Search Boyer Moore Horspool Backward Nondeterministic Dawg Matching Chang Tseng Lin Extended Backward Oracle Matching Embedding Rate Franek Jennings Smyth Fastest Optimal Parity Assignment Forward SBNDM Hashing High Capacity of Information Hiding Long BNDM Longest Common Subsequence Least Signi cant Bit Maximal Secret Data Ratio Mean Square Error Nondeterministic Polynomial Optimal Parity Assignment Parity Assignment Pan Chen Tseng Peak Signal to Noise Ratio Red Green Blue Shift Add Searchable Asymmetric Encryption Simpli ed BNDM Searchable Encryption Searchable Symmetric Encryption Thathoo Virmani Sai Balakrishnan Sekar Wagner Fischer Wu Lee iv LIST OF FIGURES Figure Figure Figure Figure Figure Figure 1.1 A simple graph 1.2 A spanning tree of the graph given in Figure 1.1 1.3 The transition diagram of A in Example 1.3 1.4 The basic diagram of digital image steganography 1.5 The degree of appearance of the pattern p 12 2.1 The nine commonly used 8-bit gray cover images sized 512 512 pixels 35 Figure 2.2 The nine commonly used 8-bit palette cover images sized 512 512 pixels Figure 2.3 The binary cover image sized 2592 1456 pixels 36 36 Figure 3.1 Sliding window mechanism Figure 3.2 The basic idea of the proposed approach Figure 3.3 The transition diagram of the automaton M p, p = abcba 41 45 47 v LIST OF TABLES Table Table Table Table 1.1 An adjacency list representation of the simple graph given in Figure 1.1 1.2 The performing steps of the BF algorithm 11 1.3 The dynamic programming matrix L 13 2.1 Elements of the Galois eld GF (2 ) represented by binary strings and decimal numbers Table 2.2 Operations + and on the Galois eld GF (2 ) Table 2.3 The representation of E and the arc weights of G for the gray image Table 2.4 The payload, ER and PSNR for the optimal data hiding scheme 30 30 31 n 37 (1; 1; n) for palette images with qcolour = Table 2.5 The payload, ER and PSNR for the near optimal data hiding scheme 37 (2; 9; 8) for gray images with qcolour = Table 2.6 The payload, ER and PSNR for the near optimal data hiding scheme 38 (2; 9; 8) for palette images with qcolour = Table 2.7 The comparisons of embedding and extracting time between the chapter’s and Chang et al.’s approach for the same optimal data hiding scheme (1; N; blog2(N + 1)c), where N = n 1, for the binary image with qcolour = Time is given in second unit 38 Table 3.1 The performing steps of the MR algorithm Table 3.2 Experimental results on rand4 problem Table 3.3 Experimental results on rand8 problem Table 3.4 Experimental results on rand16 problem Table 3.5 Experimental results on rand32 problem Table 3.6 Experimental results on rand64 problem Table 3.7 Experimental results on rand128 problem Table 3.8 Experimental results on rand256 problem Table 3.9 Experimental results on a genome sequence (with j j = 4) Table 3.10 Experimental results on a protein sequence (with j j = 20) 47 Table 4.1 The Refp of p = bacdabcad Table 4.2 The comparisons of the lcs(p; x) computation time for n = 50666 Table 4.3 The comparisons of the lcs(p; x) computation time for n = 102398 60 vi 52 52 53 53 54 54 55 55 56 67 68