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, February 03, 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 Page LIST OF SYMBOLS iii LIST OF ABBREVIATIONS iv LIST OF FIGURES v LIST OF TABLES vi INTRODUCTION CHAPTER PRELIMINARIES 1.1 Basic Structures 1.1.1 Strings 1.1.2 Graph 1.1.3 Deterministic Finite Automata m 1.1.4 The Galois Field GF (p ) 1.2 Digital Image Steganography 1.3 Exact Pattern Matching 11 1.4 Longest Common Subsequence 12 1.5 Searchable Encryption 15 CHAPTER 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 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 AUTOMATA TECHNIQUE FOR THE LONGEST COMMON SUBSEQUENCE PROBLEM 56 4.1 Introduction 56 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 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 BIBLIOGRAPHY LIST OF PUBLICATIONS ii 57 61 66 67 68 68 70 74 76 78 80 81 88 LIST OF SYMBOLS Σ Σ∗ An alphabet The set of all strings on Σ ∅ The empty set ǫ The empty string |S| The number of elements of a set S |u| The length of a string u m GF (p ) The Galois field is constructed from the polynomial ring Zp [x], where p is prime and m is a positive integer n m (GF (p ), +, ·) A vector space over the field GF (pm ) LCS(p, x) A longest common subsequence of p and x lcs(p, x) The length of a LCS(p, x) LeftID(u) The least element the leftmost location of u Rmp (u) The last component of LeftID(u) in p (I, M, K, Em, Ex) A data hiding scheme I A set of all image blocks with the same size and image format M A finite set of secret elements K A finite set of secret keys 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 qcolour The number of different ways to change the colour of each 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 c block A string of length c Pos p (z) The last position of appearance of z in p Mp An automaton accepting the pattern p Config(p) The set of all the configurations of p Wp (u) The weight of u in p Wp (C) The weight of C WConfig(p) The set of the weights of all the configurations of p i Wp (a) The weight of a at the location i in p Wmp (a) The heaviest weight of a in p W (a) 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 Significant 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 Simplified BNDM Searchable Encryption Searchable Symmetric Encryption Thathoo Virmani Sai Balakrishnan Sekar Wagner Fischer Wu Lee iv LIST OF FIGURES Figure Figure Figure Figure Figure 1.1 1.2 1.3 1.4 1.5 A simple graph A spanning tree of the graph given in Figure 1.1 The transition diagram of A in Example 1.3 The basic diagram of digital image steganography The degree of appearance of the pattern p 12 Figure 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 35 Figure 2.3 The binary cover image sized 2592 × 1456 pixels 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 Mp , p = abcba v 40 44 46 LIST OF TABLES Table 1.1 An adjacency list representation of the simple graph given in Figure 1.1 Table 1.2 The performing steps of the BF algorithm 11 Table 1.3 The dynamic programming matrix L 13 Table 2.1 Elements of the Galois field GF (22 ) represented by binary strings and decimal numbers Table 2.2 Operations + and · on the Galois field GF (22 ) 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 (1, 2n − 1, n) for palette images with qcolour = Table 2.5 The payload, ER and PSNR for the near optimal data hiding scheme (2, 9, 8) for gray images with qcolour = Table 2.6 The payload, ER and PSNR for the near optimal data hiding scheme (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, ⌊log2 (N + 1)⌋), where N = 2n − 1, for the binary image with qcolour = Time is given in second unit Table Table Table Table Table Table Table Table Table Table 3.1 The performing steps of the MR1 algorithm 3.2 Experimental results on rand4 problem 3.3 Experimental results on rand8 problem 3.4 Experimental results on rand16 problem 3.5 Experimental results on rand32 problem 3.6 Experimental results on rand64 problem 3.7 Experimental results on rand128 problem 3.8 Experimental results on rand256 problem 3.9 Experimental results on a genome sequence (with |Σ| = 4) 3.10 Experimental results on a protein sequence (with |Σ| = 20) 36 37 37 37 46 51 51 52 52 53 53 54 54 55 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 59 66 67 vi 30 30 31 or c39 9!28 !218t1 29tt2 = c39 9!28 !218t1 +9tt2 for palette images (5.10) Remark 5.3 For two algorithms eK′′ and dK′′ given as above, an arbitrary image block I in the input image F can be used many times in process of encrypting and decrypting the secret data So, for a give input image F , the secret data encrypted is not limited by the size of the input image F 5.3 Automata Technique Encrypted Data for Exact Pattern Matching on This section applies the automata approach proposed in Chapter to designing an algorithm for exact pattern matching on secret data encrypted by the cryptosystem proposed in Section 5.2 (Theorem 5.2) Suppose that Alice has a secret data and prefers to outsource this data to a cloud provider Bob As the provider is semi-trusted, Alice needs to encrypted her plaintext and wishes to only store ciphertext in the cloud Assume that Alice uses the cryptosystem (P, C, K′ , E, D) proposed in Section 5.2 to encrypt data with a pair of two secret parameters (S, k) in the cryptosystem, where S is a 2-Generators for GF (22 ) with elements and k = (f, K, I) ∈ K′ Because of limited storage space and computing ability, instead of downloading ciphertext, decrypting it and searching locally, Alice may ask Bob to perform pattern matching tasks on the ciphertext directly with a trapdoor of the pattern received from her Consider Σ to be an alphabet of size 256 Suppose that the secret data is a string over Σ x = x1 x2 xt3 for xi ∈ P, ∀i = 1, t3 , t3 ≥ and t3 is often a large natural number, where P = Σ Before uploading the secret data x to Bob, Alice use the encrypting function ek ∈ E to encrypt each xi Then Alice computes yi = ek (xi ), ∀i = 1, t3 , and the encrypted secret data is a string over Σ′ y = y1 y2 yt3 which is sent to Bob, where Σ′ is an alphabet Σ′ = {a′ |a′ = ek (a), a ∈ Σ} In general case, for x is any string over the alphabet Σ and a string y is obtained from x by the above way Then we can write y = ek (x) for short and y is a string over the alphabet Σ′ Remark 5.4 By using only one pair of two secret parameters (S, k), then the security of process of encrypting and decrypting the secret data x is similar to Formulas (5.3) (for gray images) or (5.4) (for palette images) Suppose that Bob needs to perform exact pattern matching task of an arbitrary pattern p on encrypted data y Based on previously introduced results in Chapter 3, here continues using automata technique to meet the requirement 74 Propostion 5.2 Let p be a pattern over the alphabet Σ Then P osp′ (a′ ) = P osp (a) for ∀a′ ∈ Σ′ , a = dk (a′ ), where p′ = ek (p) Proof Set i = P osp (a), then a = pi , hence a′ = p′i Without loss of generality, suppose P osp′ (a′ ) > i, then ∃i′ > i, p′i′ = a′ by Definition 3.3, then a = pi′ = dk (p′i′ ) Then i < P osp (a), a contradiction So, we complete the proof Propostion 5.3 Let a pattern p and a text x be two strings over the same alphabet Σ and the function Sign be given by ∀a′ ∈ Σ′ , Sign(a′ ) = If a ∈ p, Otherwise Then ∀a′ ∈ Σ′ , a′ ∈ p′ if and only if Sign(a′ ) = Proof Suppose ∀a′ ∈ Σ′ , a′ ∈ p′ if and only if ∃i, i = |p′ |, a′ = p′i if and only if a = pi if and only if Sign(a′ ) = Propostion 5.4 Let a pattern p and a text x be two strings over the same alphabet Σ Then p occurs at any position i in x if and only if p′ occurs at the position i in y, where y = ek (x) Proof Suppose that p occurs at any position i in x if and only if p = xi xi+1 xi+|p|−1 if and only if yi yi+1 yi+|p|−1 = p′ if and only if p′ occurs at the position i in y Propostion 5.5 Let p be a pattern over the alphabet Σ Nextp′ (l) = Nextp (l), where p′ = ek (p) Then ∀l, ≤ l ≤ |p|, Proof Without loss of generality, suppose that lm = Nextp (l) < Nextp′ (l) for ∀l, ≤ l ≤ |p| Since p′i = ek (pi ), ∀i = |p|, then p′1 p′2 p′lm +1 is both a proper prefix and suffix of p′ [1 l] by Definition 3.1 Hence, p1 p2 plm +1 is also both a proper prefix and suffix of p[1 l] by Definition 3.1 Then Nextp (l) > lm This is a contradiction to the supposition So, the proof is complete Propostion 5.6 Let p be a pattern over the alphabet Σ Then for ∀l, ≤ l ≤ |p| and ∀a′ ∈ Σ′ , a = dk (a′ ), Appearancep′ (l, a′ ) = Appearancep (l, a), where p′ = ek (p) Proof Clearly, |p| = |p′ | and for ∀i, ≤ i ≤ |p′ |, ∀a′ , a′ ∈ Σ′ , a′ = p′i if and only if a = pi By Lemma 3.1 and Proposition 5.5, Appearancep′ (l, a′ ) = Appearancep (l, a) Theorem 5.2 Let p be a pattern over the alphabet Σ Let two automata ′ Mp = (Σ, Qp , q0 , δp , Fp ) and Mp′ = (Σ , Qp′ , q0 , δp′ , Fp′ ) be determined as in Theorem 3.2 Then Qp′ = Qp , Fp′ = Fp , ∀q ∈ Qp′ , ∀a′ ∈ Σ′ , a = dk (a′ ), δp′ (q, a′ ) = δp (q, a), where p′ = ek (p) Proof It is easy to verify that |p| = |p′ | In addition, by Theorem 3.2 and Proposition 5.6, then Qp′ = Qp , Fp′ = Fp and δp′ (q, a′ ) = δp (q, a) Remark 5.5 The meaning of Theorem 5.2 in practice is to compute δp′ from δp 75 Let a pattern p and a text (secret data) x be two strings over the same alphabet Σ and assume |p| ≪ |x| For assuming that we have only the encrypted secret data y which is not decrypted to the secret data x, from Propositions 5.2, 5.3 and 5.4, Theorem 5.2, based on the MRc algorithm for c = and using the type a breaking point and the concept of Posp in Chapter 3, and by using the automaton Mp′ given as in Theorem 3.2, we have an exact pattern matching algorithm immediately that finds all occurrences of the pattern p in x as follows Note that the trapdoor according to the search pattern p is computed based on p, which includes the length of p, the functions Sign, Posp′ and the automaton Mp′ jump = |p|; While (jump ≤ |y|) { If (sign(yjump ) == 1) { q = 0; i = jump − P osp′ (yjump ) + 1; Do { q = δp′ (q, yi ); If (q == |p|) Mark an occurrence of p at i − |p| + in x; i + +; } While (q = and i ≤ |y|); jump = i − 1; } jump = jump + |p|; } Remark 5.6 Obviously, the above algorithm is the same as the MRc algorithm in Chapter for c = 1, using the type a breaking point and concepts of P osp′ , and where P osp and δp of the MRc algorithm are replaced with P osp′ and δp′ Furthermore, by Propositions 3.2 and 5.2, and Theorem 5.2, clearly in the worst case, this algorithm’s time complexity is O(n) 5.4 Automata Technique for Approximate Pattern Matching on Encrypted Data Suppose that Bob wants to approximate pattern matching of any pattern p on the ciphertext y From results proposed in Chapter 4, automata technique is still applied to meeting the requirement (Theorem 5.3) Propostion 5.7 Let p be a pattern over the alphabet Σ Then WConfig(p′ ) = WConfig(p), where p′ = ek (p) Proof Obviously, W0 belongs to WConfig(p′ ) and WConfig(p) Consider any ′ ′ ′ ′ ′ ′ W ∈ WConfig(p )\{W0 }, then we can set W = {w1 , w2 , , wl } for ≤ l ≤ |p′ | Then 76 ∃C ′ = {u′1 , u′2 , , u′l } ∈ Config(p′ ) by Definition 4.2, where Wp′ (u′i ) = wi′ for ≤ i ≤ l Then ∃!C = {u1 , u2 , , ul } ∈ Config(p), where ui = dk (u′i ) for ≤ i ≤ l Set W = Wp (C), then W ∈ WConfig(p) by Definition 4.2 It easy to verify that Rmp′ (u′i ) = Rmp (ui ) for ≤ i ≤ l by Definition 1.10, then Wp′ (u′i ) = Wp (ui ) for ≤ i ≤ l by Definition 4.1 Hence, W ′ = W , then WConfig(p′ ) ⊂ WConfig(p) Similarly, we have WConfig(p) ⊂ WConfig(p′ ) So, the proof is complete Propostion 5.8 Let p be a pattern over the alphabet Σ Then Refp′ (i, a′ ) = Refp (i, a) for ∀i, ≤ i ≤ |p′ | and ∀a′ ∈ Σ′ , a = dk (a′ ), where p′ = ek (p) Proof Clearly, Wpi′ (a′ ) = Wpi (a) by Definition 4.3 So, Refp′ (i, a′ ) = Refp (i, a) by Definition 4.4 Hence, we complete the proof Theorem 5.3 Given a pattern p on Σ and a positive integer constant c with ≤ c ≤ |p| Let two automata APp c = (Σ, Qp , q0 , δp , Fp ) and APp′c = (Σ′ , Qp′ , q0 , δp′ , Fp′ ) be determined as in Theorem 4.4 Then Qp′ = Qp , Fp′ = Fp , ∀q ∈ Qp′ , ∀a′ ∈ Σ′ , a = dk (a′ ), δp′ (q, a′ ) = δp (q, a), where p′ = ek (p) Proof By Proposition 5.7, Qp′ = Qp Evidently, ∀a′ ∈ Σ′ , a = dk (a′ ), a′ ∈ p′ if and only if a ∈ p Furthermore, by Definition 4.9 and Proposition 5.8, δp′ (W, a′ ) = δp (W, a) Then Fp′ = Fp So, the proof is complete Remark 5.7 The meaning of Theorem 5.3 in practice is to compute δp′ from δp Based on the approximate pattern matching problems considered in [66, 67, 73], the section introduces a new concept of the appearance of the pattern p in x with a given error This is a basis for giving requirements for the approximate pattern matching algorithm Definition 5.1 Given two strings p and x over Σ, and a string similarity measure d Let an error ǫ, ǫ > 0, ǫ ∈ ℜ Then p appears in x with the error ǫ if there exists a substring u of x such that d(p, u) ≤ ǫ To construct the approximate pattern matching algorithm, we need a function to measure the string similarity The most commonly used similarities are recalled in [71, 73, 77] Bakkelund [8] proposed a well known string similarity measure which is based on the longest commonly subsequence Similarly, here this section defines a new measure of similarity between two strings d(p, u) = − lcs(p, u) , min{|p|, |u|} (5.11) where p is a pattern and u is a substring of x Clearly, d given above is positive definite and symmetric Propostion 5.9 Given two strings p and x over the same alphabet Σ Then ∀u′ , u′ is an arbitrary substring of y, d(p′ , u′ ) = d(p, u), where p′ = ek (p), y = ek (x), u = dk (u′ ) Proof Clearly, |p′ | = |p|, |u′ | = |u| and lcs(p, u) = lcs(p′ , u′ ) d(p′ , u′ ) = d(p, u) So, we complete the proof 77 By Formula (5.11), By using the string similarity measure given in Formula (5.11), the automata technique proposed in Chapter for computing lcs(p′ , u′ ) will make an approximate pattern matching algorithm fast, and especially efficient for one pattern and a set of a large number of encrypted texts Given a pattern p and a text (secret data) x over the same alphabet Σ, and an arbitrary substring u of x Let ǫ, < ǫ < and d(p, u) be given as in Formula (5.11) such that d(p, u) ≤ ǫ Then by Proposition 5.9, d(p′ , u′ ) ≤ ǫ By Formula (5.11), we have lcs(p′ , u′ ) ≥ (1 − ǫ)min{|p′ |, |u′ |} (5.12) If there is u′ which is a substring of y such that lcs(p′ , u′ ) ≥ (1 − ǫ)|p|, then Formula (5.12) holds that means d(p′ , u′ ) ≤ ǫ Hence, ∃u, u is a substring of x, d(p, u) ≤ ǫ So, the constant c in Theorem 4.4 is determined by c = ⌈(1 − ǫ)|p|⌉ Without decrypting y, based on Theorem 5.3, Definition 5.1 and Formula (5.11), use the automaton APp′c given as in Theorem 4.4, we immediately have an approximate pattern matching algorithm which determines whether p appears in x with the error ǫ or not as follows, where the trapdoor responding to the pattern p is determined from p and ǫ, which consists of the constant c and the automaton APp′ c app = 0; q = W0 ; //The initial state of the automaton APp′ c is started from W0 For i = to |y| Do { q = δp′ (q, yi ); If (|q| = c) {app=1; Break;} } If (app = 1) Announce the appearance of the pattern p in x with the error ǫ; Else Announce that p does not appear in x with the error ǫ Remark 5.8 Since we can compute δp′ from δp , the above proposed algorithm is similar to the Algorithm (the parallel algorithm) in Chapter In addition, according to Theorem 4.4, δp is computed in parallel way and the Algorithm costs the worst case time complexity O(n) with the supposition that the Algorithm uses k processors for k is an upper estimate of the lcs(p, x) As an immediately consequence, in the worst case, time complexity of the algorithm is O(n) when it uses ⌈(1 − ǫ)|p|⌉ processors 5.5 Conclusions From results of steganography, pattern matching and some suggestions in the next works in Chapters 2, and 4, this chapter has completed some parts of those works Based on the data hiding scheme (2, 9, 8) in Chapter 2, the chapter constructs a novel cryptosystem with high security This method allows both of encrypting and hiding to be done at once, ciphertexts not to depend on the input image size as existing hybrid techniques of cryptography and steganography In addition, this cryptosystem can be used to encrypt and decrypt secret data by users in SSE For the ciphertext made by the cryptosystem on users side, the chapter designs two pattern matching algorithms to search for any pattern 78 in it directly on cloud servers side The outstanding feature of the algorithms is that, they can be applied well to all cryptosystems that support encrypting letters of the secret data one by one The idea of the design is to apply the automata approach for the exact pattern matching and the longest common subsequence problems in Chapters and For the assumption that the approximate algorithm uses ⌈(1 − ǫ)m⌉ processors, the time complexities of these algorithms are both O(n) in the worst case, where ǫ, m and n are the error of a measure of similarity between two strings proposed in Section 5.4 and lengths of the pattern and secret data, respectively With the proposed automata approach to pattern matching algorithms, the automata constructed are only based on search patterns Then the algorithms will have lots of advantages in case of a given pattern and a very large set of ciphertexts stored in the cloud So, future work continues studying this technique to apply in SE 79 CONCLUSION Based on the supervision of Assoc Prof Dr Sc Phan Thi Ha Duong and Dr Vu Thanh Nam, and study on results using graph theory and automata technique proposed by P T Huy et al in steganography and searchable encryption, new contributions of the dissertation in these fields can be summarized as follows: A general approach based on the Galois field GF (pm ) using graph theory and automata to designing optimal and near optimal secret data hiding schemes for binary, gray and palette images (Chapter 2); Based on the approach, Chapter shows that the data hiding scheme CTL [18] reaches an optimal data hiding scheme with N = 2n − 1, where n is a positive integer; From the approach, Chapter proposes data hiding schemes consisting of the optimal data hiding scheme (1, 2n −1, n) for binary, gray and palette images with qcolour = 1, where n is a positive integer, the near optimal data hiding scheme (2, 9, 8) and the optimal data hiding scheme (1, 5, 4) for gray and palette images with qcolour = 3; A flexible automata approach to constructing an efficient algorithm for the exact pattern matching problem in practice (Chapter 3); Mathematical basis for the development of automata technique for computing the lcs(p, x) (Chapter 4); By the above basis, Chapter proposes two efficient sequential and parallel algorithms computing the lcs(p, x) in practice The parallel algorithm takes O(n) time in the worst case if it uses k processors, where k is an upper estimate of the length of a longest common subsequence of the two strings p and x; Based on results proposed in Chapters 2, and 4, Chapter proposes two major components of SSE: a) A novel cryptosystem based on the data hiding scheme (2, 9, 8) with high security, used by users This method allows both of encrypting and hiding to be done at once, the ciphertext not to depend on the input image size as existing hybrid techniques of cryptography and steganography; b) Two exact and approximate pattern matching algorithms, using automata technique, search for any pattern in ciphertexts directly, performed by cloud servers For the assumption that the approximate algorithm uses ⌈(1 − ǫ)m⌉ processors, the time complexities of these algorithms are both O(n) in the worst case, where ǫ, m and n are the error of the proposed measure of similarity between two strings and lengths of the pattern and secret data, respectively Because the problem of development of methods of graph theory and automata in steganography and searchable encryption is 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