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

throughout the dissertation strings, graph, deterministic finite automata, digital images, the basic model of digital image steganography, some parameters to determine the quality of dig

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

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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:

1 Assoc Prof Dr Sc Phan Thi Ha Duong

2 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, 2020PhD Student

Nguyen Huy Truong

Supervisors

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

Page

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 38

CHAPTER 3 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 4 AUTOMATA TECHNIQUE FOR THE LONGEST COMMON SUBSEQUENCE PROBLEM 57

4.1 Introduction 57

4.2 Mathematical Basis 58

4.3 Automata Models for Solving The LCS Problem 62

4.4 Experimental Results 67

4.5 Conclusions 68

CHAPTER 5 CRYPTOGRAPHY BASED ON STEGANOGRAPHY AND AUTOMATA METHODS FOR SEARCHABLE ENCRYPTION 69

5.1 Introduction 69

5.2 A Novel Cryptosystem Based on The Data Hiding Scheme (2, 9, 8) 71

5.3 Automata Technique for Exact Pattern Matching on Encrypted Data 75

5.4 Automata Technique for Approximate Pattern Matching on Encrypted Data 77 5.5 Conclusions 79

CONCLUSION 81

LIST OF PUBLICATIONS 82

BIBLIOGRAPHY 83

LIST OF SYMBOLS

where p is prime and m is a positive integer

(I, M, K, Em, Ex) A data hiding scheme

block

from an image block

qcolour The number of different ways to change the colour of each

pixel in an arbitrary image block

LIST OF ABBREVIATIONS

LIST OF FIGURES

Figure 1.1 A simple graph 5

Figure 1.2 A spanning tree of the graph given in Figure 1.1 6

Figure 1.3 The transition diagram of A in Example 1.3 7

Figure 1.4 The basic diagram of digital image steganography 9

Figure 1.5 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 36

Figure 2.3 The binary cover image sized 2592 × 1456 pixels 36

Figure 3.1 Sliding window mechanism 41

Figure 3.2 The basic idea of the proposed approach 45

Figure 3.3 The transition diagram of the automaton Mp, p = abcba 47

LIST OF TABLES

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 30

Table 2.2 Operations + and · on the Galois field GF (22) 30

Table 2.3 The representation of E and the arc weights of G for the gray image 31 Table 2.4 The payload, ER and PSNR for the optimal data hiding scheme (1, 2n− 1, n) for palette images with qcolour = 1 37

Table 2.5 The payload, ER and PSNR for the near optimal data hiding scheme (2, 9, 8) for gray images with qcolour = 3 37

Table 2.6 The payload, ER and PSNR for the near optimal data hiding scheme (2, 9, 8) for palette images with qcolour = 3 38

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 = 2n − 1, for the binary image with qcolour = 1 Time is given in second unit 38

Table 3.1 The performing steps of the MR1 algorithm 47

Table 3.2 Experimental results on rand4 problem 52

Table 3.3 Experimental results on rand8 problem 52

Table 3.4 Experimental results on rand16 problem 53

Table 3.5 Experimental results on rand32 problem 53

Table 3.6 Experimental results on rand64 problem 54

Table 3.7 Experimental results on rand128 problem 54

Table 3.8 Experimental results on rand256 problem 55

Table 3.9 Experimental results on a genome sequence (with |Σ| = 4) 55

Table 3.10 Experimental results on a protein sequence (with |Σ| = 20) 56

Table 4.1 The Refp of p = bacdabcad 60

Table 4.2 The comparisons of the lcs(p, x) computation time for n = 50666 67

Table 4.3 The comparisons of the lcs(p, x) computation time for n = 102398 68

In the modern life, when the use of computer and Internet is more and more essential,

information security becomes increasingly important There are two popular methods to

provide security, which are cryptography and data hiding [2, 5, 6, 20, 56, 62, 81]

Cryptography is used to encrypt data in order to make the data unreadable by a third

party [5] Data hiding is used to embed data in digital media Based on the purpose of

the application, data hiding is generally divided into steganography that hides the

existence of data to protect the embedded data and watermarking that protects the

copyright ownership and authentication of the digital media carrying the embedded data

integrating cryptography with steganography is as a third choice for data security

[2, 5, 6, 12, 19, 61, 62, 81, 86, 93]

With the rapid development of applications based on Internet infrastructure, cloud

computing becomes one of the hottest topics in the information technology area Indeed, it

is a computing system based on Internet that provides on-demand services from application

and system software, storage to processing data For example, when cloud users use the

storage service, they can upload information to the servers and then access it on the Internet

online Meanwhile, enterprises can not spend big money on maintaining and owning a

benefits for individuals and organizations, cloud security is still an open problem when cloud

providers can abuse their information and cloud users lose control of it Thus, guaranteeing

privacy of tenants’ information without negating the benefits of cloud computing seems

necessary [28, 38, 40, 41, 60, 95, 102] In order to protect cloud users’ privacy, sensitive

data need to be encoded before outsourcing them to servers Unfortunately, encryption

makes the servers perform search on ciphertext much more difficult than on plaintext To

solve this problem, many searchable encryption techniques have been presented since 2000

Searchable encryption does not only store users’ encrypted data securely but also allows

information search over ciphertext [26, 28, 29, 38, 40, 60, 71, 85, 102]

Searchable encryption for exact pattern matching is a new class of searchable encryption

techniques The solutions for this class have been presented based on algorithms for [26]

or approaches to [41, 89] exact pattern matching

As in retrieving information from plaintexts, the development of searchable encryption

with approximate string matching capability is necessary, where the search string can

be a keyword determined, encrypted and stored in cloud servers or an arbitrary pattern

[28, 40, 71]

From the above problems, together with the high efficiency of techniques using graph and

automata proposed by P T Huy et al for dealing with problems of exact pattern matching

(2002), longest common subsequence (2002) and steganography (2011, 2012 and 2013), as

well as potential applications of graph theory and automata approaches suggested by Late

Assoc Prof Phan Trung Huy in steganography and searchable encryption, and under

the direction of supervisors, the dissertation title assigned is research on development

of methods of graph theory and automata in steganography and searchable

encryption

The purpose of the dissertation is to research on the development of new and quality

solutions using graph theory and automata, suggesting their applications in, and applying

them to steganography and searchable encryption

Based on results published and suggestions presented by Late Assoc Prof Phan Trung

Huy in steganography and searchable encryption, the dissertation will focus on following

four problems in these fields:

- Digital image steganography;

- Exact pattern matching;

- Longest common subsequence;

- Searchable encryption

The first problem is stated newly in Chapter 2, the three remaining problems are recalled

and clarified in Chapter 1 In addition, background related to these problems is presented

clearly and analysed very carefully in Chapters of the dissertation

For the first three problems, the dissertation’s work is to find new and efficient solutions

using graph theory and automata Then they will be used and applied to solve the last

problem

Introduction at the beginning and Conclusion at the end of the dissertation, the main

content of it is divided into five chapters

throughout the dissertation (strings, graph, deterministic finite automata, digital images,

the basic model of digital image steganography, some parameters to determine the

quality of digital image steganography, the exact pattern matching problem, the longest

concepts and results used and researched on development in remaining chapters of the

dissertation (adjacency list, breadth first search, Galois field, the fastest optimal parity

assignment method, the module method and the concept of the maximal secret data

ratio, the concept of the degree of fuzziness (appearance), the Knapsack Shaking

approach, and the definition of a cryptosystem)

graph theory and automata Firstly, from some proposed concepts of optimal and

near optimal secret data hiding schemes, this chapter states the interest problem in digital

image steganography Secondly, the chapter proposes a new approach based on the Galois

field using graph theory and automata to design a general form of steganography in binary,

gray and palette images, shows sufficient conditions for existence and proves existence of

some optimal and near optimal secret data hiding schemes, applies the proposed schemes

to the process of hiding a finite sequence of secret data in an image and gives security

analyses Finally, the chapter presents experimental results to show the efficiency of the

proposed results

Chapter 3 An automata approach to exact pattern matching This chapter

proposes a flexible approach using automata to design an effective algorithm for exact

pattern matching in practice In given cases of patterns and alphabets, the efficiency of

the proposed algorithm is shown by theoretical analyses and experimental results

Chapter 4 Automata technique for the longest common subsequence

computing the length of a longest common subsequence of two strings in practice, using

automata technique Theoretical analysis of parallel algorithm and experimental results

confirm that the use of the automata technique in designing algorithms for solving the

longest common subsequence problem is the best choice

Chapter 5 Cryptography based on steganography and automata methods

for searchable encryption This chapter first proposes a novel cryptosystem based on

a data hiding scheme proposed in Chapter 2 with high security Additionally, ciphertexts

do not depend on the input image size as existing hybrid techniques of cryptography and

steganography, encoding and embedding are done at once The chapter then applies results

using automata technique of Chapters 3 and 4 to constructing two algorithms for exact

and approximate pattern matching on secret data encrypted by the proposed cryptosystem

These algorithms have O(n) time complexity in the worst case, together with an assumption

that the approximate algorithm uses d(1 − )me processors, where , m and n are the error

of the string similarity measure proposed in this chapter and lengths of the pattern and

secret data, respectively In searchable encryption, the cryptosystem can be used to encode

and decode secret data on users side and pattern matching algorithms can be used to

perform pattern search on cloud providers side

The contents of the dissertation are written based on the paper [T1] published in 2019,

the paper [T4] accepted for publication in 2020 in KSII Transactions on Internet and

Information Systems (ISI), and the papers [T2, T3] published in Journal of Computer

Science and Cybernetics in 2019 The main results of the dissertation have been presented

at:

- Seminar on Mathematical Foundations for Computer Science at Institute of

Mathematics, Vietnam Academy of Science and Technology,

- Seminar at School of Applied Mathematics and Informatics, Hanoi University of

Science and Technology

CHAPTER 1

PRELIMINARIES

This chapter will attempt to recall terminologies, concepts, algorithms and results which

are really needed in order to present the dissertation’s new results clearly and logically,

knowledge re-presented here consists of basic structures (Section 1.1: strings (Subsection

1.1.1), graph (Subsection 1.1.2), deterministic finite automata (Subsection 1.1.3), and the

pattern matching (Section 1.3), longest common subsequence (Section 1.4) and searchable

encryption (Section 1.5)

1.1 Basic Structures

1.1.1 Strings

In this dissertation, secret data are considered as strings So, some terms related to

strings will be recalled here [11, 24, 83]

A finite set Σ is called an alphabet The number of elements of Σ is denoted by |Σ|

An element of Σ is called a letter A string (also referred to as a text) x of length n on the

alphabet Σ is a finite sequence of letters of Σ and we write

x = x[1]x[2] x[n], x[i] ∈ Σ, 1 ≤ i ≤ n,where n is a positive integer

A special string is the empty string having no letters, denoted by  The length of the

string x is the number of letters in it, denoted by |x| Then || = 0

Notice that for the string x = x[1]x[2] x[n], we can also write x = x[1 n] in short

of the string x The prefix (resp suffix) p is called proper if p 6= x Note that the prefix

or the suffix can be empty

1.1.2 Graph

Besides some basic concepts in graph theory, this subsection recalls the way representing

a graph by adjacency lists and breadth first search [82] These are used in Chapter 2

A finite undirected graph (hereafter, called a graph for short) G = (V, E) consists of a

nonempty finite set of vertices V and a finite set of edges, where each edge has either one

or two vertices associated with it A graph with weights assigned to their edges is called a

weighted graph

An edge connecting a vertex to itself is called a loop Multiple edges are edges connecting

the same vertices A graph having no loops and no multiple edges is called a simple graph

In a simple graph, the edge associated to an unordered pair of vertices {i, j} is called the

edge {i, j}

Two vertices i and j in a graph G are called adjacent if they are vertices of an edge of

G

A graph without multiple edges can be described by using adjacency lists, which specify

adjacent vertices of any vertex of the graph

Example 1.1 Using adjacency lists, the simple graph given in Figure 1.1 can be

Stego Image

Cover Image

Given a simple graph G, a subgraph of G that is a tree including every vertex of G is

called a spanning tree of G A spanning tree of a connected simple graph can be built by

using breadth first search (BFS) This algorithm is shown in pseudo-code as follows

Breadth First Search:

Output: A spanning tree T

Set L to be an empty list;

For each adjacent vertex j of i

If (j is not in L and T ){

Add j to the end of L;

Add j and the edge {i, j} to T ;}

}Return T ;End

Example 1.2 For a graph given in Figure 1.1, a spanning tree of this graph is found byusing BFS as in Figure 1.2

Stego Image

Cover Image

Figure 1.2 A spanning tree of the graph given in Figure 1.1

A graph with directed edges (or arcs) is called a directed graph Each arc is associatedwith the ordered pair of vertices In a simple directed graph, the arc associated with theordered pair (i, j) called the arc (i, j) And the vertex i is said to be adjacent to the vertex

j and the vertex j is said to be adjacent from the vertex i

1.1.3 Deterministic Finite AutomataStudy on the problem of the construction and the use of deterministic finite automata

is one of objectives of the dissertation Hence, this subsection will clarify this model ofcomputation [44, 82]

Definition 1.1 ([44]) Let Σ be an alphabet A deterministic finite automaton (hereafter,

• A finite set Q of elements called states,

• A set F of final states The set F is a subset of Q,

• A state transition function (or simply, transition function), denoted by δ, that takes

as arguments a state and a letter, and returns a state, so that δ : Q × Σ → Q,

• The transition function δ can be extended so that it takes a state and a string, andreturns a state Formally, this extended transition function δ can be defined recursively by

luan van hay luan van tot nghiep do an to nghiep docx 123docz luan van hay luan van tot nghiep do an to nghiep docx 123docz luan van hay luan van tot nghiep do an to nghiep docx 123docz luan van hay luan van tot nghiep

An alternative and simple way presenting an automaton is to use the notation “transition

given as follows [44]

a) Each state of Q is a vertex

vertex

d) States not in F have a single circle Vertices corresponding to final states are marked

by a double circle

diagram of A is shown in Figure 1.3

Stego Image

Cover Image

Figure 1.3 The transition diagram of A in Example 1.3

a state in F

will be used in Chapter 2

in the usual way and then reduce the coefficients modulo p at the end

7

polynomials f1(x), f2(x) ∈ Zp[x] such that

unique

1.2 Digital Image Steganography

The interest problem in Chapter 2 is digital image steganography This section will

recall the concept of digital images, the basic model of digital image steganography, some

parameters to determine the efficiency of digital image steganography and lastly re-present

results researched on development and used in Chapter 2 such as the fastest optimal parity

assignment (FOPA) method, the module method and the concept of the maximal secret

data ratio (MSDR) [18, 20, 21, 39, 49, 50, 51, 53, 61, 63, 65, 76, 78, 104]

A digital image is a matrix of pixels Each pixel is represented by a non negative integer

number in the form of a string of binary bits This value indicates the colour of the pixel

[39]

Note that based on the way representing of colours of pixels, digital images can be

divided into following different types [78]

1 Binary image: Each pixel is represented by one bit In this image type, the colour of

a pixel is white, “1” value, or black, “0” value

2 Gray image: Each pixel is typically represented by eight bits (called 8-bit gray image)

Then the colour of any pixel is a shade of gray, from black corresponding to colour value

“0” to white corresponding to colour value “255”

3 Red green blue image: Each pixel is usually represented by a string of 24 bits (called

24-bit RGB image), where the first 8 bits, the next 8 bits and the last 8 bits corresponds

to shades of red, green and blue, specifying the red, green and blue colour components

of the pixel, respectively Then the colour of the pixel is a combination of these three

components

representing the pixel as for RGB images Instead, this number is a colour index of the

colour of the pixel existed in the colour table (the palette), an ordered set of values (strings

of 24 bits) which represent all colours as in RGB images used in the image and contained

in the file with the image The size of the palette is the same as the length of a bit string

representing a pixel and is limited by 8 bits For a string of 8 bits, call palette images 8-bit

palette images

The objective of digital image steganography is to protect data by hiding the data in

a digital image well enough so that unauthorized users will not even be aware of their

existence [21, 18] Figure 1.4 shows the basic model of digital image steganography, where

the cover image is a digital image used as a carrier to embed secret data into, the stego

image is digital image obtained after embedding secret data into the cover image by the

function block Embed with the secret key on the Sender side For steganography generally,

the secret data needs to be extracted fully by the block Extract with the secret key on

the Receiver side [20, 61, 63, 76]

The total number of the secret data sequence bits embedded in the cover image is called

a Payload Corresponding to a certain Payload, to measure the embedding capacity of the

cover image, the embedding rate (ER) is used and defined as follows [104]

Stego Image

Cover Image

Figure 1.4 The basic diagram of digital image steganography

The peak signal to noise ratio (PSNR) is used to evaluate quality of stego image Based

on the value of PSNR, we can know the degree of similarity between the cover image and

stego image If the PSNR value is high, then quality of stego image is high Conversely,

quality of stego image is low In general, for the digital image, PSNR is defined by the

Green and Red components of a pixel at position (i, j) in the cover and stego image,

respectively For human’s eyes, the threshold value of PSNR value is 30dB [20, 53, 65, 104],

it means that the PSNR value is higher than 30dB, it is hard to distinguish between the

cover image and its stego image

of a pixel of G corresponding to the colour index i Each colour c in P is considered as a

vector consisting of red, green and blue components Suppose d is a distance function on P

where two conditions are satisfied for all c ∈ P as follows

1 d(c, Next(c)) = minv6=c∈Pd(c, v),

all arcs (v, Next(v)), the vertex v has the weightVal(v) for all v ∈ V The construction of

a algorithm determining F is the essence of the FOPA method

Algorithm for FOPA:

Output: A rho forest F = (V, E)

Choose a vertext c ∈ P , set V = {c}, and set C = P \{c};

SetVal(c) = 0; // Or 1 randomly

While (C is not empty) // Update F

{

a) Take one element v ∈ C;

h is a surjective function from I to U In the module method, d is considered as a secret

data, embedded in and extracted from the image block I with the key K by the blocks

Embed and Extract as follows [49, 51]

The block Embed (embedding d in I):

Step 2) Case d = m: Keep I intact;

Case d 6= m: Find v ∈ U such that d + (−m) = v Based on v and h, determine

in an image block of N pixels by changing colours of at most k pixels in the image block,

where k, N are positive integers

colour of each pixel in an arbitrary image block of N pixels According to [49]

1.3 Exact Pattern Matching

This section will restate the exact pattern matching problem, and recall the concept of

the degree of fuzziness (appearance) used in Chapter 3 [24, 52, 68]

Let x be a string of length n Denote the substring x[i]x[i + 1] x[j] of x by x[i j]

p be a substring of length m of x, where m is a positive integer, then there exists i for

1 ≤ i ≤ n − m + 1 such that p = x[i i + m − 1] And say that i is an occurrence of p in x

or p occurs in x at position i

Definition 1.5 ([68]) Let p be a pattern of length m and x be a text of length n over

the alphabet Σ Then the exact pattern matching problem is to find all occurrences of the

pattern p in x

The following example uses the Brute Force (BF) algorithm [24] to demonstrate the

most original way solving this problem

Table 1.2 The performing steps of the BF algorithm

Example 1.4 Given a pattern p = fah and a text x = dfahfkfaha Then there are two

occurrences of p in x as shown below: dfahfkfaha The BF algorithm is performed by the

following steps presented in Table 1.2, the bold letters correspond to the mismatches, the

underlined letters represent the matches when comparing the letters of the pattern and

the text We know that many letters scanned will be scanned again by the BF algorithm

because each time either a mismatch or a match occurs, the pattern is only moved to the

right one position

Chapter 3 uses the degree of fuzziness in [52] to determine the longest prefix of the

fuzziness will be replaced with the degree of appearance The concept of the degree of

appearance is restated as follows

Definition 1.6 ([52]) Let p be a pattern and x be a text of length n over the alphabet

Σ Then for each 1 ≤ i ≤ n, a degree of appearance of p in x at position i is equal to the

length of a longest substring of x such that this substring is a prefix of p, where the right

end letter of the substring is x[i]

Notice that obviously, if the degree of appearance of p in x at an arbitrary position i

equals |p|, then a match for p in x occurs at position i − |p| + 1 Figure 1.3 illustrates the

concept of the degree of appearance of the pattern p in x

Figure 1.5 The degree of appearance of the pattern p

1.4 Longest Common Subsequence

This section will recall the longest common subsequence (LCS) problem, and the

Knapsack Shaking approach addressing the problem studied on development in Chapter 4

[24, 47, 94, 101]

Definition 1.7 ([101]) Let p be a string of length m and u be a string over the alphabet

Definition 1.8 ([101]) Let u, p and x be strings over the alphabet Σ Then u is a common

subsequence of p and x if u is a subsequence of p and a subsequence of x

Definition 1.9 ([101]) Let u, p and x be strings over the alphabet Σ Then u is a longest

common subsequence of p and x if two following conditions are satisfied

(i) u is a common subsequence of p and x,

(ii) There does not exist any common subsequence v of p and x such that |v| > |u|

Denote an arbitrary longest common subsequence of p and x by LCS(p, x) The length

of a LCS(p, x) is denoted by lcs(p, x)

By convention, if two strings p and x does not have any longest common subsequences,

then the lcs(p, x) is considered to equal 0

Example 1.5 Let p = bgcadb and x = abhcbad Then string bcad is a LCS(p, x) and

lcs(p, x) = 4

Let p and x be two strings of lengths m and n over the alphabet Σ, m ≤ n The longest

common subsequence problem for two strings (LCS problem) can be stated in two following

forms [24, 47]

Problem 1 Find a longest common subsequence of p and x

Problem 2 Compute the length of a longest common subsequence of p and x

The simple way to solve the LCS problem is to use the algorithm introduced by

Wagner and Fischer in 1974 (called the Algorithm WF) This algorithm defines a dynamic

programming matrix L(m, n) recursively to find a LCS(p, x) and compute the lcs(p, x) as

where L(i, j) is the lcs(p[1 i], x[1 j]) for 1 ≤ i ≤ m, 1 ≤ j ≤ n

Example 1.6 Let p = bgcadb and x = abhcbad Use the Algorithm WF, the L(m, n)

is obtained below Then lcs(p, x) = L(6, 7) = 4 In Table 1.3, by traceback procedure,

starting from value 4 back to value 1, a LCS(p, x) found is a string bcad

Table 1.3 The dynamic programming matrix L

From Definition 1.10, the subsequence u has at least a location in p If all the different

locations of u are arranged in the dictionary order, then call the least element the leftmost

[47]

Example 1.7 Let p = aabcadabcd and u = abd Then u is a subsequence of p and has

seven different locations in p, in the dictionary order they are

(1, 3, 6), (1, 3, 10), (1, 8, 10), (2, 3, 6), (2, 3, 10), (5, 8, 10), (7, 8, 10)

Definition 1.11 ([47]) Let p be a string of length m Then a configuration C of p is

defined as follows

1 Or C is the empty set Then C is called the empty configuration of p, denoted by

that the two following conditions are satisfied

Set of all the configurations of p is denoted by Config(p)

Definition 1.12 ([47]) Let p be a string of length m on the alphabet Σ, C ∈ Config(p)

ϕ : Config(p) × Σ → Config(p) defined as follows

determined by a loop using the loop control variable i whose value is changed from t down

to 0:

a) For i = t, if the letter a appears at a location index in p such that index is greater

b) Loop from i = t − 1 down to 1, if the letter a appears at a location index in p such

c) For i = 0, if the letter a appears at a location index in p such that index is smaller

4 To accept an input string, the state transition function ϕ is extended as follows

Example 1.8 Let p = bacdabcad and C = {c, ad, bab} Then C is a configuration of p

In 2002, P T Huy et al introduced a method to solve the Problem 1 by using the

automaton given as in the following theorem In this way, they named their method the

Knapsack Shaking approach [47]

Theorem 1.1 ([47]) Let p and x be two strings of lengths m and n over the alphabet

Σ, where

• The set of states Q = Config(p),

• The initial state q0 = C0,

• The transition function ϕ is given as in Definition 1.12,

following conditions are satisfied

1.5 Searchable Encryption

This section clarifies the term of searchable encryption (SE) and recalls the definition

of a cryptosystem They will be studied and used in Chapter 5 [26, 40, 60, 85, 88, 102]

Consider a problem to occur in cloud security as follows [60, 85, 102] Cloud tenants, for

example enterprises and individuals with limited resource including software and hardware,

store data with sensitive information on cloud servers Assume that these servers cannot

be fully trusted This means they may not only be curious about the users’ information

but also abuse the data received Then users wish to encrypt their data before uploading

them to servers Because of limitations of cloud users’ information technology system,

users also wish that cloud providers can help them perform information search directly

on ciphertexts However, encryption brings difficulties for servers to do search on the

encrypted data These lead to a problem that is to find a solution to satisfy the two wishes

of cloud users when they choose cloud storage service

main components, a cryptosystem is used to encode and decode on cloud users side and

algorithms for searching on encrypted data are done on cloud providers side [40, 102]

In cryptography, SE can be either searchable symmetric encryption (SSE) or searchable

asymmetric encryption (SAE) In SSE, only private key holders can create encrypted data

ciphertexts but only private key holders can generate trapdoors [26, 102]

Since the dissertation proposes a new symmetric encryption system for SSE in Chapter

5, the correctness of this system needs to prove In this dissertation, the components and

properties of a cryptosystem defined in [88] will be considered as a standard form to verify

Here recalls this definition

Definition 1.13 ([88]) A cryptosystem is a five-tuple (P, C, K, E , D) such that the

following properties are satisfied

1 P is a finite set of plaintexts,

2 C is a finite set of ciphertexts,

3 K is a finite set of secret keys,

each x ∈ P

CHAPTER 2

DIGITAL IMAGE STEGANOGRAPHY BASED ON

THE GALOIS FIELD USING GRAPH THEORY

AND AUTOMATA

This chapter first proposes concepts of optimal and near optimal secret data hiding

schemes The chapter then proposes a new digital image steganography approach based

assumptions, where k, m, n, N are positive integers and p is prime, shows sufficient

conditions for existence and proves existence of some optimal and near optimal secret

data hiding schemes These results are derived from the concept of the maximal secret

data ratio of embedded bits, the module method and the FOPA method proposed by

application of the schemes to the process of hiding a finite sequence of secret data in an

image is also considered Security analyses and experimental results confirm that the

proposed approach can create steganographic schemes which achieve high efficiency in

embedding capacity, visual quality, speed as well as security, which are key properties of

steganography

The results of Chapter 2 have been published in [T1]

2.1 Introduction

In steganography, depend on the type of digital media there are many types of

steganography such as image, audio and video steganography [4, 5, 20, 61, 62, 75, 76, 96]

However, image steganography is used the most popularly because digital images are

often transmitted on Internet and they have high degree of redundancy Furthermore, the

technique of image steganography is mainly image steganography in spatial domain,

steganography is achieved by changing colours of some pixels directly in the image

[17, 57, 62, 76, 100] The chapter’s work focuses on steganography in digital images in

spatial domain

Digital image steganography studies the steganographic schemes, where each scheme

consists of an embedding function and extracting function The embedding function shows

how to embed secret data in the digital image and the extraction function describes how

to extract the data from the digital image carrying the embedded data [46, 87]

In digital image steganography, a few main factors must be taken in consideration when

we design a new secret data hiding scheme, which are embedding capacity of the cover

image, quality of stego image and security However, as well known, embedding capacity

of the cover image and quality of its stego image are irreconcilable conflict A balance

achieved of the two factors can be done according to different application requirements In

addition to the three main factors, speed of the embedding and extracting functions also

plays an important role in steganographic schemes It is considered as a last constraint to

determine efficiency of schemes [46, 53, 65, 69, 87, 104]

The simplest and most popular spatial domain image steganography method is the least

significant bit (LSB) substitution (called LSB based method) For 24-bit RGB and 8-bit

gray images, in this method the data is embedded in the cover image by changing the least

significant bits of the image directly, therefore it becomes vulnerable to security attacks

[18, 62, 72, 75, 76, 97, 104] EZ Stego method for palette images is similar to the commonly

used LSB based method However, this method does not guarantee quality of stego images

[36, 37, 97] To alleviate this problem, in 1999, Fridrich proposed a new method based

on the parity bits of colour indexes of pixels in palette cover images, called the parity

assignment (PA) method Then EZ Stego method can be considered as an example of

PA method [36, 50] In 2000, Fridrich et al improved the method by investigating the

problem of optimal parity assignment for the palette and this version is called the optimal

parity assignment (OPA) method [37] To easily control quality of stego images, Huy et

al introduced another OPA method, called the FOPA method, in 2013 [50] Unlike the

colour and gray images, each pixel in binary images only requires one bit to represent colour

values (black and white), therefore, modifying pixels can be easily detected So, binary

block based method is usually used to maintain quality of stego images In this method,

the cover and stego images are partitioned into individual image blocks of the same size,

embedding and extracting secret data are based on the characteristic values calculated for

the blocks WL (Wu et al., 1998), PCT (Pan et al., 2000), modified PCT (Tseng et al.,

2001), CTL (Chang et al., 2005) schemes are all well known and block based for binary

images [21, 18, 48, 75, 92]

an arbitrary image block, and use the concept of the maximal secret data ratio of embedded

bits proposed by Huy et al in 2011 [49], the chapter introduces concepts of optimal

and near optimal secret data hiding schemes Actually, the optimality of steganographic

schemes has been considered in [37, 46] However, the authors used the time complexity

of embedding and extracting functions, or the concept of optimal parity assignment that

minimizes the energy of the parity assignment for the colour palette to determine whether

a steganographic scheme is optimal

By the block based method, call a secret data hiding scheme a data hiding scheme

(k, N, r), where k, N, r are positive integers, if the embedding function can embed r bits

of secret data in each image block of N pixels by changing colours of at most k pixels in

the image block The chapter’s work is concerned with the problem of designing optimal

or near optimal data hiding schemes (k, N, r) for digital images (binary, gray and palette

images)

Based on the module approach and the (FOPA) method using graph theory proposed

by Huy et al in 2011 and 2013 [49, 50], the chapter proposes a new approach based on the

Galois field using graph and automata in order to solve the problem By this approach,

optimal data hiding scheme (2, 9, 8) and the optimal data hiding scheme (1, 5, 4) for gray

schemes to the process of hiding a finite sequence of secret data in an image can avoid

detection from brute-force attacks.

The experimental results reveal that the efficiency in embedding capacity and visual

indeed better than the efficiency of the HCIH scheme [104] The embedding and extracting

time of the proposed approach are faster than that of the Chang et al.’s approach [18] For

can be selected suitably to achieve acceptable quality of the stego images

The rest of the chapter is organized as follows Section 2.2 gives some new concepts

and states the chapter’s digital image steganography problem Section 2.3 consists of two

Subsections 2.3.1 and 2.3.2 Subsection 2.3.1 introduces mathematical basis based on the

m is a positive integer Subsection 2.3.2 firstly proposes a digital image steganography

k, m, n, N are positive integers and p is prime Secondly, the subsection gives sufficient

Subsection 2.4 proves that there exist the near optimal data hiding scheme (2, 9, 8) and

Section 2.5 shows experimental results in order to evaluate the efficiency of the proposed

data hiding schemes and approach Lastly, some conclusions are drawn from the proposed

approach and experimental results in Section 2.6

2.2 The Digital Image Steganography Problem

This section gives some new concepts and states the chapter’s digital image

steganography problem

Definition 2.1 A block based secure data hiding scheme in digital images (for short, called

a data hiding scheme) is a five-tuple (I, M, K, Em, Ex), where the following conditions are

satisfied

1 I is a set of all image blocks with the same size and image type,

2 M is a finite set of secret elements,

3 K is a finite set of secret keys,

Definition 2.2 A data hiding scheme (I, M, K, Em, Ex) is called a data hiding scheme

(k, N, r), where k, N, r are positive integers, if each image block in I has N pixels and the

embedding function Em can embed r bits of secret data in an arbitrary image block by

changing colours of at most k pixels in the image block

The chapter’s digital image steganography problem Design optimal or near optimal

data hiding schemes (k, N, r) for digital images (binary, gray and palette images)

2.3 A New Digital Image Steganography Approach

This section introduces mathematical basis based on the Galois field for the digital

image steganography problem (Subsection 2.3.1), proposes a digital image steganography

approach based on the Galois field using graph theory and automata to design the data

k, m, n, N are positive integers and p is prime, shows sufficient conditions for existence and

proves existence of some optimal data hiding schemes (Subsection 2.3.2) Security analyses

and an application of these data hiding schemes to the process of hiding a finite sequence

of secret data in an image are considered in Subsection 2.3.2

2.3.1 Mathematical Basis based on The Galois Field

the digital image steganography problem, where p is prime and m is a positive integer

(Propositions 2.2, 2.4 and Theorem 2.1)

i = 1, n}, where n is a positive integer, with two operations of vector addition + and scalar

multiplication · are defined as follows

Proof Suppose [x] ∩ [y] 6=

∅

and hence [x] = [y]

i=1aivi]|

integer, then S does not depend on the choice of representatives of classes

To prove that S does not depend on the choice of representatives of classes, it suffices to

So, A = B

Definition 2.7 Let V be a vector space over a field K, S ⊂ V Then S is called a

k-Generators for V , where k is a positive integer, if the two following conditions are satisfied

Proof Since S is a k-Generators for GFn(pm), then for all v, v0 ∈ S, there does not exists

The proof is complete

Proof This is deduced immediately from Lemmas 2.2 and 2.3

Propostion 2.4 Let c be the number of k-[Generators] of N elements for the set

depend on the choice of representatives of classes by Proposition 2.3, the number of ways

2.1

Graph Theory and Automata

This subsection firstly proposes a digital image steganography approach based on the

subsection gives sufficient conditions for existence of the optimal data hiding schemes

(1,ppmnm −1−1, blog2pmnc) and (2,

pm−3

2 + (pm−3)24 +2(2 blog2 pmnc −1)

qcolour = pm− 1 (Theorems 2.3 and 2.4) Thirdly, the subsection shows that there exists

qcolour = 1, where n is a positive integer (Proposition 2.6) And finally, the subsection

Security analysis (2.27))

Let I be a set of all image blocks with the same size and image type and assume that

each image block in I has N pixels, where N is a positive integer For simplicity, the

structure of an arbitrary image block I in I can be represented by

values or indexes of pixels of I

Let K be a finite set of secret keys For all K ∈ K, also assume that the structure of

the key K is the same as the structure of the image block I So, we can write

Definition 2.8 A weighted directed graph G = (V, E) is called a flip graph over the Galois

1 V = C and for all v ∈ V , the vertex v is assigned a weight by a functionVal such

Assume that we build a flip graph G = (V, E)

From the way to determine the arc set E in Definition 2.8, assume that

Definition 2.10 Let Σ2 = GFn(pm), N = {1, 2, , N }, 2N ×GF (p )\{0} - the set of all

Remark 2.1 For the case v 6= q, then v + (−q) 6= 0 Since S is a k-Generators for

0

Definition 2.11 Let I ∈ I, M ∈ M and K ∈ K The automaton A(I, M, K) is a

Remark 2.2 The set of states Q and the transition function δ given in Definition 2.11

A(I, M, K) is constructed accurately in Definition 2.11

Let an image block I ∈ I, a secret element M ∈ M, a key K ∈ K By using the

automaton A(I, M, K) and the flip graph G, two functions Em and Ex in the data hiding

scheme (I, M, K, Em, Ex) are designed as follows

The function Em (embedding M in I):

From Definition 2.1, the correctness of the data hiding scheme (I, M, K, Em, Ex) is

confirmed by the following proposition

Propostion 2.5 For all (I, M, K) ∈ I × M × K, Ex(Em(I, M, K), K) = M

implementing (2.4) we consider two cases of q:

N = |S|

graph G is built, we offer the way to construct the data hiding scheme (I, M, K, Em, Ex)

Em changes colours of at most k pixels I to embed M in I for all I ∈ I, M ∈ M by

Definition 2.10 and Statement (2.5)

Em is used to embed b ∈ B in I as follows

b will be determined accurately based on f

Since B and M are finite sets, thus to exist the injective function f , we let |B| ≤ |M|, it

bits of the secret data b can be embedded in I By Definition 2.2, the data hiding scheme

by the extracting function Ex as follows

of elements in S also affects the formula (2.11)) The number of choices for the key K

f, f : B → GF By (2.10), to decrypt the secret element M to the secret data b, we need to

for a brute force attack, an attacker has to try every possible combination of S, K and f

in the given data hiding scheme The number of combinations of S, K and f is

c(pm− 1)NN !pmNCp2mnblog2 pmnc2blog2 p mn c! (2.12)Theorem 2.3 Suppose that a flip graph G is built Then there exists the optimal data

mn− 1

So, by Definition 2.3, Theorem 2.2 and from Lines (2.1), (2.14) and (2.15), there exists the

Propostion 2.6 For n is a positive integer, there exists the optimal data hiding scheme

images as follows

• V = C and for all v ∈ V , the vertex v is assigned a weight by a functionVal such

thatVal(v) = v;

• V = C and for all v ∈ V , the vertex v is assigned a weight by a functionVal such

thatVal(v) = v mod 2;

the same weight 1

notations throughout this dissertation, here changes the name of the functionVal in the

• Consider G to be the rho forest built by the algorithm for FOPA and assign the same

weight 1 to all arcs of G However, all colours of the rho forest are replaced with their

colour indexes

By Definition 2.8, it is not difficult to verify that the graphs G for binary, gray and palette

hiding scheme CTL [18] So, Proposition 2.6 shows that the data hiding scheme CTL

is found and a flip graph G is built Then there

Proof For the assumption of the theorem, by Theorem 2.2, there exists the data hiding

Suppose the data hiding scheme (2, N, r) is optimal for qcolour = pm− 1, then

Given an image F used as a carrier to embed a secret data sequence into, partition F

Let Jump be a bijective function used to determine the order of blocks in F in the

in the decimal system Then there exists a bijective function f, f : B → GF

the proposed approach to the process of hiding D in F , use the secret key set K,

t = 1;

{

}

Propostion 2.7 For a cover image F , a secret data sequence D, a bijective function Jump,

Proof By (2.21) and (2.23), EmDF in (2.22) and ExDF in (2.24) use the same secret key

Jump(i) 6= Jump(j), it means that an arbitrary image block in F is only used at most

one time in the process of hiding By Proposition 2.5, M extracted by (2.24) is the same

the proof

Security analysis of process of hiding D in F : Assume that parameters k, N , Em, Ex,

Then for a brute force attack, an attacker has to try every possible combination of S, K,

Jump and f in the given process of hiding The number of combinations of S, K, Jump

and f is

c(pm− 1)NN !pmt1 Nt2!Cp2mnblog2 pmnc2blog2 pmnc! (2.27)

2.4 The Near Optimal and Optimal Data Hiding Schemes for

Gray and Palette Images

This section shows that there exist the near optimal data hiding scheme (2, 9, 8)

(Theorem 2.5 and Security analyses (2.45), (2.46)) and the optimal data hiding scheme

(1, 5, 4) (Corollary 2.1 and Security analyses (2.47), (2.48)) for gray and palette images

reduction modulo g(x)

Notice that the polynomial g(x) is irreducible in Z2[x] Indeed, if g(x) has factors being

different from the constant, then the factors of g(x) are only polynomials of degree 1 and

their coefficients and then denote the sequence of any polynomial’s coefficients by a binary

string and a decimal number as in Table 2.1

Table 2.2 Operations + and · on the Galois field GF (22)

Next, consider the case k = 2 and for p = m = 2 and n = 4 the data hiding

scheme (2, N, 8) exists if the hypothesis of Theorem 2.2 is satisfied, it means that find