Nghiên cứu phát triển các phương pháp của lý thuyết đồ thị và otomat trong giấu tin mật và mã hóa tìm kiếm

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:
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, 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
m
1.1.4 The Galois Field GF (p ) . . . . . . . . . . . . . . . . . . . . . . .
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
i

4.2
4.3
4.4
4.5

Mathematical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Automata Models for Solving The LCS Problem . . . . . . . . . . . . . . .
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CHAPTER 5 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

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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 . . . .

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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 5
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 = 1 . . . . . . . . . . . . .
Table 2.5. The payload, ER and PSNR for the near optimal data hiding scheme
(2, 9, 8) for gray images with qcolour = 3 . . . . . . . . . . . . . . . . . .
Table 2.6. The payload, ER and PSNR for the near optimal data hiding scheme
(2, 9, 8) for palette images with qcolour = 3 . . . . . . . . . . . . . . . .
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 = 1. 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)

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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 . .

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vi

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31


INTRODUCTION
In the modern life, when the use of computer and Internet is more and more essential,
digital data (information) can be copied as well as accessed illegally. As a result,
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.
Steganography can be used as an alternative way to cryptography.
However,
steganography will become weak if attackers detect existence of hidden data. Hence
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
system consisting of hardware and software. Although cloud computing brings many
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 methods using graph theory and automata
proposed by P. T. Huy et al. of solving problems of exact pattern matching (2002), longest
common subsequence (2002) and steganography (2011, 2012 and 2013), and their potential
applications in steganography and searchable encryption, as well as under the direction of
supervisors, the dissertation title assigned is research on development of methods of
1


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 the results and suggestions introduced by P. T. Huy et al., the dissertation
will focus on following four problems in steganography and searchable encryption:
- 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.
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.
The dissertation has been completed with structure as follows.
Apart from
Introduction at the beginning and Conclusion at the end of the dissertation, the main
content of it is divided into five chapters.
Chapter 1.
Preliminaries. This chapter recalls basic knowledge indicated
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
common subsequence problem, and searchable encryption), re-presents important
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).
Chapter 2. Digital image steganography based on the Galois field using
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
problem. This chapter proposes two efficient sequential and parallel algorithms for
computing the length of a longest common subsequence of two strings in practice, using
automata technique. Theoretical analysis of parallel algorithm and experimental results
2


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 ⌈(1 − ǫ)m⌉ processors, where ǫ, 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,
the revised manuscript [T4] submitted to 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,
- The 9th Vietnam Mathematical Congress, Nha Trang, August 14-18, 2018,
- Seminar at School of Applied Mathematics and Informatics, Hanoi University of
Science and Technology.

3


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,
as well as help readers follow the content of the dissertation easily. The background
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
Galois field GF (pm ) (Subsection 1.1.4)), digital image steganography (Section 1.2), exact
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 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 ǫ. The length of the
string x is the number of letters in it, denoted by |x|. Then |ǫ| = 0.
Notice that for the string x = x[1]x[2]..x[n], we can also write x = x[1..n] in short.
The set of all strings on the alphabet Σ is denoted by Σ∗ . The operator of strings is
concatenation that writes strings as a compound. The concatenation of the two strings u1
and u2 is denoted by u1 u2 .
Let x be a string. A string p is called a substring of the string x, if x = u1 pu2 for some
strings u1 and u2 . In case u1 = ǫ (resp. u2 = ǫ), the string p is called a prefix (resp. suffix)
of the string x. The prefix (resp. suffix) p is called proper if p = 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.
4


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
represented as in Table 1.1.

2

4

1
5

3
Figure 1.1. A simple graph

Table 1.1. An adjacency list representation of the simple graph given in Figure 1.1

Vertex

Adjacent vertices

1

2, 3

2

1, 3, 4

3

1, 2, 4, 5

4

2, 3, 5

5

3, 4

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:
Input: A connected simple graph G with vertices ordered as i1 , i2 , . . . , in .
Output: A spanning tree T .
Set T to be a tree consisting only i1 ;
Set L to be an empty list;
Put i1 in L
While (L is not empty)
{
Remove the first vertex i from L;
5


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 by
using BFS as in Figure 1.2.

1

2

3

4

5

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 associated
with the ordered pair of vertices. In a simple directed graph, the arc associated with the
ordered 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 Automata
Study 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 of

computation [44, 82].
Definition 1.1 ([44]). Let Σ be a finite alphabet. A deterministic finite automaton
(hereafter, called an automaton for short) A = (Σ, Q, q0 , δ, F ) over Σ consists of:
• A finite set Q of elements called states,
• An initial state q0 , one of the states in Q,
• 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, and
returns a state. Formally, this extended transition function δ can be defined recursively by
δ : Q × Σ∗ → Q
such that ∀q ∈ Q, δ(q, ǫ) = q, ∀s ∈ Σ∗ , ∀a ∈ Σ, δ(q, as) = δ(δ(q, a), s).
6


An alternative and simple way presenting an automaton is to use the notation “transition
diagram”. A transition diagram of an automaton A = (Σ, Q, q0 , δ, F ) is a directed graph
given as follows [44].
a) Each state of Q is a vertex.
b) Let q ′ = δ(q, a), where q is a state of Q and a is a letter of Σ. Then the transition
diagram has an arc (q ′ , q) labeled a. If there are several letters that cause transitions from
q ′ to q, then the arc (q ′ , q) is labeled by a list of these letters.
c) There is an arrow into the initial state q0 . This arrow does not originate at any
vertex.
d) States not in F have a single circle. Vertices corresponding to final states are marked
by a double circle.
Example 1.3. Consider an automaton A = (Σ, Q, q0 , δ, F ) over Σ = {a, b}, where
Q = {q0 , q1 , q2 }, F = {q2 }, and δ is given by the following table. Then the transition
diagram of A is shown in Figure 1.3.
δ

a

b

q0

q0

q1

q1

q2

q1

q2

q2

q2

a

q0

a, b

b

q1

a

q2

b
Figure 1.3. The transition diagram of A in Example 1.3

Definition 1.2 ([82]). A string p is said to be accepted by the automaton
A = (Σ, Q, q0 , δ, F ) if it takes the initial state q0 to a final state, it means that δ(q0 , p) is
a state in F .
1.1.4 The Galois Field GF (pm )
This subsection describes how to construct a finite field with pm elements, called the
Galois field GF (pm ), where p is prime and m ≥ 1 is an integer [88]. The algebraic structure
will be used in Chapter 2.
Let p be a prime number. Define Zp [x] to be the set of all polynomials with the variable
x, whose coefficients belong to the field Zp . Addition and multiplication in Zp [x] are defined
in the usual way and then reduce the coefficients modulo p at the end.
For f (x) ∈ Zp [x], the degree of f (x), denoted by deg(f ), is the largest exponent of
x in f (x). A polynomial f (x) ∈ Zp [x] is called to be irreducible if there does not exist
7


polynomials f1 (x), f2 (x) ∈ Zp [x] such that
f (x) = f1 (x)f2 (x),
where deg(f1 ) > 0 and deg(f2 ) > 0.
Let f (x) ∈ Zp [x] be an irreducible polynomial with deg(f ) = m ≥ 1. Define
Zp [x]/(f (x)) to be the set of pm polynomials of degree at most m − 1 in Zp [x]. Addition

and multiplication in Zp [x]/(f (x)) are given as in Zp [x], followed by a reduction modulo
f (x). Then Zp [x]/(f (x)) with these operations is a field having pm elements, called the
Galois field GF (pm ). Note that for p is prime and m ≥ 1, the Galois field GF (pm ) is
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.
4. Palette image: The colour of each pixel is not shown directly by the number
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
8


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].
ER =

Payload
(bpp),
W ×H

(1.1)

where W and H are the cover image’s width and height.

Secret Data

Secret Data
Communication
Channel

Cover
Image

Embed

Stego
Image

Stego
Image

Extract

Send to

Secret Key

Secret Key

Sender

Receiver

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
following formula [20, 53]
2552
(dB),
(1.2)
PSNR = 10 log10
MSE
where
MSE =

W −1
i=0

H−1
′
2
i=0 ((B(i, j) − B (i, j))

+ (G(i, j) − G′ (i, j))2 + (R(i, j) − R′ (i, j))2 )
,
3×W ×H

where B(i, j), G(i, j), R(i, j), B ′ (i, j), G′ (i, j) and R′ (i, j) are the colour value of the Blue,
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.
Let G be a palette image and P = {c1 , c2 , . . . , cn } be its palette, where ci is the colour

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 .
The FOPA method [50] tries to get functions Next, Next: P → P , and Val, Val: P → Z2 ,
where two conditions are satisfied for all c ∈ P as follows.
9


1. d(c, Next(c)) = minv=c∈P d(c, v),
2. Val(c) =Val(Next(c)) + 1 on the field Z2 .
Call GP = (VP , EP ) a weighted complete undirected graph of the palette image G, where
VP = P and the weight of the edge {c, c′ } is d(c, c′ ). The function Nearest, Nearest: P → P ,
is given by Nearest(c) = c′ holding d(c, c′ ) = minv=c∈P d(c, v). A rho forest F = (V, E) is
a directed graph with vertices weighted by the functionVal, where V = VP , E is a set of
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:
Input: A weighted complete undirected graph GP , the function Nearest.
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;
b) Initialize v0 = v, setVal(v0 ) = 0 (or 1 randomly), by a finite loop, find a longest
sequence of k + 1 different elements in P consecutively, v0 , v1 , . . . , vk , such that
Nearest(vi ) = vi+1 for ∀i = 0, k − 1, vi ∈ C, vk ∈ C or vk ∈ V , and set
Next(vi ) = vi+1 , i = 0, k − 1;
b1) Case vk ∈ C: SetVal(vi ) = 1+Val(vi−1 ), i = 1, k and Next(vk ) = vk−1 ;
Set V = V ∪ {v0 , v1 , . . . , vk } and C = C\{v0 , v1 , . . . , vk };
b2) Case vk ∈ F : SetVal(vi ) = 1+Val(vi+1 ), i = k − 1, . . . , 1, 0;

Set V = V ∪ {v0 , v1 , . . . , vk−1 } and C = C\{v0 , v1 , . . . , vk−1 };
}
Return F ;
End.
Definition 1.3 ([51]). Let M be a module over the ring Zm , k > 0 be a natural number,
and U be a subset of M \{0}. Call U a k-base of M if for any v in M \{0}, there
exist t elements v1 , v2 , . . . , vt ∈ U, t ≤ k, together with a1 , a2 , . . . , at ∈ Zm such that
v = v1 a1 + v2 a2 + .. + vt at .
Let G be a digital image, call CG the set of all colours of pixels in G. Consider the
case m = 2 and G is a binary image. Then CG = {0, 1}, and for n is a positive integer,
the set M = Z2n = {(x1 , x2 , . . . , xn )|xi ∈ Z2 , i = 1, n} with element addition and scalar
multiplication defined as usual is a module over the ring Z2 [49]. For k = 1, the set
U = M \{0} is an unique 1-base of M [51]. Two functions Next, Next: CG → CG , and
Val, Val: CG → Z2 , satisfying the condition Val(c) =Val(Next(c)) + 1 on the ring Z2 , are
defined in [49]. Suppose that for N ≥ |U |, I = {I1 , I2 , . . . , IN } is an arbitrary image block
of G, K = {K1 , K2 , . . . , KN |Ki ∈ Z2 , i = 1, N } is a secret key, d is any element in M , and
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].
10


The block Embed (embedding d in I):
N
Step 1) Compute m = i=1 h(Ii )(Val(Ii ) + Ki );
Step 2) Case d = m: Keep I intact;
Case d = m: Find v ∈ U such that d + (−m) = v. Based on v and h, determine
an element Ii of I. Then change Ii to Ii′ = Next(Ii );
Return I ′ ;
N

The block Extract (extracting d from I ′ ): d = i=1 h(Ii′ )(Val(Ii′ ) + Ki );
Definition 1.4 ([49]). MSDR k (N ) is the largest number of embedded bits of secret data
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.
Given a positive integer qcolour , call qcolour the number of different ways to change the
colour of each pixel in an arbitrary image block of N pixels. According to [49]
1
2
2
k
k
MSDR k (N ) = ⌊log2 (1 + qcolour CN
+ qcolour
CN
+ · · · + qcolour
CN
)⌋.

(1.3)

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]
for ∀1 ≤ i ≤ j ≤ n, the ith element of x by x[i] and i is called a position in x. Let
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

Step

x

d

f

a

h

1

p

f

a

h

2

p

f

a

h

3

p

f

a

h

4

p

f

a

h

5

p

f

a

h

6

p

f

a

h

7

p

f

a

h

8

p

f

a

11

f

k

f

a

h

a

h


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

pattern in the text at any position. However, this terminology can lead to several
misunderstandings for the readers. So throughout this dissertation, the degree of
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 |x|, 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.

The degree of appearance of p in x at the position being scanned is equal to 4

.... c d b c b a d b c c ....
x
p

b c b a c
(a prefix of p)

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
Σ. Then u is a subsequence of p if there exists a integer sequence j1 , j2 , . . . , jt such that
1 ≤ j1 < j2 < . . . < jt ≤ m and u = p[j1 ]p[j2 ]..p[jt ].

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|.
12


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
follows [94].


i = 0 or j = 0,
0
L(i, j) = L(i − 1, j − 1) + 1
p[i] = x[j],




max{L(i, j − 1), L(i − 1, j)} otherwise,

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

p=
b
g
c
a
d
b

x=
i, j 0
0 0
1 0
2 0
3 0
4 0
5 0
6 0

a
1
0
0
0
0
1
1
1

b
2
0
1
1
1
1
1
2

h
3
0
1
1
1
1
1
2

c
4
0
1
1
2
2
2
2

b
5
0
1
1
2
2
2
3

a
6
0
1
1
2
3
3
3

d
7
0
1
1
2
3
4
4

Definition 1.10 ([47]). Let u = p[j1 ]p[j2 ] . . . p[jt ] be a subsequence of p. Then an element
of the form (j1 , j2 , . . . , jt ) is called a location of u in p.
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
location of u, denoted by LeftID(u). Denote the last component of LeftID(u) by Rmp (u)
[47].

13


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).
It follows that LeftID(u) = (1, 3, 6) and Rmp (u) = 6.
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
C0 .
2. Or C = {x1 , x2 , . . . , xt } is an ordered set of t subsequences of p for 1 ≤ t ≤ m such
that the two following conditions are satisfied.

(i) ∀i, 1 ≤ i ≤ t, |xi | = i,
(ii) ∀xi , xj ∈ C, if |xi | > |xj |, then Rmp (xi ) >Rmp (xj ).
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)
and a ∈ Σ. Then a state transition function ϕ on Config(p) × Σ such that
ϕ : Config(p) × Σ → Config(p) defined as follows.
1. ϕ(C, a) = C if a ∈
/ p.
2. ϕ(C0 , a) = {a} if a ∈ p.
3. Set C ′ = ϕ(C, a). Suppose a ∈ p and C = {x1 , x2 , . . . , xt } for 1 ≤ t ≤ m. Then C ′ is
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
than Rmp (xt ), then xt+1 = xt a;
b) Loop from i = t − 1 down to 1, if the letter a appears at a location index in p such
that index ∈ (Rm p (xi ), Rm p (xi+1 )), then xi+1 = xi a;
c) For i = 0, if the letter a appears at a location index in p such that index is smaller
than Rmp (x1 ), then x1 = a;
d) C ′ = C.
4. To accept an input string, the state transition function ϕ is extended as follows
ϕ : Config(p) × Σ∗ → Config(p)
such that ∀C ∈ Config(p), ∀s ∈ Σ∗ , ∀a ∈ Σ, ϕ(C, as) = ϕ(ϕ(C, a), s) and ϕ(C, ǫ) = C.
Example 1.8. Let p = bacdabcad and C = {c, ad, bab}. Then C is a configuration of p
and C ′ = ϕ(C, a) = {a, ad, ada, baba}.
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, m ≤ n. Let Ap = (Σ, Q, q0 , ϕ, F ) corresponding to p be an automaton over the alphabet
Σ, where

• The set of states Q = Config(p),
14


• The initial state q0 = C0 ,
• The transition function ϕ is given as in Definition 1.12,
• The set of final states F = {Cn }, where Cn = ϕ(q0 , x).
Suppose Cn = {x1 , x2 , . . . , xt } for 1 ≤ t ≤ m. Then
1. For every subsequence u of p and x, there exists xi ∈ Cn , 1 ≤ i ≤ t such that the two
following conditions are satisfied.
(i) |u| = |xi |,
(ii) Rm p (xi ) ≤ Rm p (u).
2. A LCS(p, x) equals xt .

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.
SE is a way to solve the above problem. It is indeed a system consisting of two
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
and produce trapdoors for search. In SAE, users who have the public key can make
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,
4. For ∀k ∈ K, there exists an encrypting function ek ∈ E and a corresponding
decrypting function dk ∈ D, where ek : P → C and dk : C → P holds dk (ek (x)) = x
with ∀x ∈ P.

15