SECRET IMAGE SHARING APPROACHES USING NUMBER THEORY AND CHAOTIC SYSTEMS /Bishoy Kamal Gad Sharobim

Supervisor: Heba Kamal Aslan

Thesis (M.A.)—Nile University, Egypt, 2024 .

"Includes bibliographical references"

Contents:
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVII
List of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXI
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXII
1. Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Contributions Summary . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Background and Survey 4
2.1 Evaluation Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Numerical Solvers Overview . . . . . . . . . . . . . . . . . . . 9
2.2.2 Chaotic Generators Simulation . . . . . . . . . . . . . . . . . 11
2.2.3 PRNGs Results and Tests . . . . . . . . . . . . . . . . . . . . 15
2.2.4 Fractional-Order Rossler System . . . . . . . . . . . . . . . . 19
2.2.5 Generalized Tent Map . . . . . . . . . . . . . . . . . . . . . . 20
2.2.6 Generalized Arnold Transform . . . . . . . . . . . . . . . . . . 27
2.3 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 The Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.3 Euler’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.4 Cryptographic Applications of Basic Number Theory . . . . . 41
2.4 Secure Hash Algorithm: SHA-256 . . . . . . . . . . . . . . . . . . . . 44
2.5 Secret Sharing Literature . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.5.1 VSS Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5.2 SIS Using Polynomial Interpolation . . . . . . . . . . . . . . . 52
2.5.3 XOR-Based MSIS . . . . . . . . . . . . . . . . . . . . . . . . 54
3. Number Theory Based Secret Image Sharing 56
3.1 A (k, n)-Secret Image Sharing With Steganography Using Generalized
Tent Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.1 Proposed System . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 58
3.2 An Efficient Multi-Secret Image Sharing System Based on Chinese
Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1 Proposed System . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.2 Security Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 74
IV
3.2.3 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . 80
3.2.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3 A Unified System for Encryption and Multi-Secret Image Sharing Using S-box and CRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3.1 Proposed System . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 86
4. Chaos Based Secret Image Sharing 97
4.1 Different Designs of Chaos-Based Secret Image Sharing Systems . . . 97
4.1.1 The VSS System . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.1.2 The First SIS System . . . . . . . . . . . . . . . . . . . . . . 100
4.1.3 The Second SIS System . . . . . . . . . . . . . . . . . . . . . 102
4.1.4 Results and Comparisons . . . . . . . . . . . . . . . . . . . . 103
4.2 Multi-Secret Image Sharing Using Fractional-Order Rossler System . 111
4.2.1 Proposed MSIS System . . . . . . . . . . . . . . . . . . . . . 112
4.2.2 Security Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2.3 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3 Progressive Multi-Secret Sharing of Color Images Using Lorenz Chaotic
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.3.1 Proposed MSIS system . . . . . . . . . . . . . . . . . . . . . . 127
4.3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 129
4.3.3 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5. Conclusions and Future Work 141
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Abstract:
Securing the transmission of secret information is important. When sending a
secret image to mutually suspicious receivers, they must be together to recover the
image. Secret Image Sharing (SIS) provides this feature by sending meaningless shares
to n participants, and k or more shares must be available to recover the secret image, where k ≤ n. Furthermore, Multi-Secret Image Sharing (MSIS) was introduced,
where multiple images are simultaneously shared instead of only one image. This research proposed different, simple, and efficient approaches for SIS and MSIS using the
number theory concepts, such as polynomial interpolation, and Chinese Remainder
Theorem (CRT), and various chaotic systems, such as Lorenz and Rossler systems
as Pseudo-Random Number Generator (PRNG). In this research, several tests have
shown good randomness of the different PRNGs. The proposed approaches worked
on sharing any number or type of images, such as binary, grayscale, and color images, with lossless recovery. Security analysis and comparisons with related literature
were also introduced with good results, including statistical tests such as Root Mean
Square Error (RMSE), correlation, entropy, and the National Institute of Standards
and Technology (NIST) SP-800-22 test suite. In addition, they passed several attacks, such as differential attacks, noise and crop attacks, and other tests, such as
key sensitivity tests and performance analysis, where the systems used long sensitive
system keys.
Keywords: Chaos theory, Number theory, Pseudo-Random Number Generator
(PRNG), Secret Image Sharing (SIS), Visual Secret Sharing (VSS)

Text in English, abstracts in English and Arabic