SECRET SHARING FOR COLOR IMAGES by Mohsen Heidarinejad A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto Copyright © 2008 by Mohsen Heidarinejad Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-44991-2 Our file Notre reference ISBN: 978-0-494-44991-2 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Plntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. 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Canada Abstract Secret Sharing For Color Images Mohsen Heidarinejad Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto 2008 Digital data security and integrity preservation are important issues in modern com­ munication systems. The confidentiality of the transmitted visual data over digital com­ munication networks is usually obtained by encryption. However, the main disadvantage of the encryption frameworks arises from their dependency on the utilized key. In this thesis, we will investigate practical methodologies to solve these challenges regarding secure visual data transmission over communication channels. Secret sharing schemes have been employed for storage and transmission of visual data. The primary motivation of this work is to propose novel secret sharing schemes for color images which can be considered as cost effective candidates for transmission of color images over communication channels. We present Secret Sharing for Visual Data (SSVD) schemes which are followed by algebraic operations to preserve the input image content. They are capable of transmission of color images over bandwidth limited communication channels. n Acknowledgements First and foremost, I would like to thank my supervisor Prof. Konstantinos N. Pla- taniotis for his detailed and constructive comments. His wide knowledge and vision on signal processing and mathematics have been of great value for me. His understanding, encouraging and personal guidance have provided a good basis for my graduate research career. He has not only been an outstanding teacher and advisor, but also a great friend. His work ethics and dedication to ensuring the success of his students are certainly ex­ ceptional. My deepest gratitude goes to him for always believing in my work. I would also like to thank him and department of Electrical and Computer Engineering for their financial support. My graduate study at University of Toronto has been a really rewarding experience. Thank you to my defence committee, Prof. Hatzinakos, Prof. Valaee, and the chair Prof. Gulak for taking the time to serve and provide useful insight. This work would have been impossible without my friends who helped me weather the storm. Specially, I would like to convey my regards to Peyman Razzaghy, Karl Martin, Hamed Hassani, Mansour Yousefi, Mohammad Shahin Mahanta, Behrouz Khoshnevis, Amirhossein Shokouh Aghaei and Azadeh Koushki. Their inspiring discussions and valu­ able suggestions have definitely had a great impact on my work. As always, I am deeply indebted to my parents, Maryam and Esmaeil and my brothers Meisam, Milad and Mohammad for their love and support throughout this degree and my life. I would like to thank them for their understanding and unwavering belief in me. Their contribution to my life is too large to fit on this page. in Contents 1 Introduction 1 1.1 Need for Security and Integrity of Information 1 1.2 The Importance of Visual Cryptography and Secret Sharing For Visual Data 5 1.3 Key Technical Challenges in Secret Sharing For Visual Data (SSVD) . 8 1.4 Thesis Contributions 9 1.4.1 SSVD Schemes With No Expansion Factor 10 1.4.2 SSVD Schemes With Reduced Expansion Factor 11 1.5 Thesis Outline 13 2 Preliminaries and Background 15 2.1 RGB Color Space 15 2.2 Permutation 17 2.3 Implementational Issues Regarding Permutation Procedure 20 2.3.1 On Adaptive Generation of Permutation Matrices 20 2.3.2 On Compression of Permutation Matrices 21 2.3.3 On Secure Transmission of Permutation Matrices 22 2.4 Preliminaries 23 2.4.1 Karnaugh Map 24 2.4.2 Bezout's Lemma and Extended Euclidean Algorithm 25 2.4.3 Maximum Distance Separable (MDS) Codes 26 iv 2.5 Chapter Summary 27 3 Prior Works 28 3.1 {k,n} Secret Sharing Scheme 28 3.2 Bit Level Based SSVD Scheme 29 3.2.1 Encryption 31 3.2.2 Decryption 33 3.3 A Review of Prior Works In VSS 34 3.4 Matrix Projection Based VSS Scheme 38 3.4.1 Encryption 38 3.4.2 Decryption 42 3.5 Chapter Summary 44 4 SSVD Schemes With No Expansion 45 4.1 Karnaugh Map Based {2,2} SSVD Schemes 45 4.1.1 Scheme A 46 Encryption 46 Decryption 49 4.1.2 Scheme B 49 Encryption 51 Decryption 52 4.1.3 Input dependent and input independent solutions 56 4.2 Number Theory Based {k, n} SSVD Scheme 57 4.2.1 Encryption of the Permuted Image 58 4.2.2 Decryption of the Permuted Image 60 4.2.3 Number Generation Algorithm 63 4.2.4 Implementational Issues and Analysis 65 On Implementation Using 8-bit Processors 65 v On Sharing of Permutation Matrices 66 On Number Theory and Secret Sharing 67 On Security Analysis 68 On {2,2} SSVD Scheme 72 On the Optimality of the Secret Sharing Scheme 72 4.3 Experimental Results 74 4.4 Chapter Summary 82 5 SSVD Schemes With Reduced Expansion 85 5.1 Algebraic SSVD Scheme For Color Images 85 5.1.1 Encryption 87 5.1.2 Decryption 88 5.1.3 On Pixel Expansion 89 5.2 A Second Generation SSVD Scheme For Color Images 92 5.2.1 Segmentation of the input image 93 5.2.2 Encryption 95 5.2.3 Decryption 96 5.2.4 On Pixel Expansion 97 5.3 Experimental Results 99 5.4 Chapter Summary 104 6 Conclusions 106 6.1 Research Summary 106 6.2 Future Work 108 A Proofs 110 B An Example of Number Generation Algorithm 113 VI Bibliography 114 vn List of Tables 1.1 Comparison of different hash functions 4 4.1 The summary of the notations in the proposed Karnaugh map based scheme B 54 4.2 Summary of schemes implemented for comparison of experimental results 74 5.1 Summary of the schemes implemented for comparison in experimental re­ sults 100 6.1 Summary of the proposed schemes in this thesis 107 vm List of Figures 1.1 System level diagram of encryption framework 2 1.2 System level diagram of data hiding procedure 5 1.3 General description of secret sharing schemes 6 2.1 Representation of each color in RGB space as a combination of its Red, Green and Blue components 16 2.2 The RGB color model mapped to a cube 17 2.3 Applying permutation on each color channel 19 3.1 Images corresponding to the bit-levels of the color image Lena: (a) k = 8 (b) jfc = 7 (c) k = 6 (d) k = 5 (e) k = 4 (f) k = 3 (g) k = 2 (h) k = 1 . 31 3.2 Pseudocode description of the encryption phase of the proposed bit-level based SSVD scheme 33 3.3 Pseudocode description of the decryption phase of the proposed bit-level based SSVD scheme 34 3.4 The proposed bit-level based method {2,2} (tested for a color image): (a) Original image (b) Share 1 (c) Share 2 (d) Decoded image 35 3.5 Pseudo-code description of the encryption phase of the proposed SSVD scheme 40 3.6 Pseudocode description of the decryption phase of the proposed SSVD scheme 42 IX 3.7 The proposed SSVD scheme in {2,2} (tested for a color image): (a) Orig­ inal image (b) Share 1 (c) Share 2 (d) Decoded image 43 4.1 The pseudo-code description of the proposed scheme A 47 4.2 The pseudo-code description of the proposed scheme B 50 4.3 A mapping of minterms on a Karnaugh map 53 4.4 System level diagram of the proposed number theory based SSVD scheme 59 4.5 Pseudocode description of the encryption phase of the proposed number theory based SSVD scheme 61 4.6 Pseudocode description of the decryption phase of the proposed number theory based SSVD scheme 62 4.7 The log2 of the number of possible values of Xj (8) as a function of 7, indicating the security of X; against exhaustive search attack 70 4.8 The proposed Karnaugh map based scheme A (tested for a color image) {2,2} 75 4.9 The proposed Karnaugh map based scheme B (tested for a color image) {2,2} 75 4.10 The proposed number theory based method (tested for a color image) {2,2}.