Obfuscation of Abstract Data-Types - Rowan

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II An Alternative View 43 3 Techniques for Obfuscation 44 3.1 Data-types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.2 Modelling in Haskell . . . . . . . . . . . . . . . . . . . . . 46 3.2 Obfuscation as data refinement . . . . . . . . . . . . . . . . . . . 48 3.2.1 Homogeneous operations . . . . . . . . . . . . . . . . . . . 49 3.2.2 Non-homogeneous operations . . . . . . . . . . . . . . . . 50 3.2.3 Operations using tuples . . . . . . . . . . . . . . . . . . . 51 3.2.4 Refinement Notation . . . . . . . . . . . . . . . . . . . . . 51 3.3 What does it mean to be obfuscated? . . . . . . . . . . . . . . . 52 3.4 Proof Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4.1 Measuring . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4.2 Comparing proofs . . . . . . . . . . . . . . . . . . . . . . 56 3.4.3 Changing the measurements . . . . . . . . . . . . . . . . . 59 3.4.4 Comments on the definition . . . . . . . . . . . . . . . . . 60 3.5 Folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.6 Example Data-Types . . . . . . . . . . . . . . . . . . . . . . . . . 61 III Data-Type Case Studies 62 4 Splitting Headaches 63 4.1 Generalising Splitting . . . . . . . . . . . . . . . . . . . . . . . . 63 4.1.1 Indexed Data-Types . . . . . . . . . . . . . . . . . . . . . 63 4.1.2 Defining a split . . . . . . . . . . . . . . . . . . . . . . . . 64 4.1.3 Example Splits . . . . . . . . . . . . . . . . . . . . . . . . 65 4.1.4 Splits and Operations . . . . . . . . . . . . . . . . . . . . 66 4.2 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.1 Array-Split properties . . . . . . . . . . . . . . . . . . . . 69 4.3 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3.1 List Splitting . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.2 Alternating Split . . . . . . . . . . . . . . . . . . . . . . . 73 4.3.3 Block Split . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4 Random List Splitting . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4.1 Augmented Block Split . . . . . . . . . . . . . . . . . . . 87 4.4.2 Padded Block Split . . . . . . . . . . . . . . . . . . . . . . 89 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5 Sets and the Splitting 92 5.1 A Data-Type for Sets . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2 Unordered Lists with duplicates . . . . . . . . . . . . . . . . . . . 92 5.2.1 Alternating Split . . . . . . . . . . . . . . . . . . . . . . . 94 5.3 Strictly Ordered Lists . . . . . . . . . . . . . . . . . . . . . . . . 95 5.3.1 Alternating Split . . . . . . . . . . . . . . . . . . . . . . . 96 5.3.2 Block Split . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.4 Comparing different definitions . . . . . . . . . . . . . . . . . . . 99 5.4.1 First Definition . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4.2 Second Definition . . . . . . . . . . . . . . . . . . . . . . . 101 3
5.4.3 Third Definition . . . . . . . . . . . . . . . . . . . . . . . 102 5.4.4 Last Definition . . . . . . . . . . . . . . . . . . . . . . . . 104 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6 The Matrix Obfuscated 107 6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2 Matrix data-type . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2.1 Definitions of the operations . . . . . . . . . . . . . . . . 109 6.3 Splitting Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.3.1 Splitting in squares . . . . . . . . . . . . . . . . . . . . . . 110 6.3.2 Modelling the split in Haskell . . . . . . . . . . . . . . . . 112 6.3.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.3.4 Deriving transposition . . . . . . . . . . . . . . . . . . . . 113 6.3.5 Problems with arrays . . . . . . . . . . . . . . . . . . . . 115 6.4 Other splits and operations . . . . . . . . . . . . . . . . . . . . . 116 6.4.1 Other Splits . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.4.2 A more general square splitting . . . . . . . . . . . . . . . 117 6.4.3 Extending the data-type of matrices . . . . . . . . . . . . 118 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 7 Cultivating Tree Obfuscations 119 7.1 Binary Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.1.1 Binary Search Trees . . . . . . . . . . . . . . . . . . . . . 120 7.2 Obfuscating Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2.1 Ternary Trees . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2.2 Particular Representation . . . . . . . . . . . . . . . . . . 124 7.2.3 Ternary Operations . . . . . . . . . . . . . . . . . . . . . 126 7.2.4 Making Ternary Trees . . . . . . . . . . . . . . . . . . . . 127 7.2.5 Operations for Binary Search Trees . . . . . . . . . . . . . 128 7.2.6 Deletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.3 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.3.1 Abstraction Functions . . . . . . . . . . . . . . . . . . . . 130 7.3.2 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 IV Conclusions 133 8 Conclusions and Further Work 134 8.1 Discussion of our approach to Obfuscation . . . . . . . . . . . . . 134 8.1.1 Meeting the Objectives . . . . . . . . . . . . . . . . . . . 134 8.1.2 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.1.3 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.1.4 Randomness . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.2 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.2.1 Deobfuscation . . . . . . . . . . . . . . . . . . . . . . . . . 138 8.2.2 Other techniques . . . . . . . . . . . . . . . . . . . . . . . 138 4
Page 1 and 2: Obfuscation of Abstract Data-Types
Page 3: Contents I The Current View of Obfu
Page 7 and 8: List of Figures 1 An example of obf
Page 9 and 10: 8 public class c { d t1 = new d();
Page 11 and 12: 10 • Obfuscations can be proved c
Page 13 and 14: Chapter 1 Obfuscation Scully: “No
Page 15 and 16: CHAPTER 1. OBFUSCATION 14 These thr
Page 17 and 18: CHAPTER 1. OBFUSCATION 16 by changi
Page 19 and 20: CHAPTER 1. OBFUSCATION 18 We could
Page 21 and 22: CHAPTER 1. OBFUSCATION 20 B 1 [0] =
Page 23 and 24: Chapter 2 Obfuscations for Intermed
Page 25 and 26: CHAPTER 2. OBFUSCATIONS FOR INTERME
Page 45 and 46: Chapter 3 Techniques for Obfuscatio
Page 47 and 48: CHAPTER 3. TECHNIQUES FOR OBFUSCATI
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CHAPTER 3. TECHNIQUES FOR OBFUSCATI
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Part III Data-Type Case Studies 62
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CHAPTER 4. SPLITTING HEADACHES 64 (
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CHAPTER 4. SPLITTING HEADACHES 66 a
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CHAPTER 4. SPLITTING HEADACHES 68 T
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CHAPTER 4. SPLITTING HEADACHES 70 L
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CHAPTER 4. SPLITTING HEADACHES 72 S
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CHAPTER 4. SPLITTING HEADACHES 74 C
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CHAPTER 4. SPLITTING HEADACHES 76 L
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CHAPTER 4. SPLITTING HEADACHES 78 t
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CHAPTER 4. SPLITTING HEADACHES 80 C
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CHAPTER 4. SPLITTING HEADACHES 82 4
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CHAPTER 4. SPLITTING HEADACHES 84 E
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CHAPTER 4. SPLITTING HEADACHES 86 b
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CHAPTER 4. SPLITTING HEADACHES 88 W
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CHAPTER 4. SPLITTING HEADACHES 90 T
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Chapter 5 Sets and the Splitting Bu
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CHAPTER 5. SETS AND THE SPLITTING 9
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CHAPTER 6. THE MATRIX OBFUSCATED 10
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CHAPTER 7. CULTIVATING TREE OBFUSCA
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Chapter 8 Conclusions and Further W
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CHAPTER 8. CONCLUSIONS AND FURTHER
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CHAPTER 8. CONCLUSIONS AND FURTHER
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BIBLIOGRAPHY 140 [13] Thomas H. Cor
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BIBLIOGRAPHY 142 [43] Simon Peyton
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Appendix A List assertion We want t
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APPENDIX A. LIST ASSERTION 146 Case
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APPENDIX A. LIST ASSERTION 148 For
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APPENDIX A. LIST ASSERTION 150 Case
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APPENDIX A. LIST ASSERTION 152 1 +
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APPENDIX A. LIST ASSERTION 154 0 +
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APPENDIX A. LIST ASSERTION 156 The
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APPENDIX A. LIST ASSERTION 158 The
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APPENDIX B. SET OPERATIONS 160 Now
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APPENDIX B. SET OPERATIONS 162 Proo
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APPENDIX B. SET OPERATIONS 164 B.2
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Appendix C Matrices We prove the as
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APPENDIX C. MATRICES 168 Base Case
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APPENDIX D. PROVING A TREE ASSERTIO
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APPENDIX E. OBFUSCATION EXAMPLE 186
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