Obfuscation of Abstract Data-Types - Rowan

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CHAPTER 3. TECHNIQUES FOR OBFUSCATION 51 3.2.3 Operations using tuples Let us define a function cross that applies a tuple of functions to a tuple of arguments cross (f 1 ,f 2 , . . . ,f n ) (x 1 ,x 2 , . . . ,x n ) = (f 1 x 1 ,f 2 x 2 , . . . ,f n x n ) Note that we cannot write this function in Haskell without writing different definition for each n since we are not able to define arbitrary tuples (also we cannot use a list as the functions may have different types). If we want to implement this function then we could use a dependently typed language such as Cayenne [4]. So, for an operation f :: S 1 × S 2 × . . . × S n → T an obfuscation f O of f (with respect to the relevant abstraction functions) satisfies: f · cross (af S1 ,af S2 , . . . ,af Sn ) = af T · f O (3.8) If we have a conversion function cf T for the data-type T such that: cf T · af T = id then we can pre-compose (3.8) by cf T to give f O = cf T · f · cross (af S1 ,af S2 , . . . ,af Sn ) (3.9) 3.2.4 Refinement Notation If the data-type D is refined by a data-type O then we write New-Name REFINEMENT [af ] : D declaration dti 〈[f 1 ,f 2 , . . .,f n ]〉 ❀ 〈[g 1 ,g 2 , . . . ,g n ]〉 [additional operations] [additional assertions] ∣ The notation 〈[f 1 ,f 2 , . . . ,f n ]〉 ❀ 〈[g 1 ,g 2 , . . .,g n ]〉
CHAPTER 3. TECHNIQUES FOR OBFUSCATION 52 is shorthand for: f 1 ❀ g 1 ∧ f 2 ❀ g 2 ∧ . . . ∧ f n ❀ g n When we state “REFINEMENT” in a data-type we assume that we have a functional refinement and so we give the abstraction function (af ) for the refinement. We consider this refinement to be an implementation (analogous to the implementation of interfaces in object-oriented programming) rather than an overriding. This means that O still has access to the operations f 1 , . . .,f n as well as the refined operations g 1 ,g 2 , . . .,g n . The ability to access the original operations will prove useful when defining obfuscations. The type of the refined operations can be inferred from the original operations and the set of assertions for O contains the set of assertions for f 1 , . . . ,f n with the appropriate name changes. We can add extra operations to the refined data-type if we wish and if we do so, then we must give assertions for these extra operations. Section 4.3 gives an example of a data-type for lists and Section 4.3.1 shows how we can refine it. Note that as the list of assertions give away knowledge about a data-type we should ensure that we hide the assertions when implementing an obfuscated data-type. 3.3 What does it mean to be obfuscated? In Section 1.2, we saw that if P ′ is an obfuscation of P then • P ′ should have the same functionality as P. • The running time and length of P ′ should be at most polynomially larger than that of P. • P ′ should be “harder to understand” than P. Since we are going to model our operations in Haskell many of the metrics for object-oriented programs mentioned in [10] are not directly applicable. Thus we must state when we consider an operation to be obfuscated (i.e. when an operation is “harder to understand”). Suppose that we have an operation g and we obfuscate it to obtain an operation g O . Using cyclomatic complexity [37] and data-structure complexity [40] for inspiration, we could consider g O to be more obfuscated than g if: • g O uses a more complicated data structure than g • g O has more guards in the definition than g When considering what an operation does we often see what properties the operation has. If an operation is obfuscated then it should be harder to prove what properties it has. For our data-types we give a list of assertions — properties that the operations of the data-type satisfy. We should make sure that the definitions of each operation satisfy the assertions and so we will have to construct a proof for each assertion. If an operation is obfuscated then it should be harder to prove that the assertions are true — we will have to qualify what it means for a proof to be “harder”.
Page 1 and 2: Obfuscation of Abstract Data-Types
Page 3 and 4: Contents I The Current View of Obfu
Page 5 and 6: 5.4.3 Third Definition . . . . . .
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
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CHAPTER 5. SETS AND THE SPLITTING 1
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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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Obfuscation of Abstract Data-Types - Rowan

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