Obfuscation of Abstract Data-Types - Rowan

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CHAPTER 7. CULTIVATING TREE OBFUSCATIONS 127 The details for this proof are in [21]. We can define modify 3 as follows: modify 3 p q Null 3 = Null 3 modify 3 p q (Fork 3 v lt ct rt) v == p ∧ mod (p−q) 2 == 0 = Fork 3 q lt ′ ct ′ rt ′ v == p ∧ mod (p−q) 2 == 1 = Fork 3 q rt ′ ct ′ lt ′ ∣ otherwise = Fork 3 v lt ′ ct ′ rt ′ where lt ′ = modify 3 p q lt rt ′ = modify 3 p q rt ct ′ = modify 3 p q ct In this definition we have collapsed some cases; for example, we can combine the predicates even p ∧ even q and odd p ∧ odd q into one predicate: mod (p−q) == 0. A fuller discussion of modify 3 can be found in [21]. 7.2.4 Making Ternary Trees When using binary trees, we defined an operation mktree to convert a list into a binary tree. We can also define a corresponding operation mktree 3 that converts a list into a ternary tree and satisfies the relationship: mktree = to2 · mktree 3 (7.8) However as with the definition of a conversion function, we have many choices of mktree 3 that satisfy the above equation. The general form of a function that satisfies Equation (7.8) is: mktree 3 [ ] = Null 3 mktree 3 xs ∣ even z = Fork 3 z (mktree 3 ys) (mktree 3 zs) jt otherwise = Fork 3 z kt (mktree 3 zs) (mktree 3 ys) where (ys, (z : zs)) = splitAt (div |xs| 2) xs where the expressions jt and kt represent arbitrary ternary trees. As an example, we define mktree 3 so that mktree 3 = to3 · mktree where to3 as defined in Section 7.2.2. This equation gives the following definition: mktree 3 [ ] = Null 3 mktree 3 xs ∣ even z = Fork 3 z (mktree 3 ys) (mktree 3 zs) (mktree’ 3 ys) otherwise = Fork 3 z (mktree’ 3 zs) (mktree 3 zs) (mktree 3 ys) where (ys, (z : zs)) = splitAt (div (|xs| 2) xs mktree’ 3 [ ] = Null 3 mktree’ 3 xs = Fork 3 (3 ∗ z + 2) (mktree 3 ys) (mktree’ 3 ys) (mktree’ 3 zs) where (ys, (z : zs)) = splitAt (div (|xs| 2) xs We can construct this function by using Equation (7.6) and so this function is specialised to the choice of conversion of binary to ternary. Computing mktree 3 [1, 5, 2, 7, 3, 6, 4]
CHAPTER 7. CULTIVATING TREE OBFUSCATIONS 128 produces the ternary tree (7.5). Since the input for this operation is a list, we can also perform an obfuscation on the input list. In Appendix D we use a split list as well as ternary trees to obfuscate the definition of mktree. We prove an assertion for different versions of mktree and we find that using ternary trees increases the cost but not the height of proof trees, but using a split list increases both. 7.2.5 Operations for Binary Search Trees Now let us consider representing binary search trees with our specific abstraction function; for a binary search tree t 2 , t 2 ❀ t 3 if t 2 = (to2 t 3 ) ∧ inc (flatten (to2 t 3 )) We define operations corresponding to memBST and insert routinely: memBST 3 p Null 3 = False memBST 3 p (Fork 3 v lt ct rt) p == v = True p < v ∧ even v = memBST 3 p lt p < v ∧ odd v = memBST 3 p rt ∣ p > v = memBST 3 p ct insert 3 x Null 3 = Fork 3 x Null 3 Null 3 Null 3 insert 3 x (Fork 3 y lt ct rt) x < y ∧ even y = Fork 3 y (insert 3 x lt) ct jt x < y ∧ odd y = Fork 3 y kt ct (insert 3 x rt) x == y = Fork 3 y lt ct rt ∣ x > y = Fork 3 y lt (insert 3 x ct) rt where jt and kt are arbitrary ternary trees. Note that the case x > y does not require a test to see whether y is even. This is because we chose to always map the right subtree of a binary tree to the centre subtree of a ternary tree. 7.2.6 Deletion To define delete for our ternary tree, we first need to define headTree and tailTree for ternary trees which satisfy analogues of Equations (7.2) and (7.3): headTree 3 = head · flatten 3 flatten 3 · tailTree 3 = tail · flatten 3 Using these equations, we obtain: headTree 3 (Fork 3 v lt ct rt) ∣ even v = if lt == Null 3 then v else headTree 3 lt otherwise = if rt == Null 3 then v else headTree 3 rt tailTree 3 (Fork 3 v lt ct rt) even v ∣ = if lt == Null 3 then ct else Fork 3 v (tailTree 3 lt) ct rt otherwise = if rt == Null 3 then ct else Fork 3 v lt ct (tailTree 3 rt)
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Obfuscation of Abstract Data-Types
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Contents I The Current View of Obfu
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5.4.3 Third Definition . . . . . .
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List of Figures 1 An example of obf
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8 public class c { d t1 = new d();
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10 • Obfuscations can be proved c
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Chapter 1 Obfuscation Scully: “No
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CHAPTER 1. OBFUSCATION 14 These thr
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CHAPTER 1. OBFUSCATION 16 by changi
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CHAPTER 1. OBFUSCATION 18 We could
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CHAPTER 1. OBFUSCATION 20 B 1 [0] =
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Chapter 2 Obfuscations for Intermed
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CHAPTER 2. OBFUSCATIONS FOR INTERME
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Chapter 3 Techniques for Obfuscatio
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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 74 C
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Page 143 and 144: BIBLIOGRAPHY 142 [43] Simon Peyton
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Page 147 and 148: APPENDIX A. LIST ASSERTION 146 Case
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Page 169 and 170: 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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