Obfuscation of Abstract Data-Types - Rowan

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CHAPTER 6. THE MATRIX OBFUSCATED 109 Matrix (α) Matrix α ::= List (List α) ∧ (∀ mss :: Matrix α • valid mss) scale :: α → Matrix α → Matrix α add :: (Matrix α,Matrix α) → Matrix α transpose :: Matrix α → Matrix α mult :: (Matrix α,Matrix α) → Matrix α (!!!) :: Matrix α → N × N → α transpose · transpose = id transpose · add = add · cross(transpose, transpose) transpose · (scale s) = (scale s) · transpose (add (M,N)) !!! (r,c) = (M !!! (r,c)) + (N !!! (r,c)) transpose (mult (M,N)) = mult (transpose N, transpose M) add (M,N) = add (N,M) ∣ Figure 6.1: Data-type for Matrices it into four n×n matrices but by padding a matrix with zeros, this method can be adapted for more general matrices. We must ensure that when we obfuscate these operations we do not change the complexity. 6.2.1 Definitions of the operations We now define operations to match the data-type given in Figure 6.1. Two of our definitions will use the function zipWith — the definition below contains a catch-all to deal with the situation where one of the lists is empty: zipWith f (x : xs) (y : ys) = f x y : (zipWith f xs ys) zipWith f = [ ] For brevity, we define the infix operation (⊗) to correspond to zipWith ( +). We define the matrix operations functionally as follows: scale a mss = map (map (a×)) mss add (mss,nss) = zipWith (zipWith (+)) mss nss transposemss = foldr1 (⊗) (map (map wrap) mss) where wrap x = [x] mult (mss,nss) = map (row nss) mss where row nss xs = map (dotp xs) (transpose nss) dotp ms ns = sum (zipWith (×) ms ns) mss !!! (r,c) = (mss !! r) !! c
CHAPTER 6. THE MATRIX OBFUSCATED 110 We can see that using lists of lists to model matrices allows us to write succinct definitions for our matrix operations. 6.3 Splitting Matrices Since matrices are an IDT we can use the Function Splitting Theorem (Theorem 1 from Section 4.1.4). Can we express our matrix operations using functions h and φ e ? For scale (×s), we can take h = (×s) and φ 1 = id; for transpose, h = id and φ 1 (i,j) = (j,i) and for add, we can take h = (+) and φ 1 = id = φ 2 . We cannot define mult using h and φ e — in the next section, we use a split in which the components of the split of mult(A,B) are calculated using two components from the split of A and two from the split of B. For add and scale the φ functions are equal to id (as are the θ functions) and so Equation (4.5) is satisfied for any split. If, for some split sp, A ❀ B ❀ 〈A 0 , . . . ,A n−1 〉 sp 〈B 0 , . . . ,B n−1 〉 sp then add sp (A,B) = 〈add (A 0 ,B 0 ), . . . , add (A n−1 ,B n−1 )〉 sp and scale sp s A = 〈scale s A 0 , . . . , scale s A n−1 〉 sp Note that it is not surprising that scale satisfies the Function Splitting Theorem as it is defined in terms of map. 6.3.1 Splitting in squares A simple matrix split is one which splits a square matrix into four matrices — two of which are square. Using this split we can give definitions of our operations for split matrices. Suppose that we have a square matrix M r×r and choose a positive integer k such that k < n. The choice function ch(i,j) is defined as ⎧ ⎪⎨ ch(i,j) = ⎪⎩ 0 (0 ≤ i < k) ∧ (0 ≤ j < k) 1 (0 ≤ i < k) ∧ (k ≤ j < r) 2 (k ≤ i < n) ∧ (0 ≤ j < k) 3 (k ≤ i < n) ∧ (k ≤ j < r) which can be written as a single formula ch(i,j) = 2sgn (i div k) + sgn (j div k) where sgn is the signum function. The family of functions F is {f 0 ,f 1 ,f 2 ,f 3 } where f 0 = (λ(i,j).(i,j)) f 1 = (λ(i,j).(i,j − k)) f 2 = (λ(i,j).(i − k,j)) and f 3 = (λ(i,j).(i − k,j − k))
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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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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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