Obfuscation of Abstract Data-Types - Rowan

More documents

Recommendations

Info

CHAPTER 4. SPLITTING HEADACHES 87 Base Case Suppose that xs = [ ] unsplit A (split A d [ ]) = {definition of split A } unsplit A 〈d, [ ], [ ]〉 A = {definition of unsplit} [ ] Step Case Suppose that xs = y : ys and that ys satisfies Equation (4.26). Let n = 2d + m so that (d,m) = divMod n 2 then unsplit A (split A n (y : ys)) = {definition of split A } unsplit A (cons A m y (split d ys)) = {definition of unsplit A and Equation (4.25)} y : (unsplit A (split A d ys)) = {induction hypothesis} y : ys Consider this definition: filter A p 〈n, [ ], [ ]〉 A = 〈n, [ ], [ ]〉 A filter A p xsp = if p h then cons A b h ysp else ysp where ysp = filter A p t (b,h,t) = headTail A xsp We would like this operation to correspond to the filter operation for lists, i.e. xs ❀ xsp ⇒ filter p xs ❀ filter A p xsp To show this, we can prove that filter p ys = unsplit A (filter A p (split A n ys)) by structural induction on ys. The complexities of the augmented split operations are the same as for normal lists however most operations require some extra computations. Operations such as cons A and tail A need to perform some arithmetic and cat A and filter A need to compute headTail A . Using augmented split lists we develop two versions of the block split that allows us to introduce some randomness in the splits. 4.4.1 Augmented Block Split We now briefly describe (without proofs) a variation of the block split in which we store the decision value and the split components. We call this the augmented block splits — denoted by a subscript B. We would like xs ❀ 〈l,r〉 b(k) ⇒ xs ❀ 〈k,l,r〉 B
CHAPTER 4. SPLITTING HEADACHES 88 We can define a split function as follows: split B k [ ] = 〈k, [ ], [ ]〉 B split B 0 (x : xs) = 〈0, [ ],x : xs〉 B split B (k + 1) (x : xs) = 〈k + 1,x : l,r〉 B where 〈m,l,r〉 B = split B k xs The unsplitting function is the same as the k-block split: unsplit B 〈n,l,r〉 B = l + r The representation xs ❀ 〈n,l,r〉 B satisfies the invariant dti ≡ (|r| = 0 ∧ |l| < n) ∨ (|l| = n) (4.27) By storing the decision value, we will see that the operations that we define can use this value. Hence, we can choose the decision value randomly and so on different execution runs, the same list could be split differently. Also, when performing operation on this split, we can change the decision value as well the lists themselves. For instance, we can define a cons operation as follows: cons ∣ B a 〈k,l,r〉 B ∣ |l| < k = 〈k,a : l,r〉 B otherwise = 〈k + 1,a : l,r〉 B The definition for the k-block split required keeping the length of the first list component the same and so the last of element of the first list was added to the front of the second list. The version of cons for the augmented block split is more efficient as we merely increase the decision value by one. However cons B cannot be used to build up split lists starting from an empty split list as the list r will remain empty. The straightforward list operations are defined as follows: |〈k,l,r〉 B | B = |l| + |r| null B 〈k,l,r〉 B = null l ∧ null r elem B a 〈k,l,r〉 B = elem a l ∨ elem a r { head r if k = 0 head B 〈k,l,r〉 B = head l otherwise { 〈k,l, tail r〉B if k = 0 tail B 〈k,l,r〉 B = 〈k − 1, tail l,r〉 B otherwise map B f 〈k,l,r〉 B = 〈k, map f l, map f r〉 B All these definitions, with the exception of tail, match the definitions for the k-block split. The definition of tail is more efficient as we just decrease the decision value instead of joining the head of the second component to the end of the first component. Here is a possible definition for a concatenation operation: 〈k, [ ], [ ]〉 B + B 〈k ′ ,l ′ ,r ′ 〉 B = 〈k ′ ,l ′ ,r ′ 〉 B 〈k,l,r〉 B + B 〈k ′ ,l ′ ,r ′ 〉 B = 〈|l + r|,l + r,l ′ + r ′ 〉 B
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 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38: CHAPTER 2. OBFUSCATIONS FOR INTERME
Page 45 and 46: Chapter 3 Techniques for Obfuscatio
Page 47 and 48: CHAPTER 3. TECHNIQUES FOR OBFUSCATI
Page 63 and 64: Part III Data-Type Case Studies 62
Page 65 and 66: CHAPTER 4. SPLITTING HEADACHES 64 (
Page 67 and 68: CHAPTER 4. SPLITTING HEADACHES 66 a
Page 69 and 70: CHAPTER 4. SPLITTING HEADACHES 68 T
Page 71 and 72: CHAPTER 4. SPLITTING HEADACHES 70 L
Page 73 and 74: CHAPTER 4. SPLITTING HEADACHES 72 S
Page 75 and 76: CHAPTER 4. SPLITTING HEADACHES 74 C
Page 77 and 78: CHAPTER 4. SPLITTING HEADACHES 76 L
Page 79 and 80: CHAPTER 4. SPLITTING HEADACHES 78 t
Page 81 and 82: CHAPTER 4. SPLITTING HEADACHES 80 C
Page 83 and 84: CHAPTER 4. SPLITTING HEADACHES 82 4
Page 85 and 86: CHAPTER 4. SPLITTING HEADACHES 84 E
Page 87: CHAPTER 4. SPLITTING HEADACHES 86 b
Page 91 and 92: CHAPTER 4. SPLITTING HEADACHES 90 T
Page 93 and 94: Chapter 5 Sets and the Splitting Bu
Page 95 and 96: CHAPTER 5. SETS AND THE SPLITTING 9
Page 109 and 110: CHAPTER 6. THE MATRIX OBFUSCATED 10
Page 121 and 122: CHAPTER 7. CULTIVATING TREE OBFUSCA
Page 135 and 136: Chapter 8 Conclusions and Further W
Page 137 and 138: CHAPTER 8. CONCLUSIONS AND FURTHER
Page 139 and 140:
CHAPTER 8. CONCLUSIONS AND FURTHER
Page 141 and 142:
BIBLIOGRAPHY 140 [13] Thomas H. Cor
Page 143 and 144:
BIBLIOGRAPHY 142 [43] Simon Peyton
Page 145 and 146:
Appendix A List assertion We want t
Page 147 and 148:
APPENDIX A. LIST ASSERTION 146 Case
Page 149 and 150:
APPENDIX A. LIST ASSERTION 148 For
Page 151 and 152:
APPENDIX A. LIST ASSERTION 150 Case
Page 153 and 154:
APPENDIX A. LIST ASSERTION 152 1 +
Page 155 and 156:
APPENDIX A. LIST ASSERTION 154 0 +
Page 157 and 158:
APPENDIX A. LIST ASSERTION 156 The
Page 159 and 160:
APPENDIX A. LIST ASSERTION 158 The
Page 161 and 162:
APPENDIX B. SET OPERATIONS 160 Now
Page 163 and 164:
APPENDIX B. SET OPERATIONS 162 Proo
Page 165 and 166:
APPENDIX B. SET OPERATIONS 164 B.2
Page 167 and 168:
Appendix C Matrices We prove the as
Page 169 and 170:
APPENDIX C. MATRICES 168 Base Case
Page 171 and 172:
APPENDIX D. PROVING A TREE ASSERTIO
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
Page 187:
APPENDIX E. OBFUSCATION EXAMPLE 186
show all

Obfuscation of Abstract Data-Types - Rowan

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?