Logical Analysis and Verification of Cryptographic Protocols - Loria

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78 CHAPTER 3. PROTOCOLS WITH VULNERABLE HASH FUNCTIONS Figure 3.2 Ordered satisfiability for Intruder. deduction system Input - A IAU symbolic derivation C = (V, S, K, In, Out); - a finite set of ground terms KI representing the initial knowledge; - an ordering ≺ on V ar(C) ∪ all.Cons(C), ≺ is compatible with < defined on Ind, i.e. i < j implies xi ≺ xj. Step 1: For each variable x ∈ V, guess the set of constants Px = {a1, . . . , ak} that may occur in the solution, where ai ≺ x for i ∈ {1, . . . , k}; Step 2: Check the satisfiability of S with these ordering constraints. Step 3: Check for each variable x ∈ In, that Px ⊆ all.Cons(KI) ∪ � x ′ ∈Out,x ′
3.5. DECIDABILITY RESULTS 79 Theorem 2 The ordered IAU satisfiability problem is decidable. Related results. The decidability result obtained in this section is interesting in its own right as it is the first decidability result that we are aware of for an intruder system for which unification is infinitary, and that permits to consider an associative concatenation of messages instead of their pairing. A related decidability result has been obtained by Y. Chevalier et al. [72] for an intruder on multi-set of terms, that is an intruder with an associativecommutative-unit equational theory. In [58], S. Bursuc et al. obtain another decidability result in presence of associative and commutative symbols under some restrictions. This result is more recent than ours and does not cover the case of [72]. 3.5.2 Decidability of ordered If and Ig satisfiability problems Theorem 3 The ordered If satisfiability problem is decidable. PROOF. We have If = 〈{f}, {x1, x2, x3, x4 → f(x1, x2, x3, x4)}, ∅〉. It is easy to see that If is local (Definition 38). Since the equational theory in If is empty, by [70], we deduce that given E, t with E a set of ground terms and t a ground term, t ∈ E If is decidable, and hence, we can see easily that the ordered If satisfiability problem is decidable. � Following the same reasoning as before, we can deduce the following theorem. Theorem 4 The ordered Ig satisfiability problem is decidable. 3.5.3 Decidability of ordered Ifree satisfiability problem Theorem 5 The ordered Ifree satisfiability problem is decidable. PROOF. Ifree is the disjoint union of IAU, Ig and If intruder deduction systems. We have that ordered-IAU (respectively Ig and If) satisfiability problem, is decidable (respectively Theorem 2, Theorem 4, and Theorem 3). The result obtained in [72] prove that the ordered satisfiability problem for the disjoint union of decidable intruder deduction systems is also decidable. Thus ordered Ifree satisfiability problem is decidable. �
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Logical Analysis and Verification o
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Abstract This thesis is developed i
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x CONTENTS 2.1.7 Unification . . .
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xii CONTENTS 5.8 Decidability of re
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2 CHAPTER 1. INTRODUCTION Figure 1.
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4 CHAPTER 1. INTRODUCTION Secrecy T
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20 CHAPTER 2. PROTOCOL ANALYSIS USI
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128 CHAPTER 5. SATURATED DEDUCTION
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178 CHAPTER 7. VOTER VERIFIABILITY
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200 CHAPTER 8. CONCLUSION AND PERSP
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202 BIBLIOGRAPHY [12] M. Abadi and
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204 BIBLIOGRAPHY [36] M. Bellare, A
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206 BIBLIOGRAPHY [60] U. Hustadt Ch
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208 BIBLIOGRAPHY [83] H. Comon-Lund
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210 BIBLIOGRAPHY [108] B. Donovan,
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212 BIBLIOGRAPHY [132] T. Kohno, A.
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214 BIBLIOGRAPHY [160] J. C. Mitche
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216 BIBLIOGRAPHY [187] H. Seidl and
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Logical Analysis and Verification of Cryptographic Protocols - Loria

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