Logical Analysis and Verification of Cryptographic Protocols - Loria

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76 CHAPTER 3. PROTOCOLS WITH VULNERABLE HASH FUNCTIONS � • If σ |=HAU � m ? = x1 · g(x1, x2, y1, y2) · x2, m ′ ? � = y1 · f(x1, x2, y1, y2) · y2 then � m1σ =HAU x1σ · f(x1σ, x2σ, y1σ, y2σ) · x2σ m2σ =HAU y1σ · g(x1σ, x2σ, y1σ, y2σ) · y2σ Since E → ∗ IAU (m1σ)↓, we have E → ∗ IAU (x1σ · f(x1σ, x2σ, y1σ, y2σ) · x2σ)↓ and thus, E → ∗ IAU (x1σ)↓ · f((x1σ)↓, (x2σ)↓, (y1σ)↓, (y2σ)↓) · (x2σ)↓ which implies that E → ∗ IAU (x1σ)↓, (x2σ)↓, f((x1σ)↓, (x2σ)↓, (y1σ)↓, (y2σ)↓). By hypothesis, f((x1σ)↓, (x2σ)↓, (y1σ)↓, (y2σ)↓) /∈ SubE, Lemma 19 implies E → ∗ IAU (x1σ)↓, (x2σ)↓, (y1σ)↓, (y2σ)↓ and thus E → ∗ IAU g((x1σ)↓, (x2σ)↓, (y1σ)↓, (y2σ)↓) which implies that E → ∗ IAU (y1σ)↓ · g((x1σ)↓, (x2σ)↓, (y1σ)↓, (y2σ)↓) · (y2σ)↓. We conclude that E → ∗ IAU (m2σ)↓. 3.5 Decidability results We show next the decidability of respectively ordered IAU, ordered If, ordered Ig, ordered Ifree-satisfiability problems, and we conjecture the decidability of the ordered Ih-satisfiability problem. 3.5.1 Decidability of ordered IAU-satisfiability problem We state below some basic facts on IAU = 〈{.}, LI AU, HAU〉. Lemma 21 Let E and t be respectively a set of terms and a term in normal form with respect to the equational theory HAU. We have: 1. all.Cons(E) ⊆ E IAU , 2. E ⊆ all.Cons(E) IAU , 3. E IAU 4. t ∈ E IAU = all.Cons(E) IAU , if and only if all.Cons(t) ⊆ all.Cons(E). PROOF. Let E and t be respectively a set of ground terms and a ground term in normal form with respect to the equational theory HAU. 1. Let c ∈ all.Cons(E) be a constant, and let e be a term in E such that c ∈ all.Cons(e). We have therefore e = e1.c.e2. By associativity of ., we have e = e1.c.e2 = (e1.c).e2 = e1.(c.e2). Thus E →IAU E, (e1.c) →IAU E, (e1.c), c. This implies all.Cons(E) ⊆ E IAU .
3.5. DECIDABILITY RESULTS 77 2. By definition of IAU, it is easy to see that E ⊆ all.Cons(E) IAU . 3. (1) and (2) implies easily (3). 4. We have t ∈ E IAU is equivalent to E IAU = (E, t) IAU , which is equivalent to all.Cons(E) IAU = all.Cons(E, t) IAU by (3). By contradiction, assume that t ∈ E IAU and there is a constant c ∈ all.Cons(t) \ all.Cons(E). c /∈ all.Cons(E) implies that c /∈ all.Cons(E) IAU , and c ∈ all.Cons(t) implies that c ∈ all.Cons(E, t) IAU . This contradicts the equality all.Cons(E) IAU all.Cons(E, t) IAU , and hence we deduce that all.Cons(t) ⊆ all.Cons(E). Now, we prove the converse. We have all.Cons(t) ⊆ all.Cons(E), this implies all.Cons(t) ⊆ all.Cons(E) IAU . By (2), we have t ∈ all.Cons(t) IAU , and hence, t ∈ all.Cons(E) IAU . By (3), we conclude that t ∈ E IAU . � The consequence of Lemma 21 is that only the set of constants appearing or not in the initial knowledge and the goal of a supposed derivation are relevant to decide its feasibility. This has the important implication, with respect to decidability, that it is not necessary to know the exact instance of intruder’s knowledge (his initial knowledge and the messages in the output of the symbolic derivation up to this point) and the goal (the next input message of the symbolic derivation) to decide whether a derivation exists. It suffices to know the guessable sets of constants of the knowledge and of the goal. We give in Figure 3.2 the algorithm that solve ordered IAU-satisfiability problem and then, show its correctness, completeness and termination. The definition of satisfiability of symbolic derivations allows us to conclude automatically the completeness of the algorithm: Lemma 22 (Completeness) The algorithm described in Figure 3.2 is complete. PROOF. Let C = (V, S, K, In, Out) be a IAU symbolic derivation, KI be the initial intruder knowledge, and ≺ be a linear ordering on the variables and constants of C. Assume that C is satisfiable, this implies, by definition, that there exists a IAU-symbolic derivation CI = (VI, SI, KI, InI, OutI), a closed composition Ca of CI and C, and a substitution σ such that (1) σ |=IAU Ca and (2) for all x variable in C and c constant in C, x ≺ c implies c /∈ all.Cons(xσ). Hence, the intruder I receives all messages sent by the protocol participants, and every message they receive is send by the intruder. We have that xσ is defined for all x ∈ V, we let Px = all.Cons(xσ). Furthermore, we have that for all x ∈ In, (xσ)↓ ∈ K ′ IσIAU with K ′ I = KI ∪ {x ′ | x ′ ∈ Out and x ′ ≺ x}, and by Lemma 21, we have =
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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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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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214 BIBLIOGRAPHY [160] J. C. Mitche
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216 BIBLIOGRAPHY [187] H. Seidl and
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