Logical Analysis and Verification of Cryptographic Protocols - Loria

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42 CHAPTER 2. PROTOCOL ANALYSIS USING CONSTRAINT SOLVING � with s < t1. t1 < x implies t1 < xσ for all σ and then for σ = {x ↦→ s}, we have t1 < s which contradicts s < t1. We conclude that either t1 = t2 or t1 is the minimal ground term in T (F) for < which exists due to the wellfoundness of < and which is unique due to the fact that < is total over ground terms. Lemma 7 Let I = 〈F, TI, H〉 be an intruder deduction system, and let I ′ = 〈F, L ′ , ∅〉 be the variant intruder deduction system of I. If t1, . . . , tn → t ∈ L ′ , then V ar(t) ⊆ V ar(t1, . . . , tn). PROOF. Let t1, . . . , tn → t ∈ L ′ , we have two cases: 1. If t1, . . . , tn → t ∈ L then, by definition of L, t1, . . . , tn → t = x1, . . . , xn → f(x1, . . . , xn) where f is a public function symbol in F and hence, V ar(t) ⊆ V ar(t1, . . . , tn). 2. If t1, . . . , tn → t �∈ L, then, by definition of L ′ , there exists a rule x1, . . . , xn → f(x1, . . . , xn) ∈ L, a variant substitution θ of f(x1, . . . , xn) such that ti = xiθ for every i, and (f(x1, . . . , xn)θ)↓ = t. We have that V ar(f(x1, . . . , xn)θ) = �n i=1 V ar(xiθ), and (f(x1, . . . , xn)θ)↓ ≤ f(x1, . . . , xn)θ. Lemma 6 implies that V ar((f(x1, . . . , xn)θ)↓) ⊆ V ar(f(x1, . . . , xn)θ) and hence, V ar((f(x1, . . . , xn)θ)↓) ⊆ �n i=1 V ar(xiθ) which concludes the lemma. � We prove in what follows (Lemma 8) that when considering only deductions on ground terms in normal form and yielding ground terms in normal form, it is sufficient to consider derivations modulo the empty theory. Lemma 8 Let E and F be two sets of ground terms in normal form we have: E →I F if and only if E →I ′ F . PROOF. Let E and F be two sets of ground terms in normal form and assume there is a rule x1, . . . , xn → f(x1, . . . , xn) ∈ L such that E →x1,...,xn→f(x1,...,xn) F . By definition there exists a ground substitution σ in normal form such that (x1, . . . , xn)σ ⊆ E and F = E ∪ {(f(x1, . . . , xn)σ)↓}. Due to the finite variant property, there exists a variant substitution θ of f(x1, . . . , xn) and a ground normal substitution σ ′ such that (f(x1, . . . , xn)σ)↓ = (f(x1, . . . , xn)θ)↓σ ′ and σ = θσ ′ . The rule x1θ, . . . xnθ → (f(x1, . . . , xn)θ)↓ was added to L ′ by definition of I ′ (Definition 19). This implies that E →I ′ F . To prove the converse, notice that if (x1, . . . , xn)θ → (f(x1, . . . , xn)θ)↓ can be applied with the normal ground substitution σ ′ on E, then the rule x1, . . . , xn → f(x1, . . . , xn) can be applied with the ground substitution σ = (θσ ′ )↓ on E. �
2.1. PRELIMINARIES 43 2.1.12 Constraint systems In [156], the authors introduced the notions of “deduction constraint” and “Iconstraint systems”. They defined a deduction constraint to be an expression of the form E � t where E is a set of terms and t is a term, and they defined an I-constraint system C as follows: C = (E1 � t1, . . . , En � tn) where Ei ⊆ Ei+1, and V ar(Ei) ⊆ V ar({t1, . . . , ti−1}). They defined also a solution of C as follows: a substitution σ is a solution of C if tiσ ∈ Eiσ for every i. This notion of Iconstraint system has been defined with the ∅ equational theory in mind. Unfortunately, such definitions of I-constraint systems are not adequate in presence of non empty equational theory. For instance, let us consider the equational theory H = {f(x, x) = a} which is generated by the convergent rewrite system R = {f(x, x) → a}, and let us consider the I-constraint system C = ({a, b} � f(x, y), {a, b, x} � b). This constraint system follows the definition of constraint system given above, and the substitution σ = {x ↦→ y} is a solution of C following the definition above. When we apply this substitution σ to C then normalise, we obtain the following system C ′ = ({a, b} � a, {a, b, y} � b). It is easy to see that C ′ does not satisfy the definition of constraint systems given above. In order to avoid such problem, in [73], the authors introduced another definition of constraint systems. This definition, given below, is adequate with the non empty equational theories, and it is the definition adapted in this document. Definition 20 (I-Constraint systems) Let I be an intruder deduction system. An Iconstraint system C is denoted (E1 ⊢ v1, . . . , En ⊢ vn, U) and is defined by a finite set of expressions Ei ⊢ vi, called deduction constraints, with: • vi ∈ X for i ∈ {1, . . . , n}, • E1 ⊆ T (F), and Ei ⊆ T (F, X ) for i ∈ {2, . . . , n}, • Ei ⊆ Ei+1 for i ∈ {1, . . . , n − 1}, • V ar(Ei) ⊆ {v1, . . . , vi−1} for i ∈ {2, . . . , n}, • and a H-unification system U. An I-Constraint system C is satisfied by a substitution σ, and we write σ |=I C, if for all i ∈ {1, . . . , n} we have viσ ∈ Eiσ I and if σ |=H U. We call such a substitution a solution of C. It is easy to see that if a substitution σ is a solution of a constraint system C, the substitution (σ)↓ is also a solution of C.
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178 CHAPTER 7. VOTER VERIFIABILITY
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202 BIBLIOGRAPHY [12] M. Abadi and
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204 BIBLIOGRAPHY [36] M. Bellare, A
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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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