Logical Analysis and Verification of Cryptographic Protocols - Loria

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30 CHAPTER 2. PROTOCOL ANALYSIS USING CONSTRAINT SOLVING Definition 7 (most general unifiers) Let H be an equational theory, and let U be a Hunification system. A substitution σ is a most general H-unifier of U if and only if there exists a minimal complete set Σm of H-unifiers of U, such that σ ∈ Σm. Definition 8 The equational theory H is of unification type Unitary if for every satisfiable H-unification system U, there exists a minimal complete set of H-unifiers with cardinality 1. finitary if for every satisfiable H-unification system U, there exists a minimal complete set of H-unifiers with finite cardinality. infinitary if for every satisfiable H-unification system U, there exists a minimal complete set of H-unifiers, and there exists an H-unification system for which this set is infinite. zero if there exists an H-unification system that does not have a minimal complete set of H-unifiers. The equational theory ∅ is of type unitary, that is, when H = ∅, given a satisfiable set of equations U = {ui ? =∅ vi}i∈{1,...,n}, U has a unique most general unifier, denoted by mgu(U) [21]. If H is finitary, then the set of all H-unifiers of a given H-unification system can always be represented as H-instances of finitely many unifiers. The commutativity is a finitary equational theory that is not unitary [21]. A finite representation of all H-unifiers via H-instantiation is not always possible for equational theories of type infinitary and zero. The associativity theory is an � example of infinitary equational theory [21], and the equational theory H = x.(y.z) · = (x.y).z, x.x · � = x is of type zero [20, 184]. 2.1.8 Finite variant property We define in this section what means an equational theory H has the finite variant property. The finite variant property has been intially introduced in [86]. Definition 9 (finite variant property) Let H be an equational theory generated by a convergent rewrite system R. We say that H has the finite variant property if for any term t, there is a finite set of substitutions Σ(t) such that for any substitution σ, there exists a substitution θ ∈ Σ(t), and a substitution τ verifying (σ)↓ = θτ and (tσ)↓ = (tθ)↓τ. The substitutions in Σ(t) are called variant substitutions of t. We say that R has the finite variant property if H has that property. It is easy to see that the variant substitutions of any term t are in normal form. The finite variant property and its application for cryptographic protocols are analysed in more details in Chapter 5.
2.1. PRELIMINARIES 31 Figure 2.4 Unification algorithm based on the finite variant property Input: An arbitrary equational theory H having the � finite variant property, and a Hunification system U = . � s1 ? =H t1, . . . , sn ? =H tn Step 1: Compute the term M = g(s1, t1, . . . , sn, tn) where g is a new function symbol not appearing in F playing the role of cartesian product. Step 2: Compute the set of variant substitutions Σ(M). Output: • “U is H-unifiable” if there is a substitution θ ∈ Σ(M) and a substitution τ such that (sjθ)↓τ = (tjθ)↓τ for every j ∈ {1, . . . , n}; • “U is not H-unifiable” otherwise. Finite variant property and unification We introduce in this section a general H-unification procedure, where H is an arbitrary equational theory having the finite variant property. This algorithm is given in Figure 2.4. It is easy to see that the finite variant property reduces the H-unifiability problem to the syntactic unifiability problem. We show next the correctness and completeness of this unification procedure. Lemma 1 Let H be an equational theory having the finite variant property, and let s and t be two terms. Let M = g(s, t) where g is a new function symbol playing the role of cartesian product. If there exists a variant substitution θ of M and a substitution τ such that (sθ)↓τ = (tθ)↓τ then s and t are H-unifiable and σ = θτ is one of their possible H-unifiers. PROOF. Let s, t be two terms such that (sθ)↓τ = (tθ)↓τ where θ is a variant substitution of g(s, t) and τ an arbitrary substitution. (sθ)↓τ = (tθ)↓τ implies (sθ)↓τ =H (tθ)↓τ. Let σ = θτ, we have (sσ)↓ = (sθτ)↓ = ((sθ)↓τ)↓ and similarly, (tσ)↓ = ((tθ)↓τ)↓. Since (sθ)↓τ =H (tθ)↓τ, we conclude that ((sθ)↓τ)↓ = ((tθ)↓τ)↓ and hence (sσ)↓ = (tσ)↓ which implies that σ is an H-unifier of s and t. � Lemma 2 Let H be an equational theory having the finite variant property, and let s and t be two terms. Let M = g(s, t) where g is a new function symbol playing the
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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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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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