Logical Analysis and Verification of Cryptographic Protocols - Loria

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158 CHAPTER 6. ON THE GROUND ENTAILMENT PROBLEMS clause CS where I is the unique predicate symbols used in CI ∪CS and such that CI ∪ CS is saturated by ordered resolution. This insecurity problem outputs unsat if and only if CPθ ∪ ICI∪CS is unsatisfiable, for a substitution θ grounding for CP. S. Delaune, H. Lin and Ch. Lynch showed the decidability of this problem. While in the above sections (Section 6.3.1, 6.3.2 and 6.3.3), the secrecy property is the unique studied security property, other security properties can also be encoded as Horn Clauses. B. Blanchet has encoded authentication [47], and strong secrecy [51]. 6.4 Our contribution In Section 6.3, we have presented some decidable fragments of first order logic, and showed their use in the analysis of security protocols. To this end, intruder rules, protocol description and security property are represented by Horn clauses and the insecurity problem is reduced to the satisfiability problem of a fragment of first order logic. In this section, we present our model to analyse security protocols using Horn clauses. As opposite to the models represented in Section 6.3, we do not represent protocol description by Horn clauses. As showed in Chapter 2, one line of research reduces the insecurity problem of cryptographic protocols to an intruder reachability problem, also called I-reachability problem, and in this chapter we reduce the I-reachability problem to the entailment problem (or satisfiability problem) in first order logic. We remark that Horn clauses and a unique unary predicate symbol I are sufficient to analyse security protocols. The predicate symbol I represents the knowledge of the intruder: I(m) means that the intruder knows the term (or message) m. Thus a clause I(u1), . . . , I(un) → I(v) should be read as “if the intruder knows some messages of the form u1, . . . , un, then he knows a message of the form v”. For example, let us consider the clause I(x), I(y) → I(< x, y >), and assume that the intruder knows the messages a, b, then, due to the clause given above, the intruder knows also the message < a, b >. There is thus a natural correspondence between constraint systems (respectively solving of constraint systems) and Horn clauses (respectively entailment problems). 6.4.1 Model for cryptographic protocols Intruder clauses Let I be an intruder and let LI be its deduction system (LI represents the intruder capacities). LI is a set of rules of the form u1, . . . , un → v where u1, . . . , vn, v are terms in the given algebra. The set of clauses associated with
6.4. OUR CONTRIBUTION 159 I is CI def = {I(u1), . . . , I(un) → I(v) such that u1, . . . , un → v ∈ LI} Example 23 The Dolev-Yao rules presented in Chapter 2, in Example 8, more exactly the rules without explicit destructors are represented by the following set of Horn clauses: ⎧ I(x), I(y) → I(< x, y >) I(x), I(y) → I({x} ⎪⎨ CDY = ⎪⎩ s y) I(x), I(y) → I({x} p y) I(< x, y >) → x I(< x, y >) → I(y) I({x} s y), I(y) → I(x) I({x} p y), I(y−1 ) → I(x) I({x} p y−1), I(y) → I(x) Given a set of terms E, we define CE def = {I(u) such that u ∈ E}. Lemma 56 Let CI be the set of clauses associated with the intruder I, E a set of ground terms and m a ground term. If m ∈ ĒI then CE ∪ CI |= I(m). PROOF. Let E and m be respectively a set of ground terms and a ground term, and assume that m ∈ ĒI . This implies that there is a derivation d starting from E of goal m, d = E →I E1 →I . . . →I En−1, m, where Ei = Ei−1 ∪ mi, E0 = E and mn = m. We reason by induction on the lenghth of the derivation d. • lenghth(d) = 0: then m ∈ E and thus I(m) ∈ CE which implies that CE |= I(m) and then CE ∪ CI |= I(m). • lenghth(d) = 1: let u1, . . . , un → v be the intruder deduction rule applied on E in the derivation d, we have {uiσ}1≤i≤n ⊆ E and m = vσ for a ground substitution σ. Hence, for all i ∈ {1, . . . , n}, I(ui)σ ∈ CE which implies that CE |= I(ui)σ. Consider an arbitrary model Int of CE ∪ CI, Int satisfies all I(ui)σ. Let us prove that I(m) which is equal to I(v)σ is true in Int by contradiction. If I(v)σ is false in Int then the clause I(u1)σ, . . . , I(un)σ → I(v)σ is not satisfied by Int, and hence neither the clause I(u1), . . . , I(un) → I(v) which belong to CI. But this contradicts Int being a model of CE ∪ CI. Thus, I(m) = I(v)σ is true in Int. We conclude that CE ∪ CI |= I(m). • Assume that CE ∪ CI |= I(m) for any length k of the derivation d, k ≤ n and n ≥ 0, and let us prove it for lenghth(d) = n + 1. Assume that u1, . . . , un → v be the last applied rule in the derivation d. Then all uiσ are in En and vσ = m for a ground substitution σ. By induction we have CE ∪ CI |= I(ui)σ for all i. By the same reasoning as above, we deduce that CE ∪ CI |= I(v)σ, and thus CE ∪ CI |= I(m).
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Logical Analysis and Verification of Cryptographic Protocols - Loria

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