Logical Analysis and Verification of Cryptographic Protocols - Loria

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174 CHAPTER 6. ON THE GROUND ENTAILMENT PROBLEMS Lemma 68 Let S be a saturated set of clauses, C be a ground clause and Π be a nonempty set of proofs of S |= C. Let π ∈ Π be such that δS(π, C) is minimal. If A is an A. atom in the proof π then there is an atom B ∈ µ(C) such that B → ∗ R g S PROOF. By Lemma 67, the minimality of δS(π, C) implies that δS(π, C) = ∅. This implies that if A is an atom in π (A ∈ µ(π)) then A ∈ µ(C) ↓S, and hence, by A. � definition, there is an atom B ∈ µ(C) such that B → ∗ R g S Theorem 14 If the set of clauses S is saturated by selected resolution up to redundancy with respect to ≻a then ground entailment problem for S is decidable. PROOF. Let S be a saturated set of clauses. Given a ground clause C, we prove that we can decide if S |= C.We make use of the fact that S |= C if and only if S ∪ ¬C is unsatisfiable. By Lemma 59 and Lemma 60, S ∪¬C is unsatisfiable if and only if the set of T of ground refutational proofs of S ∪ ¬C |= ∅ is not empty. Let π be a proof in T having the minimal δS(π, C). By lemma 68, the atoms appearing in the proof π belongs to the set µ(C) ↓S and since the clause C is ground, Lemma 64 implies that the set µ(C) ↓S is finite.Now, the strategy for deciding the problem S |= C is to construct first the set µ(C) ↓S, then to construct the set S ′ of ground instances of clauses of S such that for every ground clause C ′ ∈ S ′ , and for every atom A ∈ µ(C ′ ), A ∈ µ(C) ↓S. By finiteness of the set µ(C) ↓S, the set S ′ is finite. We construct now the set of trees Π such that for each tree, the leaves are labelled by ¬L with L is a literal in C and by clauses in S ′ , and internal nodes are labelled by the result of selected resolution from their direct descendents. Since S ′ is finite, the set Π is finite. Moreover, by definition of selected resolution and since S ′ and S are ground, we have that each tree in Π is finite. Finally, we have S |= Cl if and only if there is a tree in the set Π whose root is labelled by the empty clause, and as the existence of such tree is decidable then, the problem S |= C for a ground clause C is decidable, and hence the ground entailment problem for S is decidable. � Example 25 We consider the Needham-Schroeder symmetric key protocol described in Example 24. We analyse the security of this protocol in the presence of an intruder I to which we associate the following set of clauses, CI = CDY ∪ Cpre. The set of clauses CDY is given in Example 23 and the clause Cpre is given in Example 24. As shown in Section 6.4.1, the insecurity problem for this protocol is reduced to the ground entailment problem for CI. We show that the decidability of insecurity problem for this protocol can be deduced from Theorem 14. In fact, CI is saturated by selected resolution up to redundancy with respect to a complete atom ordering compatible with a complete simplification term ordering. Theorem 14 implies that the ground entailment problem for CI is decidable, and hence the insecurity problem of Needham-Schroeder symmetric
6.6. DISCUSSIONS AND CONCLUSIONS 175 key protocol is decidable under a bounded number of sessions and under the hypothesis that the protocol runs correctly. 6.6 Discussions and conclusions The main contribution of this chapter is a decidability result of the ground entailment problem for a set S of clauses under the hypothesis that S is saturated by selected resolution up to redundancy with respect to a complete atom ordering compatible with a complete simplification ordering over terms. Many decidability results have been obtained for this problem, and we discuss some of them in Section 6.5. Among the works cited above, the closest to ours is [28]. In [28], D. Basin and H. Ganzinger proved the decidability of the ground entailment problem for a set S of clauses under the assumptions that (i) S is saturated up to redundancy under ordered resolution with respect to a complete well-founded ordering over atoms and that (ii) each maximal atom in each clause in S contains all variables appearing in the other atoms in the clause. In this chapter, we relax the condition (ii), and in order to get the decidability, we use an order more restrictive than the one used in [28]. Another contribution of this chapter is the reduction of insecurity problem of cryptographic protocols to a ground entailment problem in the first order logic. In this line of research, many results have been obtained, and some of them have been discussed in Section 6.3. We remark that our result is different than the results discussed in that section. In fact, we define a new model to analyse cryptographic protocols using Horn clauses, which is different that the models given in [84, 180, 205]. Moreover, in [84, 180, 205], the decidability results are obtained for Horn clauses and not full clauses as in this chapter. while in [84, 205], they can deal with an unbounded number of sessions, we consider only a bounded number of sessions but our result allows us to deal with a class of cryptographic protocols less restrictive than [84, 205]. In future works, we plan implement the decision procedure with respect to the result of D. Basin and H. Ganzinger [28], we remark that for ground entailment problems, it suffices to consider inferences by selected resolution in which one of the atoms is ground. Thus we do not need to construct the set of ground atoms before solving an entailment problem. Furthermore, we plan to extend the result to the case where the clauses are considered modulo an equational theory. We thanks Michaël Rusinowitch for all the discussions we had concerning the results of this chapter
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Logical Analysis and Verification of Cryptographic Protocols - Loria

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