Logical Analysis and Verification of Cryptographic Protocols - Loria

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144 CHAPTER 5. SATURATED DEDUCTION SYSTEMS with increasing rules. Thus, after a right choice of at most |Sub(E) \ V ar(E)| decreasing rules, all terms deducible from the obtained knowledge set can be deduced using only increasing rules, hence the tagging with inc of the final deduction constraint E ∪ {r1, . . . , rk} ⊲inc t, k = |Sub(E) \ V ar(E)|. Termination of Step 2. First let us notice that if a unification is chosen, it unifies two subterms of the constraint system in the empty theory, and thus either the two terms were already equal or it reduces strictly the number of variables in the constraint system. Thus the number of unification choices is bounded by the number of variables in the constraint system. Once all unification have been performed, the termination of the first part of the iteration can easily be proved by considering the multiset of the right-hand side of the deduction constraints, ordered by the extension to multisets of the (well-founded) subterm ordering. The second part of the iteration obviously terminates. Thus each iteration terminates. Since each iteration decreases strictly the number of non-labelled deduction constraints, Step 2. terminates. 5.9 Related works Several decidability results have been obtained for cryptographic protocols in a similar setting [15, 156, 16, 178]. These results have been extended to handle algebraic properties of cryptographic primitives [68, 70, 66, 9]. In [10], a decidability result was given to the ground reachability problems in the case of subterm convergent equational theories. This result was extended in [9] and a decidability result to the ground reachability problems in the case of locally stable AC-convergent equational theories was given. Moreover, again in [9], a decidability result was given to the ground reachability problems for the theory of blind signature [136] while this theory was not included in [10]. To obtain a decidability result for the theory of blind signature, M. Abadi and V. Cortier [9] use a new extended definition of subterm. The result obtained in [10] was extended in [31] in different way than in [9] and a decidability result was obtained to the general reachability problem for the class of subterm convergent equational theory. The first result of our chapter is a decidability result to the ground reachability problems for a class of equational theories which includes the class studied in [10]. We note that the class studied in [9] is incomparable to ours and we note also that the proof used in [9] to decide the ground reachability problems for the theory of blind signature is different from the ours. Another result of this chapter is a decidability result to the general reachability problem for a class of equational theories under some conditions on the deduction systems and the class studied in [31] is incomparable with ours. In [40], a decidability result was given to the general reachability problems under some
5.10. CONCLUSION 145 syntactic conditions on the intruder deduction rules, this result is incomparable with ours. 5.10 Conclusion In [80], H. Comon-Lundh proposes a two-steps strategy for solving general reachability problems: first, decide ground reachability problems and, second, reduce general reachability problems to ground reachability ones, e.g. by providing a bound on the size of a minimal solution of a problem. Our results are in this line: for contracting deduction systems, general reachability can be reduced to ground reachability. We strongly conjecture that it permits one to provide a bound on the size of minimal solutions. Thus, this chapter adds a new criterion to the one already known for deciding reachability problems. In future works, we will investigate how the construction presented here can be extended to equational theories having the finite variant property w.r.t. a nonempty equational theory. We will also try to weaken the definition of µ(T ) for a set of terms T . In the next chapter, we employ similar techniques to solve ground entailment problems for saturated first-order theories.
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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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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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