Logical Analysis and Verification of Cryptographic Protocols - Loria

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108 CHAPTER 4. PROTOCOLS WITH VULNERABLE SIGNATURE SCHEMES � r ? =∅ t, (li � ? =∅ ei)1≤i≤n has solution. Let µ be the most general ∅-unifier. C ′ = (Cα, (E ⊲ x)x∈lx, Cβ)µ. Since σ ′ |=I” C ′ and by definition of µ, we have σ ′ µ |=I” C. � From Lemmas 33, 34, 35, 38, 39 and Lemma 36 (respectively Lemma 37), we deduce respectively the following two corollaries: Corollary 1 The IDSKS-reachability problem is decidable. Corollary 2 The IDEO-reachability problem is decidable. 4.5 Conclusion In this chapter, we study the constructive exclusive ownership vulnerability and the destructive exclusive ownership vulnerability properties for digital signature schemes, and we show the decidability of the insecurity problem for the two classes of cryptographic protocols using signature schemes vulnerable respectively to constructive exclusive ownership and destructive exclusive ownership properties: we give an algorithm that solves IDSKS- and IDEOreachability problems. This algorithm can be summarised as follows: first, we saturate IDSKS and IDEO intruder deduction rules, this saturation terminates, and yields two new intruder deduction systems I”DSKS and I”DEO. Second, we simplify IDSKS and IDEO constraint systems, using two steps, in order to get respectively a modified I”DSKS and I”DEO constraint system in solved form. This simplification is sound and complete. We proved also the termination of this simplification, which is obtained due to the termination of saturation, and to the fact that each simplification step reduces the complexity of the constraint system on which the simplification is applied. Unfortunately, the termination of such procedure is specific to our cases of cryptographic protocols. In [95], S. Delaune and F. Jacquemard showed that the I-reachability problem is decidable in non-deterministic polynomial time providing that the equational theory is generated by a convergent public-collapsing rewrite system. We remark that both hDSKS and hDEO are generated by convergent publiccollapsing rewrite systems, and hence, the result of [95] can be applied on our classes of cryptographic protocols. We also remark that in [31], M. Baudet obtained a decidability result to the reachability problem for the class of subterm convergent equational theory. Since both hDSKS and hDEO are subterm convergent equational thoeries, we can apply the result of [31] to our classes of cryptographic protocols. The method we used in this chapter is different than the methods used in [95, 31], and can be generalised to other classes of deduction systems than can not be considered by [95, 31].
4.5. CONCLUSION 109 In the next chapter, we generalise this decidability result to the class of cryptographic protocols that use cryptographic primitives represented by equational theories generated by convergent rewrite systems having the finite variant property. That decidability result is obtained by generalising the proofs of this chapter: we show that the termination of the saturation is sufficient to obtain the decidability of the ground reachability problems, and we add another condition to obtain the decidability of the non ground reachability problems.
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Logical Analysis and Verification o
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Abstract This thesis is developed i
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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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Logical Analysis and Verification of Cryptographic Protocols - Loria

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