Logical Analysis and Verification of Cryptographic Protocols - Loria

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136 CHAPTER 5. SATURATED DEDUCTION SYSTEMS The initial ⎧ set of deduction rules is given by the following set of rules: ⎪⎨ L0 = ⎪⎩ x, y → sig(x, y), x, y, z → ver(x, y, z), x, y → Sk ′ (x, y), x, y → P k ′ (x, y), ∅ → 1. The saturation (modulo the simplification introduced in Section 5.4) outputs the following ⎧ set of deduction rules: L ′ ⎪⎨ = L0 ∪ ⎪⎩ x, sig(x, Sk(y)), P k(y) → 1, x, sig(x, Sk ′ (y1, y2)), P k ′ (y1, y2) → 1, x, sig(x, Sk(y)), P k ′ (P k(y), sig(x, Sk(y))) → 1, x, Sk ′ (P k(y), sig(x, Sk(y))) → sig(x, Sk(y)), Sk(y), P k(y) → 1, Sk ′ (y1, y2), P k ′ (y1, y2) → 1, x, Sk(y), P k ′ (P k(y), sig(x, Sk(y))) → 1, x, P k(y), sig(x, Sk(y)) → sig(x, Sk(y)), x, P k(y), Sk(y) → sig(x, Sk(y)), y1, y2, P k ′ (y1, y2) → 1, x, y1, y2, sig(x, Sk ′ (y1, y2)) → 1, y1, y2, Sk ′ (y1, y2) → 1, x, P k(y), sig(x, Sk(y)) → 1, x, P k(y), Sk(y), sig(x, Sk(y)) → 1, x, Sk(y), P k(y), P k ′ (P k(y), sig(x, Sk(y))) → 1, x, Sk(y), P k(y) → sig(x, Sk(y)). 5.7 Decidability of ground reachability problems for the blind signature theory Blind signature was introduced in [136], it is defined by the signature FBS = {sig, ver, Bl, ⎧ Ubl, P k, Sk} which satisfies the following set of equations: ⎨ ver(sig(x, Sk(y)), P k(y)) = x, HBS = Ubl(Bl(x, y), y) = x, ⎩ Ubl(sig(Bl(x, y), Sk(z)), y) = sig(x, Sk(z)). Let RBS be the set of rules obtained by orienting equations of HBS from left to right, RBS is convergent and it is obvious that any basic narrowing derivation (Definition 13) issuing from any of the right hand side term of the rules of RBS terminates. This implies that any narrowing derivation (and in particular basic narrowing derivation) issuing from any term terminates (Theorem 1) and thus HBS has finite variant property [86]. The initial deduction system is given by the tuple I0 = 〈FBS, L0, HBS〉 and we have:
5.7. DECIDABILITY OF GROUND REACHABILITY PROBLEMS FOR THE BLIND SIGNATURE THEORY137 ⎧ ⎪⎨ 1 : x, y → sig(x, y), 2 : x, y → ver(x, y), L0 = ⎪⎩ 3 : x, y → Bl(x, y), 4 : x, y → Ubl(x, y). The initialisation step of saturation (Figure 5.1) outputs the following set of deduction rules: ⎧ ⎨ 5 : sig(x, Sk(y)), P k(y) → x, L = L0 ∪ 6 : Bl(x, y), y → x, ⎩ 7 : sig(Bl(x, y), Sk(z)), y → sig(x, Sk(z)). We define a new deduction system I = 〈FBS, L, ∅〉 and by lemma 31, we have: t ∈ E I0 if and only if t ∈ E I for every set of ground terms E (respectively a ground term t) in normal form. From now we remark that the equational theory employed is the empty one. Now, let us apply the first step of saturation. The closure applied on rules 1 and 5 outputs the rule 8 : x, Sk(y), P k(y) → x, the closure applied on rules 3 and 6 outputs the rule 9 : x, y → x which will be deleted by the simplification step introduced in Section 5.4. The closure applied on rules 1 and 7 outputs the rule 10 : y, Bl(x, y), Sk(z) → sig(x, Sk(z)). We prove in the next lemma that the last rule is redundant when the employed equational theory is the empty one. Lemma 51 Let L ′ 1 = L ∪ {x, Sk(y), P k(y) → x} ∪ {y, Bl(x, y), Sk(z) → sig(x, Sk(z))} and let L ′ 2 = L ′ 1 \ {y, Bl(x, y), Sk(z) → sig(x, Sk(z))}. Suppose that the employed equational theory is the empty one. For any two sets of ground terms in normal form E and F we F . have: E → ∗ L ′ 2 PROOF. F if and only if E → ∗ L ′ 1 Let E and F be two sets of normal ground terms. The direct implication is F and let us prove F . Suppose that in the L ′ 1-derivation D starting from E to F there obvious, let us prove the second one. Suppose that E → ∗ L ′ 1 that E → ∗ L ′ 2 is some steps where the applied rule is in L ′ 1 \ L ′ 2 that is, by definition of L ′ 1 and L ′ 2, the applied rule is y, Bl(x, y), Sk(z) → sig(x, Sk(z)). Let i be the first step in the derivation where the applied rule is y, Bl(x, y), Sk(z) → sig(x, Sk(z)), we prove that this step can be replaced by other steps where the respectives applied rules are in L ′ 2. D : E = E0 → . . . → Ei →y,Bl(x,y),Sk(z)→Sig(x,Sk(z)) Ei+1 . . . → F . There is a ground substitution σ in normal form such that {yσ, Bl(x, y)σ, Sk(z)σ} ⊆ Ei and Ei+1 = Ei ∪ sig(xσ, Sk(z)σ). Thus, the rule Bl(x, y), y → x ∈ L ′ 2 with the substitution σ can be applied first on Ei and outputs Ei1 = Ei ∪ xσ, then the rule x, y → sig(x, y) ∈ L ′ 2 also with the substitution σ can be applied on Ei1 and outputs Ei1 ∪ sig(xσ, Sk(zσ)) = Ei+1. We deduce that each application of the
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Logical Analysis and Verification o
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Abstract This thesis is developed i
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x CONTENTS 2.1.7 Unification . . .
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2 CHAPTER 1. INTRODUCTION Figure 1.
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4 CHAPTER 1. INTRODUCTION Secrecy T
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198 CHAPTER 8. CONCLUSION AND PERSP
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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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