Logical Analysis and Verification of Cryptographic Protocols - Loria

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92 CHAPTER 4. PROTOCOLS WITH VULNERABLE SIGNATURE SCHEMES The intruder system we consider to analyse our class of cryptographic protocols is given as follows: with: ⎧ ⎪⎨ ⎪⎩ IDSKS = 〈FDSKS, TDSKS, HDSKS〉 FDSKS = FDSKSpub ∪ FDSKSpri FDSKSpub = {sig, ver, Sk ′ , P k ′ , 1} FDSKSpri = {Sk, P k} TDSKS = {sig(x, y), ver(x, y, z), Sk ′ (x, y), P k ′ (x, y), 1} The associated set of intruder deduction rules, denoted by LDSKS is given as follows: ⎧ ⎪⎨ LDSKS = ⎪⎩ x, y → sig(x, y) x, y, z → ver(x, y, z) x, y → Sk ′ (x, y) x, y → P k ′ (x, y) ∅ → 1 In what follows, we introduce the rewrite system, RDSKS, generating the equational theory HDSKS, and we prove that RDSKS is convergent. The rewrite system RDSKS is obtained by applying Knuth-Bendix completion procedure [131] on HDSKS. This completion procedure is described in Chapter 2, at Section 2.1.6. Lemma 25 HDSKS is generated by the convergent rewriting system: ⎧ ⎪⎨ RDSKS = ⎪⎩ ver(x, sig(x, Sk(y)), P k(y)) → 1 ver(x, sig(x, Sk ′ (y1, y2)), P k ′ (y1, y2)) → 1 ver(x, sig(x, Sk(y)), P k ′ (P k(y), sig(x, Sk(y)))) → 1 sig(x, Sk ′ (P k(y), sig(x, Sk(y)))) → sig(x, Sk(y)) PROOF. The application of the Knuth-Bendix completion procedure [131] on HDSKS terminates successfully and outputs the rewrite system RDSKS, which is a convergent rewrite system generating HDSKS (Chapter 2, Section 2.1.6). � Lemma 26 Any RDSKS-narrowing derivation starting from any term t terminates. PROOF. In Lemma 25, we proved that RDSKS is a convergent rewrite system. Following the definition of basic narrowing (Definition 13, Chapter 2), it is easy to see that any right member of any RDSKS rule is not RDSKS-basic narrowable, and hence, by Theorem 1 (Chapter 2) , we conclude that any RDSKS-narrowing derivation starting from any term terminates.�
4.4. DECIDABILITY RESULTS 93 Lemma 27 HDSKS has the finite variant property. PROOF. In Lemma 25, we proved that HDSKS is generated by a convergent rewrite system RDSKS, and in lemma 26, we proved that any RDSKS-narrowing derivation starting from any term terminates. In [86], the authors showed that each convergent rewrite system R such that any R-basic narrowing derivation starting from any term terminates has the finite variant property. This allows us to conclude that RDSKS and hence HDSKS has the finite variant property. � 4.4.2 Symbolic model for destructive exclusive ownership vulnerability property In order to symbolically analyse the class of cryptographic protocols using digital signature schemes vulnerable to the destructive exclusive ownership property, we consider the following signature FDEO = {Sk, P k, sig, ver, Sk”, P k”, f, 1} where • Sk, denoting the secret key generation function which is a part of the key generation algorithm G, is a function with arity 1, • P k, denoting the public key generation function which is a part of the key generation algorithm G, is a function with arity 1, • sig, denoting the signature generation algorithm, is a function with arity 2, • ver, denoting the verification algorithm, is a function with arity 3, • Sk”, denoting the “special” intruder secret key generation function, is a function with arity 2, • P k”, denoting the “special” intruder public key generation function, is a function with arity 2, • f, denoting the “special” algorithm applied by the intruder to generate a “special” message, is a function with arity 2, • 1, denoting a possible output of ver, is a function with arity 0. The functions Sk, P k (respectively sig and ver) given above abstract the two parts of the key generation algorithm G(respectively the signature generation algorithm, and the verification algorithm) in a signature scheme. Following the same reasons as in Section 4.4.1, Sk and P k are private function symbols. The special functions Sk”, P k”, f abstract the ability of the intruder, knowing an agent’s public key (pk), and the agent’s signature s on a message m, to construct
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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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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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