Logical Analysis and Verification of Cryptographic Protocols - Loria

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90 CHAPTER 4. PROTOCOLS WITH VULNERABLE SIGNATURE SCHEMES 4.3 Destructive exclusive ownership property 4.3.1 Description of DEO property In [170], T. Pornin and J. P. Stern studied the duplicate-signature key selection property. They introduced the notion of conservative exclusive ownership defined as follows: a signature scheme with appendix is said to provide conservative exclusive ownership if it does not have duplicate-signature key selection property. T. Pornin and J.P. Stern [170] also introduced the notion of destructive exclusive ownership defined as follows: a signature scheme with appendix is said to provide destructive exclusive ownership if it is computationally infeasible for the intruder, given an agent public key P kA, a message m and A ′ s signature s of m, to produce a new pair of secret and public keys (SkI, P kI), a new message m ′ �= m, such that SkI matches P kI, and ver(m ′ , s, P kI) = 1. Example 21 In [170], the authors showed that, in certain circumstances, RSA [174] and DSS [6] do not provide destructive exclusive ownership. In what follows, we make use of “signature schemes vulnerable to constructive exclusive ownership property”, and “signature schemes vulnerable to destructive exclusive ownership property”. We say that a signature scheme is vulnerable to conservative exclusive ownership property if it does not provide that property, that is if it has duplicate-signature key selection property, and similarly, we say that a signature scheme is vulnerable to destructive exclusive ownership property if it does not provide that property. 4.4 Decidability results In this section, we show that the insecurity problem for the class of cryptographic protocols using signature schemes vulnerable to conservative exclusive ownership property (respectively for the class of cryptographic protocols using signature schemes vulnerable to destructive exclusive ownership property) is decidable. To get this decidability result, we follow the symbolic approach described in Chapter 2, that is, we reduce the insecurity problem for our two classes of cryptographic protocols to the satisfiability problem of respectively two classes of constraint systems, and we give a procedure to decide the satisfiability problem for these two classes of constraint systems.
4.4. DECIDABILITY RESULTS 91 4.4.1 Symbolic model for constructive exclusive ownership vulnerability property In order to symbolically analyse the class of cryptographic protocols using digital signature schemes vulnerable to the constructive exclusive ownership property (also called digital signature schemes having duplicate signature key selection property), we consider the following signature FDSKS = {Sk, P k, sig, ver, Sk ′ , P k ′ , 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, • 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. The key generation algorithm employs randomly generated number to perform its computation. We assume that this number is kept secret and that it is destroyed at the end of the computation. We abstract this situation by assuming the functions modelling this algorithm are private. The special functions Sk ′ , P k ′ 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 a new pair of secret and public keys (P k ′ (pk, s), Sk ′ (pk, s)) such that the verification of s with respect to m and the new public key succeeds. We assume that FDSKS = FDSKSpub ∪ FDSKSpri where FDSKSpub = {sig, ver, Sk ′ , P k ′ , 1} and FDSKSpri = {Sk, P k}. The constructive exclusive ownership vulnerability property is represented by the following equational theory, denoted by HDSKS: ⎧ ⎨ HDSKS = ⎩ ver(x, sig(x, Sk(y)), P k(y)) = 1 ver(x, sig(x, Sk ′ (y1, y2)), P k ′ (y1, y2)) = 1 sig(x, Sk ′ (P k(y), sig(x, Sk(y)))) = sig(x, Sk(y))
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Logical Analysis and Verification o
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202 BIBLIOGRAPHY [12] M. Abadi and
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216 BIBLIOGRAPHY [187] H. Seidl and
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