Logical Analysis and Verification of Cryptographic Protocols - Loria

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162 CHAPTER 6. ON THE GROUND ENTAILMENT PROBLEMS Figure 6.1 Attack on the Needham-Schroeder protocol (1).1 A ⇒ S : A, B, NA (1).2 S ⇒ A : {NA, B, KAB, {KAB, A} s KBS }KAS (2).3 I(B) ⇒ A : {NA, B}KAS (2).4 A ⇒ I(B) : {N ′ A }NA by O. Pereira and J.-J. Quisquater [169], this attack described in Figure 6.1 allows, as we will see below, the intruder to break the secrecy of NB. This attack is possible if the encryption scheme used in the implementation of the protocol is used in the cipherblock-chaining (CBC) mode. In the case of CBC encryption, the intruder may be able to get from any encrypted message the encryption of any of its prefixes, and that without knowing the encryption key: from the message {< x, y >} s z, the intruder can deduce the message {x} s z. This property, called prefix property, is encoded by the deduction rule and hence by the clause {< x, y >} s z → {x} s z Cpre = I({< x, y >} s z) → I({x} s z) In a first session (1), the intruder can listen to the message {NA, B, KAB, {KAB, A} s }KAS and then, using the prefix property, computes KBS the message {NA, B}KAS . This message is of the form Alice might expect to receive as third message of a latter session of the protocol where Bob is considered to play initiator’s role. Then once the message {NA, B} s computed, the intruder starts a KAS another session of the protocol by sending to Alice the message {NA, B}KAS . Alice thinks that Bob has started a session (2) with her and thus, Alice can be fooled into accepting the publicly known NA as a secret key shared with Bob. Let us consider an instance of the protocol PNS where Alice instantiates once the role A and once the role B, Bob instantiates only once the role B and the agent s instantiates only once the role S, Kbs denotes the secret key shared between Bob and s, and Kas denotes the secret key shared between Alice and s. The attack described above is a possible execution of this instance of PNS. This execution is given by the set of rules {A1(Alice), S1(s), B1(Alice)}. At the end of this execution, the intruder breaks the secrecy of the nonce n ′ a. This execution is then given by the following set of rules: ∅ ⇒ Alice, Bob, na v1 ⇒ {na, Bob, Kab, {Kab, Alice} s Kbs }Kas; v1 v2 ⇒ {n ′ a}na; v2 ? = {na, Bob}Kas ? = Alice, Bob, na We associate to this execution the ground constraint system C, C = (E1 ⊲ Alice, E1⊲Bob, E1⊲na, E1∪{na, Bob, Kab, {Kab, Alice} s Kbs }sKas⊲{na, Bob} s , E1∪ Kas
6.5. A DECIDABILITY RESULT 163 {na, Bob, Kab, {Kab, Alice} s Kbs }s Kas ∪ {n′ a} s na ⊲ n′ a) and E1 is the initial knowledge of the intruder, E1 = {Alice, Bob, s, Kis, na}. We associate to this ground constraint system the following set of clauses CC: CE1 → I(Alice) CE1 → I(Bob) CE1 → I(na) CE1 ∪ I({na, Bob, Kab, {Kab, Alice}Kbs }s Kas ) → I({na, Bob} s Kas ) CE1 ∪ I({na, Bob, Kab, {Kab, Alice} s Kbs }s Kas ) ∪ I({n′ a} s na ) → I(n′ a) where CE1 = {I(Alice), I(Bob), I(s), I(Kis), I(na)}. It is easy to see that CC is entailed from CI where CI = CDV ∪ Cpre. 6.5 A decidability result In this section, we show the decidability of the ground entailment problem for a new fragment of first order logic. To get this result, we consider a refinement of resolution, the selected resolution, that we present in Section 6.5.1, and for which we show the completeness and soundness. We prove our decidability result in Section 6.5.2. 6.5.1 Selected resolution In Section 6.1.2, we have presented the resolution and the standard ordering refinement of resolution (the ordered resolution). In this section, we present another refinement of resolution, the selected resolution (or resolution with selection) and then show its completeness and soundness.The selected resolution is the resolution considered in the remainder of this chapter. A selection function is a function that will be applied to each clause and selects (or marks) a possibly empty set of its atoms. A selection function is said to be valid if for each clause, either all maximal atoms are selected or, at least one atom appearing in the antecedent of the clause is selected [146]. The selection resolution has been widely studied in the literature [134, 133, 137, 146, 164]. In [134], R. Kowalski and D. Kuehner introduced the linear resolution with selection function, also called SL-resolution. This resolution is closely related to D. W. Loveland’s model elimination system [143]. In [133], R. Kowalski introduced another variation of linear resolution with selection, which considers only Horn clauses and not general clauses as in [134]. In [146], Ch. Lynch introduced the resolution with selection as we employ it. He assumed a selection function which, for each clause, selects all maximal atoms or, at least one atom appearing in the antecedent of the clause. He assumed also that the resolution includes the following deletion rules: tautology deletion and
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Logical Analysis and Verification of Cryptographic Protocols - Loria

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