Logical Analysis and Verification of Cryptographic Protocols - Loria

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52 CHAPTER 2. PROTOCOL ANALYSIS USING CONSTRAINT SOLVING ordered as follows: 2 < 1 < 3 < 5. More formally, the following sequence of rules ∅ ⇒ 〈Alice, 〈Bob, na〉〉; ∅ v1 ⇒ encs (〈π2(π2(v1)), 〈π1(π2(v1)), 〈Kπ1(v1)π1(π2(v1)), encs (〈Kπ1(v1)π1(π2(v1)), π1(v1)〉, Kπ1(π2(v1))s)〉〉〉, Kπ1(v1)s); ? v1 = X1, Y1, Z1 v2 ⇒ π2(π2(π2(decs ? (v2, Kas)))); v2 = encs (〈na, 〈Bob, 〈Y2, Z2〉〉〉, Kas) v4 ⇒ encs (nb, π1(decs ? (v4, Kbs))); v4 = encs (〈X3, Y3〉, Kbs) represents an execution of the instance given in Example 11. 2.3 From cryptographic protocols to constraint systems Constraint systems are quite common in modelling security protocols for a bounded number of sessions [156, 87, 73, 72]. Actually, many protocol security properties can be characterised as reachability problems which are converted to constraint solving problems. This happens for properties such as secrecy, where the objective of protocol analysis is to search for an execution of the protocol in which some secret data has been released publicly by the intruder. Although the reachability is undecidable for cryptographic protocols in the general case [110], some decidability results have been obtained for restricted cases, for instance, in [178, 15, 156], the authors proved the decidability of reachability for cryptographic protocols in the case of finite number of sessions. We show here how constraint systems can be used to analyse cryptographic protocols. We present in Section 2.3.1 how to built constraint system from an execution of a protocol, and in Section 2.3.2 we show how to reduce insecurity problem of a protocol with a bounded number of sessions to the satisfiability problem of constraint systems. We follow the same definitions, constructions and notations given in previous works [156, 73]. 2.3.1 From an execution of a protocol to a constraint system Given an instance of a protocol, and let exec be an execution of the given instance of protocol. Assume exec = {∅ ⇒ S1, v2 ⇒ S2; U2, . . . , vn ⇒ Sn; Un} = {∅ ⇒ S1, v2 ⇒ S2, . . . , vn ⇒ Sn; U2, . . . , Un}. The constraint system associated with the execution exec, denoted by Cexec, and with the initial intruder knowledge KI is: Cexec = (E1 ⊢ v2, . . . , En−1 ⊢ vn, U) where: • E1 is a set of ground terms in T (F) representing the initial intruder knowledge and the first sent message, E1 = KI ∪ S1,
2.3. FROM CRYPTOGRAPHIC PROTOCOLS TO CONSTRAINT SYSTEMS 53 • for every i in the set {2, . . . , n − 1}, Ei is a set of terms included in T (F, X ) defined as Ei = Ei−1 ∪ Si • U = U2 ∪ . . . ∪ Un. It is easy to see that Cexec is a constraint system in the sense of Definition 20. In the context of cryptographic protocols, the inclusion Ei ⊆ Ei−1 for every i ∈ {1, . . . , n − 1} means that the knowledge of an intruder does not decrease as the protocol progresses: after receiving a message a honest agent will respond to it, this response can then be added to the knowledge of the intruder who listens to all communications. The condition on variables, V ar(Ei) ⊆ {v1, . . . , vi−1}, stems from the fact that a message sent at some step i must be built from previously received messages recorded in the variables vj, j < i, and from the ground initial knowledge of the honest agents. An execution exec is said to be a possible execution if the correspondant constraint system Cexec is satisfiable, that is, there is a substitution σ satisfying Cexec with respect to the intruder I. Example 13 We consider the instance of Needham-Schroeder symmetric key protocol given in Example 11, and we consider an execution of that instance given by the following sequence of rules ∅ ⇒ 〈Alice, 〈Bob, na〉〉 v1 ⇒ encs (〈π2(π2(v1)), 〈π1(π2(v1)), 〈Kπ1(v1)π1(π2(v1)), encs (〈Kπ1(v1)π1(π2(v1)), π1(v1)〉, Kπ1(π2(v1))s)〉〉〉, Kπ1(v1)s); ? = 〈X1, 〈Y1, Z1〉〉 v1 The associated constraint system is defined as follows: C = (E1 � � ⊢ v1, U) and ? U = = 〈X1, 〈Y1, Z1〉〉 . The intruder deduction system considered here is the v1 Dolev-Yao intruder system with explicit destructors, Ie DY . It is easy to see that the constraint system given above has a solution which is given by the substitution σ = {X1 ↦→ Alice, Y1 ↦→ Bob, Z1 ↦→ na, x1 ↦→ Alice, x2 ↦→ Bob, x3 ↦→ na}. Thus, the execution given above is a possible execution. 2.3.2 From insecurity problem to satisfiability of constraint systems We are concerned here with the secrecy property. Secrecy. The secrecy property can be expressed by requiring that the secret data s is not deducible from the messages sent on the network. Assume S to be the set of messages sent on the network, the secrecy of the message (data) s is preserved if and only if s is not deduced from S, that is s �∈ ¯ S I where I is the given intruder system. Here, we are interested by the secrecy property in the following context:
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