Logical Analysis and Verification of Cryptographic Protocols - Loria

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54 CHAPTER 2. PROTOCOL ANALYSIS USING CONSTRAINT SOLVING “given an execution exec of an instance of a protocol, does this execution preserve secrecy?” An execution exec preserves the secrecy of the data s if and only if the intruder, at the end of that execution, is not able to deduce the data s. While the execution exec = {∅ ⇒ S1, v2 ⇒ S2, . . . , vn ⇒ Sn; U2, . . . , Un} is associated to the constraint system Cexec = (E1 ⊢ v2, . . . , En−1 ⊢ vn, U) defined as in Section 2.3.1, the knowledge of the intruder at the end of exec is En = En−1∪Sn, and hence, the secrecy of I s is preserved in that execution if and only if s �∈ En . We conclude that the secrecy property of the execution can be encoded directly by adding an additional ? constraint En ⊢ vn+1 and additional equation vn+1 =H s to Cexec, and asking for the satisfiability of the new constraint system, also called extended constraint system. We say that a protocol preserves the secrecy property if and only if each execution of that protocol preserves the secrecy property, and to verify whether the secrecy property is preserved by a protocol for a “bounded number of sessions” (i.e. an instance of a protocol), it is then sufficient to check whether the secrecy property is preserved by each execution of that instance. In [156] and [88], the authors showed that not all executions of an instance of a protocol need to be enumerated. But anyhow, here we are only interested whether an arbitrary execution of an instance of a protocol preserves the secrecy property, and deciding whether an execution preserves the secrecy property is done by deciding the satisfiability of the correspondant extended constraint system. Hence, deciding whether a protocol under a bounded number of sessions is insecure, that is does not preserve the secrecy property (also called deciding the secrecy problem of a protocol under a bounded number of sessions) is reduced to deciding the satisfiability of constraint systems (also called decidability of reachability problem). Example 14 We consider the execution given in the Example 13. We recall the constraint � system associated � to that execution: ? = 〈X1, 〈Y1, Z1〉〉 . C = (E1 ⊢ v1, U) and U = v1 Let sec be a secret data. Deciding if the given execution does not preserve the secrecy of sec under the intruder Ie DY is reduced to deciding if the following constraint system where E2 = E1 ∪ { C = (E1 ⊢ v1, E2 ⊢ v6, U ′ ) 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)}, and U ′ � � ? = U ∪ = ses is I e DY -satisfiable. v6
2.4. CONCLUSION 55 2.4 Conclusion In this chapter, we have presented the basic notions that we will use in the next chapters. Mainly, we showed how cryptographic protocols are symbolically modeled, and how to reduce the insecurity problem of cryptographic protocols with bounded number of sessions to the satisfiability problem of constraint systems. In the next chapters, we will relax the perfect cryptography assumption by taking into account some algebraic properties of cryptographic primitives, and, following the symbolic approach given in this chapter, we will analyse cryptographic protocols in the presence of some algebraic operators.
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Logical Analysis and Verification o
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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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