Logical Analysis and Verification of Cryptographic Protocols - Loria

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14 CHAPTER 1. INTRODUCTION have been obtained. These results can be divided into two classes: the decidability results with (respectively without) the perfect cryptography hypothesis. An encryption scheme is said to be perfect when no (not even partial) information about the plaintext can be obtained from a ciphertext without knowing the decryption (secret) key. This hypothesis has been introduced by R. Needham and M. Schroeder [163], and D. Dolev and A. Yao [107] in their respective works. When the perfect encryption hypothesis is generalised to the other cryptographic primitives, we talk about the perfect cryptography hypothesis. While perfect cryptography hypothesis is not realistic, (actually the intruder may use the algebraic properties of cryptographic primitives when he is attacking the protocol), several important attacks have been discovered. An example of such attacks would be the attack discovered by G. Lowe on the Needham- Schroeder public key [145]. Furthermore, several results have been obtained, we show below some of them. Unbounded number of sessions. Assuming an unbounded number of sessions, the problem is undecidable [14, 81, 110, 111], and remains undecidable even if we suppose that no nonces (fresh data) are generated [81, 111], or we bound the size of messages [14, 110]. This problem becomes DEXPTIME-complete if we suppose that no nonces are generated and bound the size of messages [71, 110]. In [106], the authors showed that the security of Ping-pong protocols is decidable in PTIME. Bounded number of sessions. Assuming a bounded number of sessions, M. Rusinowitch and M. Turuani [178] showed that the problem is co-NP-complete. Later, The perfect encryption hypothesis has been relaxed, and several algebraic properties of cryptographic primitives have been considered in the analysis of cryptographic protocols. Several results have been obtained, we show below some of them. The problem is co-NP-complete for the Ping-pong protocols with commutative equational theory [69, 195]. The problem remains decidable for the bounded number of sessions with exclusive Or [87]. It also remains decidable for the abelian groups [188]. Many researches have been focused on general classes of algebraic properties, and several results have been obtained in that field. For example, M. Baudet [32] has proved the decidability of the security problem for the class of cryptographic protocols using primitives represented by subterm convergent equational theories. S. Delaune and F. Jacquemard [95] have proved the decidability of the security problem for the class of cryptographic protocols using primitives represented by convergent public-collapsing equational theories, etc. Other results can be found in [90].
1.5. CONTRIBUTIONS AND PLAN OF THIS THESIS 15 1.5 Contributions and plan of this thesis In this thesis, we relax the perfect cryptography hypothesis by taking into account some algebraic properties of cryptographic primitives that we formulate by equations. We follow the symbolic approach to analyse security protocols, and in particular, the approach based on the resolution of constraint systems. To this end, we formulate the capacity of the intruder by deduction rules, and the verification task of the protocol by a reachability problem. The latter is the problem of determining if a certain (finite) parallel program which models the protocol and the specification can reach an erroneous state while interacting with the environment. We provide decision procedures for the reachability problem in presence of several algebraic operators. In Chapter 2, we give the basic notions and definitions we use in the most of this thesis. We define the constraint systems, and reachability problem. These notions have been initially introduced by J. Millen and V. Shmatikov [156], but as defined there, they are not adequate for the non empty equational theories. We actually follow the definitions introduced by Y. Chevalier and M. Rusinowitch [73] who generalised the initial definitions of [156] in order to capture non empty equational theories. We show how protocols are modeled in a high specification language, and we show how to reduce the insecurity problem of cryptographic protocols to the satisfiability problem of constraint systems. Several works follow this approach [156, 87, 73, 72]. 1.5.1 Decidability results in presence of algebraic operators Chapter 3: Analysis of protocols with collision vulnerable hash functions. In Chapter 3, we consider the class of cryptographic protocols that use collision vulnerable hash functions. The collision vulnerability property for a hash function means that one can construct two different messages having the same hash value. We remark that only a few years ago, it was intractable to compute collisions on hash functions, so they were considered to be collision resistant by cryptographers, and collision was considered to be a possible attack on hash functions only from the nineties when collision attacks have been proved and showed by several ressearchers [98, 103, 199, 201]. Examples of collision vulnerable hash functions are “MD5” and “SHA-0”. In this chapter, we symbolically represent how the intruder may compute collisions on hash functions. We then reduce the insecurity problem of our class of cryptographic protocols to the ordered satisfiability problem for the intruder using the collision vulnerability property of hash functions when attacking a protocol execution. The ordered satisfiability problem is a variant of the satisfiability problem presented in Chapter 2. It was initially introduced by Y. Chevalier et al. [72]. Roughly following the results obtained in [74], we conjecture that
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112 CHAPTER 5. SATURATED DEDUCTION
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148 CHAPTER 6. ON THE GROUND ENTAIL
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178 CHAPTER 7. VOTER VERIFIABILITY
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198 CHAPTER 8. CONCLUSION AND PERSP
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200 CHAPTER 8. CONCLUSION AND PERSP
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202 BIBLIOGRAPHY [12] M. Abadi and
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204 BIBLIOGRAPHY [36] M. Bellare, A
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206 BIBLIOGRAPHY [60] U. Hustadt Ch
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208 BIBLIOGRAPHY [83] H. Comon-Lund
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210 BIBLIOGRAPHY [108] B. Donovan,
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212 BIBLIOGRAPHY [132] T. Kohno, A.
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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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