Knowledge flow analysis offers a simple and flexible way to find flaws in security protocols. A protocol is described by a collection of rules constraining the propagation of knowledge amongst principals. Because this characterization corresponds closely to informal descriptions of protocols, it allows a succinct and natural formalization; because it abstracts away message ordering, and handles communications between principals and applications of cryptographic primitives uniformly, it is readily represented in a standard logic. A generic framework in the Alloy modelling language is presented, and instantiated for two standard protocols, and a new key management scheme.
Our formalism derives its simplicity from being just sufficiently expressive to enable modelling of practical cryptographic protocols. In particular, existentials @cite_23 cannot be encoded as knowledge flows; existentials are implicitly modeled in Oscar's initial knowledge. As mentioned in Section ), NP-hardness proofs which use (existential) Horn clause reduction @cite_23 or SAT3 reduction @cite_16 are not applicable to Knowledge Flow Analysis.
@cite_16 @cite_23
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We formalize the Dolev-Yao model of security protocols, using a notation based on multiset rewriting with existentials. The goals are to provide a simple formal notation for describing security protocols, to formalize the assumptions of the Dolev-Yao model using this notation, and to analyze the complexity of the secrecy problem under various restrictions. We prove that, even for the case where we restrict the size of messages and the depth of message encryption, the secrecy problem is undecidable for the case of an unrestricted number of protocol roles and an unbounded number of new nonces. We also identify several decidable classes, including a DEXP-complete class when the number of nonces is restricted, and an NP-complete class when both the number of nonces and the number of roles is restricted. We point out a remaining open complexity problem, and discuss the implications these results have on the general topic of protocol analysis. We are interested in the design of automated procedures for analyzing the (in)security of cryptographic protocols in the Dolev-Yao model for a bounded number of sessions when we take into account some algebraic properties satisfied by the operators involved in the protocol. This leads to a more realistic model in comparison to what we get under the perfect cryptography assumption, but it implies that protocol analysis deals with terms modulo some equational theory instead of terms in a free algebra. The main goal of this paper is to setup a general approach that works for a whole class of monoidal theories which contains many of the specific cases that have been considered so far in an ad-hoc way (e.g. exclusive or, Abelian groups, exclusive or in combination with the homomorphism axiom). We follow a classical schema for cryptographic protocol analysis which proves first a locality result and then reduces the insecurity problem to a symbolic constraint solving problem. This approach strongly relies on the correspondence between a monoidal theory E and a semiring S"E which we use to deal with the symbolic constraints. We show that the well-defined symbolic constraints that are generated by reasonable protocols can be solved provided that unification in the monoidal theory satisfies some additional properties. The resolution process boils down to solving particular quadratic Diophantine equations that are reduced to linear Diophantine equations, thanks to linear algebra results and the well-definedness of the problem. Examples of theories that do not satisfy our additional properties appear to be undecidable, which suggests that our characterization is reasonably tight. In formal approaches, messages sent over a network are usually modeled by terms together with an equational theory, axiomatizing the properties of the cryptographic functions (encryption, exclusive or, ...). The analysis of cryptographic protocols requires a precise understanding of the attacker knowledge. Two standard notions are usually used: deducibility and indistinguishability. Only few results have been obtained (in an ad-hoc way) for equational theories with associative and commutative properties, especially in the case of static equivalence. The main contribution of this paper is to propose a general setting for solving deducibility and indistinguishability for an important class (called monoidal) of these theories. Our setting relies on the correspondence between a monoidal theory E and a semiring SE which allows us to give an algebraic characterization of the deducibility and indistinguishability problems. As a consequence we recover easily existing decidability results and obtain several new ones. We present decidability results for the verification of cryptographic protocols in the presence of equational theories corresponding to xor and Abelian groups. Since the perfect cryptography assumption is unrealistic for cryptographic primitives with visible algebraic properties such as xor, we extend the conventional Dolev-Yao model by permitting the intruder to exploit these properties. We show that the ground reachability problem in NP for the extended intruder theories in the cases of xor and Abelian groups. This result follows from a normal proof theorem. Then, we show how to lift this result in the xor case: we consider a symbolic constraint system expressing the reachability (e.g., secrecy) problem for a finite number of sessions. We prove that such a constraint system is decidable, relying in particular on an extension of combination algorithms for unification procedures. As a corollary, this enables automatic symbolic verification of cryptographic protocols employing xor for a fixed number of sessions.