Logical Analysis and Verification of Cryptographic Protocols - Loria

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156 CHAPTER 6. ON THE GROUND ENTAILMENT PROBLEMS copying, i.e. protocols for which, at each step of the protocol, at most one message is blindly copied, fall into the class CP. While not all intruder clauses fall into the class CI, for example the intruder clause I({x} p y), I(y−1 ) → I(x) representing the asymmetric decryption does not fall in CI, H. Comon-Lundh and V. Cortier proposed a solution. As showed in [82], one needs to consider only a finite number of agents. H. Comon-Lundh and V. Cortier proposed to replace each problematic intruder clause, that is each intruder clause that does not fall in CI, by a finite set of Horn clauses where the terms dependent from the agents (for example, a secret key of an agent) are replaced by constants (agents parameters). The finiteness of this set is guaranteed by the fact that we need only a finite number of agents. For example, the Horn clause I({x} p y), I(y−1 ) → I(x) can be replaced by the Horn clause I({x} p pk(a) ), I(sk(a)) → I(x) if we consider a unique agent, a. In many cases, for example in the case of Dolev-Yao rules, the set of Horn clauses replacing a problematic Horn clause fall in CP. Furthermore, CP contains all ground Horn clauses, hence clauses modelling the initial intruder knowledge and the secrecy property fall in the class. H. Comon-Lundh and V. Cortier showed that the satisfiability problem of a set of clauses of CI ∪CP is decidable in 3-EXPTIME, and H. Seidl and K. Verma [187] have shown that satisfiability is in fact DEXPTIME-complete. 6.3.2 Zalinescu’s work In [205], E. Zalinescu extended the result obtained in [84] to a larger class of clauses. Indeed, the prefix property, that is the ability of the intruder to get from any encrypted message the encryption of any of its prefixes, which is represented by the Horn clause Cpre = I({< x, y >}z) → I({x}z) does not fall in neither CI nor CP, and can not be replaced by a set of clauses falling in CP. The same problem arises with the Horn clause representing blind signature Cbsig = I(sig(blind(x, y), z), I(y) → I(sig(x, z)). In order to extend intruder power to clauses like Cpre and Cbsig, E. Zalinescu defined a class of special clauses, CS, defined as follows. Given a strict and total ordering over function symbols, he considered a special function symbol f0 which is the smaller over the function symbols, and CS is the class of Horn clauses of the form I(f0(u(g(y1, . . . , yk)), v)), p� I(wi(g(y1, . . . , yk))), i=1 q� I(yil ) → I(f0(yj, z)) where {j, i1, . . . , iq} ⊆ {1, . . . , k}, p, q ≤ 0, u, wi are ground contexts, v is a term with var(v) = {z}, g �= f0 and I(f0(u(g(y1, . . . , yk)), v)) is greater than any other atom in the clause. He introduced also the notion of well-behaved term and well-behaved clause. A term is well-behaved if for any of its subterms of the form f0(u, v), the following two implications hold: l=1
6.3. FROM CRYPTOGRAPHIC PROTOCOLS TO LOGIC OF CLAUSES 157 if var(u) �= ∅ then v is a constant; if var(v) �= ∅ then u is a ground term. A clause in CS is well-behaved if the terms u, v and wi, for all i, are well-behaved. A clause not in CS is well-behaved if for any atom I(w) in the clause, w is wellbehaved. E. Zalinescu showed that if I, P, S are finite set of clauses included respectively in the classes CI, CP and CS, and if I ∪ S is saturated by a variant of ordered resolution with respect to a monotone atom ordering and P ∪ S is wellbehaved then the satisfiability of I ∪ P ∪ S is decidable. 6.3.3 Delaune, Lin and Lynch work In [180], S. Delaune, H. Lin and Ch. Lynch introduced flexible and rigid clauses. A rigid clause is a clause where variables are only allowed to have one instantiation, and a flexible clause is a clause where variables are allowed as many instantiations as desired. S. Delaune, H. Lin and Ch. Lynch assume a finite set of unary predicate symbols P, and a well-founded and total ordering > over P. Furthermore, they considered an embedding ordering � over term which is then extended to an atom ordering. They defined intruder clauses as defined in [84], that is with a unique function symbol, and then they extended this definition. To this end, they introduced a special flexible Horn clause of the form I0(f0(g(x1, . . . , xp), y2, . . . , yq), I1(xi1), . . . , Im(xim) → Im+1(f0(xi0, y2, . . . , yq)) where f0 �= g, xij ∈ {x1, . . . , xp} for every j ∈ {0, . . . , m}, and Ij ∈ P for all j.They defined a protocol clause to be a rigid Horn clause of the form I1(u1), . . . , In(un) → I0(u0) where var(u0) ⊆ var({u1, . . . , un}), and I0 ≥ Ii for all i. Note also that intruder knowledge and the secrecy goal, represented as before, are considered as protocol clauses. They defined intruder theory as follows. Assuming CI to be a finite set of intruder clauses, CS a special clause such that CI ∪ CS is saturated by ordered resolution, and the unique predicate symbol in CI ∪ CS is I. The intruder theory ICI∪CS is the set of clauses which contains I1(u1), . . . , In(un) → I0(u0) if and only if I0, . . . , In ∈ P and I0 ≥ Ii for all i, and I(u1), . . . , I(un) → I(u0) ∈ CI ∪ {CS, I(x) → I(x)} S. Delaune, H. Lin and Ch. Lynch reduced the insecurity problem to a satisfiability problem defined as follows. The insecurity problem takes as input a finite set CP of protocol clauses, a finite set CI of intruder clauses and a special
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