Logical Analysis and Verification of Cryptographic Protocols - Loria

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62 CHAPTER 3. PROTOCOLS WITH VULNERABLE HASH FUNCTIONS 3.3 The model To analyse cryptographic protocols, we follow in this chapter the symbolic model described in Chapter 2. To this end, we assume an infinite set of variables X , an infinite set of constants C, a set of function symbols F. In addition to what is already intoducted in Chapter 2, we make use here of some additional notions that we will show next. Given a term t, we recall that V ar(t) denotes the set of variables appearing in t, we recall also that Cons(t) denotes the set of free constants appearing in t, that is, the set of constants from C appearing in t. Now, we extend the definition of Cons(t) to take in consideration non-free constants, i.e. the function symbols in F with arity 0, appearing in t, and we defined all.Cons(t) to denote the set of free constants of t together with the set of non-free constants of t. More formally, all.Cons(t) = Cons(t) ∪ � f ∈ F such that arity(f) = 0 and there is a position p ∈ P os(t) with t|p = f � . We define the set of atoms Atoms to be the union of V ar and all.Cons, if t is a term, Atoms(t) = V ar(t) ∪ Cons(t). In the rest of this chapter, we assume a complete simplification ordering > on T (F, X ), i.e. > is a simplification ordering on T (F, X ) total over ground terms T (F), and for which the minimal element is a free constant, i.e. a constant in C, called cmin. We denote by Cspec the set consisting of cmin and of all symbols in F of arity 0, i.e. Cspec = {cmin} ∪ {f ∈ F such that arity(f) = 0}. 3.3.1 Mode in an equational theory The notion of Mode on a signature F has been initially defined in [74]. Assume H is an equational theory over a signature F. Assume also that F is partitioned into two disjoint sets F0 and F1, and that X is partitioned into two disjoint sets X0 and X1. We first define a signature function Sign on F ∪ Atoms in the following way: Sign : F⎧ ∪ Atoms → {0, 1, 2} ⎨ 0 if f ∈ F0 ∪ X0 Sign(f) = 1 if f ∈ F1 ∪ X1 ⎩ 2 otherwise, i.e. when f is a free constant (f ∈ C) The function Sign is extended to terms by taking Sign(t) = Sign(T op(t)). We also assume that there exists a Mode function with arity 2, Mode, such that Mode(f, i) is defined for every symbol f ∈ F and every integer i such that 1 ≤ i ≤ arityf. For all valid f, i we have Mode(f, i) ∈ {0, 1} and Mode(f, i) ≤ Sign(f). Thus for all f ∈ F0 and for all i we have Mode(f, i) = 0.
3.3. THE MODEL 63 3.3.2 Well-moded equational theories A position different from ε in a term t is well-moded if it can be written p · i (where p is a position and i a non negative integer) such that Sign(t|p·i) = Mode(T op(t|p), i). In other words the position in a term is well-moded if the subterm at that position is of the expected type with respect to the function symbol immediately above it. A term is well-moded if all its non root positions are well-moded. Note in particular that a well-moded term does not contain free constants (i.e. constants in C). If a position of t is not well-moded we say it is ill-moded in t. The root position of a term is always ill-moded. An equational theory H is well-moded if for all equations u · = v ∈ H the terms u and v are well-moded and Sign(u) = Sign(v), and similarly, a rule u → v with u and v are two terms is well-moded if u and v are well-moded and Sign(u) = Sign(v). In [74], the authors proved that if an equational theory is well-moded then the application of Unfailing Knuth-Bendix completion procedure (Chapter 2, Section 2.1.6) outputs a well-moded system. We remark also that if H is the union of two equational theories H0 and H1 over two disjoint signatures F0 and F1, the theory H is well-moded when assigning Mode i to each argument of each operator f ∈ Fi, for i ∈ {0, 1}. 3.3.3 Subterm values The notion of Mode also permits to define a new subterm relation in T (F, X ). This relation has been initially defined in [74]. We call a subterm value of a term t a subterm (as defined in Section 2.1.3, Chapter 2) of t that is either atomic or occurs at an ill-moded position of t. We denote Subv(t) the set of subterm values of t. We extend as expected the notion of subterm value to a set of terms E, Subv(E) = � e∈E Subv(e). We denote by factors(t), for a term t, the subset of the maximal and strict subterm values of t. Example 15 Consider two binary function symbols f and g with Sign(f) = Sign(g) = Mode(f, 1) = Mode(g, 1) = 1 and Mode(f, 2) = Mode(g, 2) = 0, and t = f(f(g(a, b), f(c, c)), d). Its subterm values are a, b, f(c, c), c, d, and its factors are a, b, f(c, c) and d. In the rest of this chapter, the notion of subterm will refer to subterm values. 3.3.4 Intruder deduction system As stated before, our concern in this chapter is the symbolic analysis of cryptographic protocols using collision vulnerable hash functions. In chapter 2, we show how to reduce the insecurity problem of cryptographic protocols to the satisfiability problem of constraint solving. This reduction is done assuming the intruder deduction rules are of the form:
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Logical Analysis and Verification o
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Logical Analysis and Verification of Cryptographic Protocols - Loria

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