On the Decidability of (ground) Reachability Problems for Cryptographic Protocols (extended version)

Analysis of cryptographic protocols in a symbolic model is relative to a deduction system that models the possible actions of an attacker regarding an execution of this protocol. We present in this paper a transformation algorithm for such deduction systems provided the equational theory has the finite variant property. the termination of this transformation entails the decidability of the ground reachability problems. We prove that it is necessary to add one other condition to obtain the decidability of non-ground problems, and provide one new such criterion

A Proof Theoretic Analysis of Intruder Theories

We consider the problem of intruder deduction in security protocol analysis: that is, deciding whether a given message M can be deduced from a set of messages Gamma under the theory of blind signatures and arbitrary convergent equational theories modulo associativity and commutativity (AC) of certain binary operators. The traditional formulations of intruder deduction are usually given in natural-deduction-like systems and proving decidability requires significant effort in showing that the rules are "local" in some sense. By using the well-known translation between natural deduction and sequent calculus, we recast the intruder deduction problem as proof search in sequent calculus, in which locality is immediate. Using standard proof theoretic methods, such as permutability of rules and cut elimination, we show that the intruder deduction problem can be reduced, in polynomial time, to the elementary deduction problem, which amounts to solving certain equations in the underlying individual equational theories. We show that this result extends to combinations of disjoint AC-convergent theories whereby the decidability of intruder deduction under the combined theory reduces to the decidability of elementary deduction in each constituent theory. To further demonstrate the utility of the sequent-based approach, we show that, for Dolev-Yao intruders, our sequent-based techniques can be used to solve the more difficult problem of solving deducibility constraints, where the sequents to be deduced may contain gaps (or variables) representing possible messages the intruder may produce.Comment: Extended version of RTA 2009 pape

O problema da dedução do intruso para teorias AC-convergentes localmente estáveis

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2013.Apresenta-se um algoritmo para decidir o problema da dedução do intruso (PDI) para a classe de teorias localmente estáveis normais, que incluem operadores associativos e comutativos (AC). A decidibilidade é baseada na análise de reduções de reescrita aplicadas na cabeça de termos que são construídos a partir de contextos normais e o conhecimento inicial de um intruso. Este algoritmo se baseia em um algoritmo eficiente para resolver um caso restrito de casamento módulo AC de ordem superior, obtido pela combinação de um algoritmo para Casamento AC com Ocorrências Distintas, e um algoritmo padrão para resolver sistemas de equações Diofantinas lineares. O algoritmo roda em tempo polinomial no tamanho de um conjunto saturado construído a partir do conhecimento inicial do intruso para a subclasse de teorias para a qual operadores AC possuem inversos. Os resultados são aplicados para teoria AC pura e a teoria de grupos Abelianos de ordem n dada. Uma tradução entre dedução natural e o cálculo de sequentes permite usar a mesma abordagem para decidir o problema da dedução elementar para teorias localmente estáveis com inversos. Como uma aplicação, a teoria de assinaturas cegas pode ser modelada e então, deriva-se um algoritmo para decidir o PDI neste contexto, estendendo resultados de decidibilidade prévios. ______________________________________________________________________________ ABSTRACTWe present an algorithm to decide the intruder deduction problem (IDP) for the class of normal locally stable theories, which include associative and commutative (AC) opera- tors. The decidability is based on the analysis of rewriting reductions applied in the head of terms built from normal contexts and the initial knowledge of the intruder. It relies on a new and efficient algorithm to solve a restricted case of higher-order AC-matching, obtained by combining the Distinct Occurrences of AC-matching algorithm and a stan- dard algorithm to solve systems of linear Diophantine equations. Our algorithm runs in polynomial time on the size of a saturation set built from the initial knowledge of the intruder for the subclass of theories for which AC operators have inverses. We apply the results to the Pure AC equational theory and Abelian Groups with a given order n. A translation between natural deduction and sequent calculus allows us to use the same approach to decide the elementary deduction problem for locally stable theories with inverses. As an application, we model the theory of blind signatures and derive an algorithm to decide IDP in this context, extending previous decidability results