A typing result for trace inclusion (for pair and symmetric encryption only)

Privacy-type properties such as vote secrecy, anonymity, or untraceability are typicallyexpressed using the notion of trace equivalence in a process algebra that models securityprotocols. In this paper, we propose some results to reduce the search space when we arelooking for an attack regarding trace equivalence. Our work is strongly inspired from [10],which establishes that, if there is a witness of non trace equivalence, then there is one that iswell-typed.Our main contribution is to establish a similar result for trace inclusion. Our motivationis twofolds: first, this small attack property is needed for proving soundness of the toolSatEquiv [13]. Second, we revisit the proof in order to simplify it. Specifically, we show tworesults. First, if there is a witness of non-inclusion then there is one that is well-typed. Weestablish this result by providing a decision procedure for trace inclusion similar to the oneproposed in [10] for trace equivalence. We also show that we can reduce the search space whenconsidering the notion of static inclusion. Acutally, if there is a witness of static non-inclusionthere is one of a specific shape.Even if our setting slightly differs from the one considered in [10], our proofs essentiallyfollow the same ideas than the existing proof for trace equivalence. Nevertheless, we hopethat this proof will be easier to extend to other primitives such as asymmetric encryption orsignatures.