## Section: New Results

### Deciding trace equivalence (Objectives 1, 3)

Participants : Vincent Cheval, Hubert Comon-Lundh, Stéphanie Delaune, Rémy Chrétien.

Most existing results focus on trace properties like secrecy or authentication. There are however several security properties, which cannot be defined (or cannot be naturally defined) as trace properties and require the notion of indistinguishably. Typical examples are anonymity, privacy related properties or statements closer to security properties used in cryptography.

In the framework of the applied pi-calculus [44] , as in similar languages based on equational logics, indistinguishability corresponds to a relation called trace equivalence. Roughly, two processes are trace equivalent when an observer cannot see any difference between the two processes. Static equivalence applies only to observations on finite sets of messages, and does not take into account the dynamic behavior of a process, whereas trace equivalence is more general and takes into account this aspect.

#### Static equivalence.

As explained above, static equivalence is a cornerstone to provide decision procedures for observational equivalence.

Stéphanie Delaune, in collaboration with Mathieur Baudet and Véronique Cortier, has designed a generic procedure for deducibility and static equivalence that takes as input any convergent rewrite system [15] . They have shown that their algorithm covers most of the existing decision procedures for convergent theories. They also provide an efficient implementation, and compare it briefly with the tools ProVerif and KiSs. This paper is a journal version of the work presented in [47] .

In [17] , Ştefan Ciobâcă, Stéphanie Delaune and Steve Kremer propose a representation of deducible terms to overcome the limitation of the procedure mentionned above. This new procedure terminates on a wide range of equational theories. In particular, they obtain a new decidability result for the theory of trapdoor bit commitment encountered when studying electronic voting protocols. The algorithm has been implemented in the KiSs tool. This paper is a journal version of the work presented in [49] .

In [18] , Stéphanie Delaune, in collaboration with
Véronique Cortier (LORIA, France), shows that existing decidability
results can be easily combined for any disjoint equational theories:
if the deducibility and indistinguishability relations are decidable
for two disjoint theories, they are also decidable for their union.
They also propose a general setting for solving deducibility and
indistinguishability for an important class (called *monoidal*)
of equational theories involving $\mathrm{\U0001d5a0\U0001d5a2}$ operators. This paper
is a journal version of the works presented
in [45] , [50] .

#### Trace equivalence.

When processes under study do not contain replication, trace equivalence can be reduced to the problem of deciding symbolic equivalence, an equivalence relation introduced by M. Baudet [46] .

Stéphanie Delaune, Steve Kremer, and Daniel Pasaila study this symbolic equivalence problem when cryptographic primitives are modeled using a group equational theory, a special case of monoidal equational theories. The results strongly rely on the correspondance between group theories and rings. This allows them to reduce the problem under study to the problem of solving systems of equations over rings. This result was published at IJCAR'12 [33] ,

When processes under study contain replication, the approach relying on symbolic equivalence does not work anymore. Moreover, since it is well-known that deciding reachability properties is undecidable under various restrictions, there is actually no hope to do better for equivalence-based properties. Rémy Chrétien, Véronique Cortier, and Stéphanie Delaune provide the first results of (un)decidability for certain classes of protocols for the equivalence problem. They consider a class of protocols shown to be decidable for reachability properties, and establish a first undecidability result. Then, they restrained the class of protocols a step further by making the protocols deterministic in some sense and preventing it from disclosing secret keys. This tighter class of protocols was then shown to be decidable after reduction to an equivalence between deterministic pushdown automata (see [42] )

To deal with replication, another approach was studied by Vincent Cheval in collaboration with Bruno Blanchet. They propose an extension of the automatic protocol verifier ProVerif. ProVerif can prove observational equivalence between processes that have the same structure but differ by the messages they contain. In order to extend the class of equivalences that ProVerif handles, they extend the language of terms by defining more functions (destructors) by rewrite rules. These extensions have been implemented in ProVerif and allow one to automatically prove anonymity in the private authentication protocol by Abadi and Fournet. This work is currently under submission [40] .