## Section: New Results

### Algebraic properties of cryptographic primitives

Participants : Sergiu Bursuc, Hubert Comon-Lundh, Stéphanie Delaune.

To enable formal and automated analysis of security protocols, one has to abstract implementations of cryptographic primitives by terms in a given algebra. However, the algebra can not be free, as cryptographic primitives have algebraic properties that are either relevant to their specification or else they can be simply observed in implementations at hand. These properties are sometimes essential for the execution of the protocol, but they also open the possibility for an attack, as they give to an intruder the means to deduce new information from the messages that he intercepts over the network.

In consequence, there was much work over the last few years towards enriching the Dolev-Yao model, originally based on a free algebra, with algebraic properties, modelled by equational theories. In this context, we have been interested in general decision procedures for the insecurity of protocols, that can be applied to classes of equational theories.

In [24] , Sergiu Bursuc and Hubert Comon-Lundh have proposed a general way to simplify an equational theory, based on an appropriate definition of alien subterms of a term. From this, they derive a decision procedure for a non-trivial combination of Abelian group properties, exponentiation and homomorphism. This theory was proposed by Stéphanie Delaune, in her PhD thesis, for modelling an electronic purse protocol by France Telecom. Previously known techniques were not applicable, as the theory was a too intricate combination of sub-theories.

Next, in [25] , Sergiu Bursuc, Hubert Comon-Lundh and Stéphanie Delaune have shown that constraint systems, that represent all possible traces of a protocol, can be simplified in an uniform and systematic way, when the equational theory does not contain Associative-Commutative symbols. This allows for a symbolic representation of all traces as a set of solved forms. The main property of the equational theory that ensures the completeness of the proposed simplification procedure is saturation with respect to ordered resolution. When the saturated theory is finite, the set of solved forms is finite as well and permits deciding any trace property, in particular the secrecy of a message. When the saturated theory is infinite, one needs to group solved forms together, by introducing a new predicate, in order to obtain a finite representation of all solutions of the original system. This has been done for the particular case of blind signatures, result that is yet to be published and is part of the PhD thesis of Sergiu Bursuc.