 Today I'm going to present to you a work that has been done in collaboration with the University of Canom and Paris Dauphine and my industrial partner, that is Kézé Depot in France. My sincere congrats for the organization of this conference. Really, really professional. Well done. Okay, I would like to introduce you to the problem of cross-chain swap among different blockchains. So we have different blockchains that has to swap assets or cryptocurrency, for instance, among each other. I will show you some preliminary results and model through game theoretical notions and analysis of the equilibria. So for this work we started from scratch, basically redefining the... So think about what is a general swap problem and then trying to define what is a blockchain swap protocol. And we did it step by step. So first of all what is a swap problem is a problem that involves assets, owners, and some method that has to be used to exchange these two. So it's our ownership transaction that changed. So what we have here is that we can define there's a problem thanks to a couple of asset owners and subjective map that denotes that with the original configuration that we want and that we have and the desired one be started that we want. And what we have is that the cardinality of the set of asset is greater with respect to the owner's one because we don't want share ownership of an asset. So we want owners to have more assets but not to share the ownership of a single asset, for instance. Okay, we can define... So we have now the first configuration, the original one and the desired one that we want. The assets involved, the owners involved, but what we need is a measure of the happiness of the two of the owners, evidently, the swap participants and we need a classical utility function that is monotonic. And of course the player is happy if he has more assets with respect to having less. So what is a protocol that involves cross-chain exchange is a sequence. So we have a sequence of transfer with the asset symbols, owners involved, and a map that is this XK that tell us who is going to own what. And of course it has been done in sequence. So we have an index that in this case is the K that runs from one to T, identifying these exchange that takes place. So the ownership change event during the time. And of course what we have is that a certain step we have a transfer of an object or multiple object but the other stays stately. So at time K, if the asset is not involved, we stay with the configuration we had in K minus one. An example. So this is an example with... So five assets, three owners, and we can see the initial and the desired configuration with... We can go through this graphical representation that we have. So we can, from our configuration, we can go through the graphical ones, but we cannot do the opposite. And we can see that, for instance, in this case the swap takes place, the exchange takes place in three steps. So we have one and two, so A and C, the change owners at step one, then we have D and then we have B and E. Okay, now we have defined an exchange protocol, a decentralized exchange protocol. What we want to define is a swap protocol. So if a swap is a situation where I can, okay, you can see from the camera, I'm standing here with two hands and I give an object and I receive another one from the other side. What we don't want to have is the passing of an object during the swap. So I can take one and I give to the first person on the row. So one time I give one and I receive one without passing it. And to define that simply, we have to simply add the constraints that AK is a partition of the set of assets. In this case, we have swap protocol. Then we have to take the swap protocol and contextualize it in the blockchain system. And of course we have to deal with the problem of all or nothing atomicity. So what we want to have is that any commitment so any actual transfer of the property of an object should be conditioned on the current asset locking. So what we need to go to us to lock the asset before transferring the ownership because we want to preserve this all or not guarantee the all or nothing atomicity property. The property is also is also that fact that consequently to failures in the asset locking the initial situation must be restored. And once an asset transfer is committed, all the other transfers as committed to. So we have to guarantee the truth, these three things. And in order to do that we need to separate two phases so the locking phases that is matches the publishing phases so we have this transfer that is locked and has to be published on the blockchain because you know that contract needs to be validated and published on the blockchain and then what we have to do is to commit so remove the locker unlock the transfer and transfer the ownership of an object. And these two phases are sequential so first the publishing one and then the commitment one. And the commitment one is coincide with the decentralized blockchain swap protocol that we have presented before. And the publishing one is a simple sequence establishing the publisher and the transfer that has to be. We want to to have so preliminary results we were able to formalize the concept of commitment requirements so the fact that we want to we can have a transfer so a commitment only if the publishing phase has been instituted in a correct manner and of course commitment requirement is an accessory condition for blockchain swap to be atomic. Second easy trivial result is the fact that if along the sequence there is a transfer that doesn't occur in a correct manner so doesn't realize we have a situation where there is someone that game and someone to lose. The fact that the first I the first commitment says that there is someone that is happier because has an asset more and someone that second I the second point said that okay there is someone that is said because he loses an asset. So we have a situation this way and this position will be useful for the final result. And then we identified the fact that each player plays a role in the commitment phase so the transfer has to be unlocked and there is someone that does that. And this someone is the decider and that is established through this decision faction F that can decide whether to unlock or not this transfer. And of course it can be as well participants so in the asset all there you can see function F or can be an external actor that stays in lies in this asset T and the scene of later what you can do. So let's take a case where F is effective that means the owner that has to unlock the commitment is the one that has to receive the asset the object. So of course, in this case, let me go just let me jump a slide. In this case we can see that if half is effective so we have a configuration in this way. Everyone is incentivized to follow so everyone so following the protocol sorry that is specified as action one is a sub game perfect equilibrium. That is of course a refinement of the Nash equilibrium. That means that of course, if I have to decide whether to take an asset I go. There is no rational intention to deviate. And of course that's first the first was the first result for a class of a protocol that is called sequential publishing and commitment protocols. And we used an extensive game to model that and of course that the strategies were follow or not to follow but it's not always the case we can have other protocols and we'll see the last example. So, first example of secretion protocol is the one proposed by Nolan and generalized by Harley. And in this case we can see that our formalization actually give actually formalize and characterize this protocol we can see from J and K that we have two steps. So it's sequential. And the results that I presented so far action one so following the protocol is a sub game perfect Nash equilibrium. But let's take another protocol for example, a concurrent snap protocols where the to the commitment phase takes place in just one step and we have a central authority T so an external one that has to decide whether to unlock or not the transfer so to allow the commitment. And of course we can see in this case that everything takes place in one step and we can see from the index J and K that is one, and that FTK is T so the the actor who decides is actually an external synchronization line in T. And for this case, we were able to prove that action one so following the protocols is Nash equilibrium. So, thanks for your attention. I had to so to this formalization we were able to identify each type of existing swap protocol among blockchain so cross chain swap protocol. So we are working on this topic for next publication and I'm here for for answering event or question. Thanks.