## Section: New Results

### Fault Tolerance in Distributed Networks

#### Wait-Freedom with Advice

Participants : Carole Delporte-Gallet, Hugues Fauconnier, Eli Gafni, Petr Kuznetsov.

In [14] , we motivate and propose a new way of thinking about failure detectors
which allows us to define, quite surprisingly, what it means to solve a distributed task
*wait-free* *using a failure detector*. In our model, the system is composed of
*computation* processes that obtain inputs and are supposed to output in a finite
number of steps and *synchronization* processes that are subject to failures and can
query a failure detector. We assume that, under the condition that *correct*
synchronization processes take sufficiently many steps, they provide the computation
processes with enough *advice* to solve the given task wait-free: every computation
process outputs in a finite number of its own steps, regardless of the behavior of other
computation processes. Every task can thus be characterized by the *weakest* failure
detector that allows for solving it, and we show that every such failure detector captures
a form of set agreement. We then obtain a complete classification of tasks, including
ones that evaded comprehensible characterization so far, such as renaming or weak
symmetry breaking.

#### Partial synchrony based on set timeliness

Participants : Markos Aguilera, Carole Delporte-Gallet, Hugues Fauconnier, Sam Toueg.

We introduce in [1] , a new model of partial synchrony for read-write shared memory systems. This model is based on the simple notion of set timeliness—a natural generalization of the seminal concept of timeliness in the partially synchrony model of Dwork et al. (J. ACM 35(2):288–323, 1988). Despite its simplicity, the concept of set timeliness is powerful enough to define a family of partially synchronous systems that closely match individual instances of the $t-$resilient $k-$set agreement problem among $n$ processes, henceforth denoted $(t,k,n)-$agreement. In particular, we use it to give a partially synchronous system that is synchronous enough for solving $(t,k,n)-$ agreement, but not enough for solving two incrementally stronger problems, namely, $(t+1,k,n)-$agreement, which has a slightly stronger resiliency requirement, and $(t,k-1,n)-$agreement, which has a slightly stronger agreement requirement. This is the first partially synchronous system that separates these sub-consensus problems. The above results show that set timeliness can be used to study and compare the partial synchrony requirements of problems that are strictly weaker than consensus.

#### Byzantine Agreement with Homonyms in Synchronous Systems

Participants : Carole Delporte-Gallet, Hugues Fauconnier, Hung Tran-The.

In [15] , [6] , we consider the Byzantine agreement problem (BA) in synchronous systems with homonyms. In this model different processes may have the same authenticated identifier. In such a system of $n$ processes sharing a set of $l$ identifiers, we define a distribution of the identifiers as an integer partition of $n$ into $l$ parts ${n}_{1},...,{n}_{l}$ giving for each identifier $i$ the number of processes having this identifier.

Assuming that the processes know the distribution of identifiers we give a necessary and sufficient condition on the integer partition of $n$ to solve the Byzantine agreement with at most $t$ Byzantine processes. Moreover we prove that there exists a distribution of $l$ identifiers enabling to solve Byzantine agreement with at most $t$ Byzantine processes if and only if $n>3t$, $l>t$ and where $r=n\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}l$.

This bound is to be compared with the $l>3t$ bound proved in Delporte-Gallet et al. (2011) when the processes do not know the distribution of identifiers.

#### Homonyms with forgeable identifiers

Participants : Carole Delporte-Gallet, Hugues Fauconnier, Hung Tran-The.

In [16] , we refine the Byzantine Agreement problem (BA) in synchronous systems with homonyms, in the particular case where some identifiers may be forgeable. More precisely, the $n$ processes share a set of $l$ ($1\le l\le n$) identifiers. Assuming that at most $t$ processes may be Byzantine and at most $k$ ($t\le k\le l$) of these identifiers are forgeable in the sense that any Byzantine process can falsely use them, we prove that Byzantine Agreement problem is solvable if and only if $l>2t+k$. Moreover we extend this result to systems with authentication by signatures in which at most $k$ signatures are forgeable and we prove that Byzantine Agreement problem is solvable if and only if $l>t+k$.