## Section: New Results

### Byzantine Resilience in autonomous networks

Participant : Sébastien Tixeuil.

We studied [22] the problem of Byzantine-robust topology discovery in an arbitrary asynchronous network. We formally stated the weak and strong versions of the problem. The weak version requires that either each node discovers the topology of the network or at least one node detects the presence of a faulty node. The strong version requires that each node discovers the topology regardless of faults. We focused on non-cryptographic solutions to these problems. We explored their bounds. We proved that the weak topology discovery problem is solvable only if the connectivity of the network exceeds the number of faults in the system. Similarly, we showed that the strong version of the problem is solvable only if the network connectivity is more than twice the number of faults. We presented solutions to both versions of the problem. Our solutions match the established graph connectivity bounds. The programs are terminating, they do not require the individual nodes to know either the diameter or the size of the network. The message complexity of both programs is low polynomial with respect to the network size.

Given a set of robots with arbitrary initial location and no
agreementon a global coordinate system, *convergence* requires
that allrobots asymptotically approach the exact same, but unknown
beforehand,location. Robots are oblivious— they do not recall the
pastcomputations — and are allowed to move in a
one-dimensionalspace. Additionally, robots cannot communicate
directly, instead theyobtain system related information only
*via* visual sensors. We draw in [32] a connection
between the convergence problem in robotnetworks, and the distributed
*approximate agreement* problem(that requires correct processes
to decide, for some constant , values distance
apart and within the range ofinitial proposed values). Surprisingly,
even though specifications are similar,the convergence implementation
in robot networks requires specific assumptions about synchrony and
Byzantine resilience. In more details, we proved necessary and
sufficient conditions for the convergence of mobile robots despite a
subset of them being Byzantine (*i.e.* they can exhibit arbitrary
behavior). Additionally, we proposed a deterministic convergence
algorithm for robot networks and analyze its correctness and
complexity in various synchrony settings.The proposed algorithm
tolerates f Byzantine robots for (2f + 1) -sized robot networks in
fully synchronous networks, (3f + 1) -sized in semi-synchronous
networks and (4f + 1) -sized in asynchronous networks. The bounds
obtained for the ATOM model are optimal for the class of
*cautious* algorithms, which guarantee that correct robots always
move inside the range of positions of the correct robots. We proposed
in [33] the first deterministic algorithm that tolerates up
to f byzantine faults in 3f + 1 -sized networks and performs in the
asynchronous CORDA model. Our solution matches the previously
established lower bound for the semi-synchronous ATOM model on the
number of tolerated Byzantine robots. Our algorithm works under
bounded scheduling assumptions for oblivious robots moving in a
uni-dimensional space. We also studied [31] the convergence
problem in fully asynchronous, uni-dimensional robot networks that are
prone to Byzantine (i.e. malicious) failures. We proposed a
deterministic algorithm that solves the problem in the most generic
settings: fully asynchronous robots that operate in the non-atomic
CORDA model. Our algorithm provides convergence in 5f + 1 -sized
networks where f is the upper bound on the number of Byzantine
robots. Additionally, we proved that 5f + 1 is a lower bound whenever
robot scheduling is fully asynchronous. This constrasts with previous
results in partially synchronous robots networks, where 3f + 1 robots
are necessary and sufficient.