In the Lattice Agreement (LA) problem, originally proposed by Attiya et al. cite{Attiya:1995}, a set of processes has to decide on a chain of a lattice. More precisely, each correct process proposes an element $e$ of a certain join-semi lattice $L$ and it has to decide on a value that contains $e$. Moreover, any pair $p_i,p_j$ of correct processes has to decide two values $dec_i$ and $dec_j$ that are comparable (e.g., $dec_i leq dec_j$ or $dec_j < dec_i$). LA has been studied for its practical applications, as example it can be used to implement a snapshot objects cite{Attiya:1995} or a replicated state machine with commutative operations cite{Faleiro:2012}. Interestingly, the study of the Byzantine Lattice Agreement started only recently, and it has been mainly devoted to asynchronous systems. The synchronous case has been object of a recent pre-print cite{Zheng:aa} where Zheng et al. propose an algorithm terminating in ${cal O}(sqrt f)$ rounds and tolerating $f < lceil n/3 rceil$ Byzantine processes. In this paper we present new contributions for the synchronous case. We investigate the problem in the usual message passing model for a system of $n$ processes with distinct unique IDs. We first prove that, when only authenticated channels are available, the problem cannot be solved if $f=n/3$ or more processes are Byzantine. We then propose a novel algorithm that works in a synchronous system model with signatures (i.e., the {em authenticated message} model), tolerates up to $f$ byzantine failures (where $f<n/3$) and that terminates in ${cal O}(log f)$ rounds. We discuss how to remove authenticated messages at the price of algorithm resiliency ($f < n/4$). Finally, we present a transformer that converts any synchronous LA algorithm to an algorithm for synchronous Generalised Lattice Agreement.
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