## Multi-Threshold Asynchronous Reliable Broadcast and Consensus

### Martin Hirt, Ard Kastrati, and Chen-Da Liu Zhang

Classical protocols for reliable broadcast and consensus provide security guarantees as long as the number of corrupted parties $f$ is bounded by a single given threshold $t$. If $f > t$, these protocols are completely deemed insecure. We consider the relaxed notion of \emph{multi-threshold} reliable broadcast and consensus where validity, consistency and termination are guaranteed as long as $f \le t_v$, $f \le t_c$ and $f \le t_t$ respectively. For consensus, we consider both variants of $(1-\epsilon)$-consensus and \emph{almost-surely terminating} consensus, where termination is guaranteed with probability $(1-\epsilon)$ and $1$, respectively. We give a very complete characterization for these primitives in the asynchronous setting and with no signatures:

-Multi-threshold reliable broadcast is possible if and only if $\max\{t_c,t_v\} + 2t_t < n$.

-Multi-threshold almost-surely consensus is possible if $\max\{t_c, t_v\} + 2t_t < n$, $2t_v + t_t < n$ and $t_t < n/3$. Assuming a global coin, it is possible if and only if $\max\{t_c, t_v\} + 2t_t < n$ and $2t_v + t_t < n$.

-Multi-threshold $(1-\epsilon)$-consensus is possible if and only if $\max\{t_c, t_v\} + 2t_t < n$ and $2t_v + t_t < n$.

## BibTeX Citation

@inproceedings{HiKaLi20b, author = {Martin Hirt and Ard Kastrati and Chen-Da Liu Zhang}, title = {Multi-Threshold Asynchronous Reliable Broadcast and Consensus}, booktitle = {International Conference on Principles of Distributed Systems --- OPODIS 2020}, year = {2020}, month = {12}, }