## Two-Threshold Broadcast and Detectable Multi-Party Computation

### Matthias Fitzi, Martin Hirt, Thomas Holenstein, and Jürg Wullschleger

Classical distributed protocols like broadcast or multi-party computation provide security as long as the number of malicious players $f$ is bounded by some given threshold $t$, i.e., $f\le t$. If $f$ exceeds $t$ then these protocols are completely insecure.

We relax this binary concept to the notion of two-threshold security: Such protocols guarantee full security as long as $f\le t$ for some small threshold $t$, and still provide some degraded security when $t<f\le T$ for a larger threshold $T$. In particular, we propose the following problems.

$\circ$ **Broadcast with Extended Validity:** Standard broadcast is achieved when $f\le t$. When $t<f\le T$, then either broadcast is achieved, or every player learns that there are too many faults. Furthermore, when the sender is honest, then broadcast is always achieved.

$\circ$ **Broadcast with Extended Consistency:** Standard broadcast is achieved when $f\le t$. When $t<f\le T$, then either broadcast is achieved, or every player learns that there are too many faults. Furthermore, the players agree on whether or not broadcast is achieved.

$\circ$ **Detectable Multi-Party Computation:** Secure computation is achieved when $f\le t$. When $t<f\le T$, then either the computation is secure, or all players detect that there are too many faults and abort.

The above protocols for $n$ players exist if and only if $t=0$ or $t+2T<n$.

## BibTeX Citation

@inproceedings{FHHW03, author = {Matthias Fitzi and Martin Hirt and Thomas Holenstein and Jürg Wullschleger}, title = {Two-Threshold Broadcast and Detectable Multi-Party Computation}, editor = {Eli Biham}, booktitle = {Advances in Cryptology --- EUROCRYPT 2003}, pages = 51--67, series = {Lecture Notes in Computer Science}, volume = 2656, year = 2003, month = 5, publisher = {Springer-Verlag}, }