**Extended Validity and Consistency in Byzantine Agreement**

*Matthias Fitzi and Martin Hirt and Thomas Holenstein and Jürg Wullschleger*

**Abstract: **A broadcast protocol allows a sender to distribute a value among a set of
players such that it is guaranteed that all players receive the same
value (consistency), and if the sender is honest, then all players
receive the sender's value (validity). Classical broadcast protocols for
$n$ players provide security with respect to a fixed threshold $t<n/3$,
where both consistency and validity are guaranteed as long as at most $t$
players are corrupted, and no security at all is guaranteed as soon as
$t+1$ players are corrupted. Depending on the environment, validity or
consistency may be the more important property.

We generalize the notion of broadcast by introducing an additional threshold $t^+\ge t$. In a {\em broadcast protocol with extended validity}, both consistency and validity are achieved when no more than $t$ players are corrupted, and validity is achieved even when up to $t^+$ players are corrupted. Similarly, we define {\em broadcast with extended consistency}. We prove that broadcast with extended validity as well as broadcast with extended consistency is achievable if and only if $t+2t^+<n$ (or $t=0$). For example, six players can achieve broadcast when at most one player is corrupted (this result was known to be optimal), but they can even achieve consistency (or validity) when two players are corrupted.

Furthermore, our protocols achieve {\em detection} in case of failure, i.e., if at most $t$ players are corrupted then broadcast is achieved, and if at most $t^+$ players are corrupted then broadcast is achieved or every player learns that the protocol failed. This protocol can be employed in the precomputation of a secure multi-party computation protocol, resulting in {\em detectable multi-party computation}, where up to $t$ corruptions can be tolerated and up to $t^+$ corruptions can either be tolerated or detected in the precomputation, for any $t,t^+$ with $t+2t^+<n$.

**Category / Keywords: **cryptographic protocols / Byzantine agreement, detectable precomputation, multi-party computation, unconditional security

**Date: **received 26 Apr 2002

**Contact author: **hirt at inf ethz ch

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Version: **20020426:165404 (All versions of this report)

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