Mose Mizrahi Erbes | Searcharxiv

arXiv2025

Asynchronous Approximate Agreement with Quadratic Communication

We consider an asynchronous network of $n$ message-sending parties, up to $t$ of which are byzantine. We study approximate agreement, where the parties obtain approximately equal outputs in the convex hull of their inputs. In their seminal work, Abraham, Amit and Dolev [OPODIS '04] solve this problem in $\mathbb{R}$ with the optimal resilience $t < \frac{n}{3}$ with a protocol where each party reliably broadcasts a value in every iteration. This takes $Θ(n^2)$ messages per reliable broadcast, or $Θ(n^3)$ messages per iteration. In this work, we forgo reliable broadcast to achieve asynchronous approximate agreement against $t < \frac{n}{3}$ faults with a quadratic communication. In a tree with the maximum degree $Δ$ and the centroid decomposition height $h$, we achieve edge agreement in at most $6h + 1$ rounds with $\mathcal{O}(n^2)$ messages of size $\mathcal{O}(\log Δ+ \log h)$ per round. We do this by designing a 6-round multivalued 2-graded consensus protocol and using it to recursively reduce the task to edge agreement in a subtree with a smaller centroid decomposition height. Then, we achieve edge agreement in the infinite path $\mathbb{Z}$, again with the help of 2-graded consensus. Finally, we show that our edge agreement protocol enables $\varepsilon$-agreement in $\mathbb{R}$ in $6\log_2\frac{M}{\varepsilon} + \mathcal{O}(\log \log \frac{M}{\varepsilon})$ rounds with $\mathcal{O}(n^2 \log \frac{M}{\varepsilon})$ messages and $\mathcal{O}(n^2\log \frac{M}{\varepsilon}\log \log \frac{M}{\varepsilon})$ bits of communication, where $M$ is the maximum non-byzantine input magnitude.

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arXiv2025

Extending Asynchronous Byzantine Agreement with Crusader Agreement

In this work, we study multivalued byzantine agreement (BA) in an asynchronous network of $n$ parties where up to $t < \frac{n}{3}$ parties are byzantine. We present a new reduction from multivalued BA to binary BA. It allows one to achieve BA on $\ell$-bit inputs with one instance of binary BA, one instance of crusader agreement (CA) on $\ell$-bit inputs and $Θ(\ell n + n^2)$ bits of additional communication. As our reduction uses multivalued CA, we also design two new information-theoretic CA protocols for $\ell$-bit inputs. In the first one, we use almost-universal hashing to achieve statistical security with probability $1 - 2^{-λ}$ against $t < \frac{n}{3}$ faults with $Θ(\ell n + n^2(λ+ \log n))$ bits of communication. Following this, we replace the hashes with error correcting code symbols and add a preliminary step based on the synchronous multivalued BA protocol COOL [DISC '21] to obtain a second, perfectly secure CA protocol that can for any $\varepsilon > 0$ be set to tolerate $t \leq \frac{n}{3 + \varepsilon}$ faults with $\mathcal{O}\bigl(\frac{\ell n}{\min(1, \varepsilon^2)} + n^2\max\bigl(1, \log \frac{1}{\varepsilon}\bigr) \bigr)$ bits of communication. Our CA protocols allow one to extend binary BA to multivalued BA with a constant round overhead, a quadratic-in-$n$ communication overhead, and information-theoretic security.

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