We study approximate agreement in an asynchronous network of n parties, up to t of which are byzantine. This an agreement task where the parties obtain approximately equal inputs in the convex hull of their inputs. In an asynchronous network, it can be solved with the optimal resilience t < n/3 by forcing the parties to reliably broadcast their messages and thus preventing inconsistent byzantine behavior. This costs Θ(n²) messages per reliable broadcast, or Θ(n³) messages per protocol iteration. In this work, we forgo reliable broadcast to achieve asynchronous approximate agreement against t < n/3 faults with quadratic communication. In a tree with the maximum degree Δ and the centroid decomposition height h, we achieve edge agreement (agreement on two adjacent vertices) in at most 6h + 1 rounds with 𝒪(n²) messages of size 𝒪(log Δ + log h) per round. We do this by designing a 6-round multivalued 2-graded consensus protocol and using it to construct a recursive edge agreement protocol. Then, we achieve edge agreement in the infinite path ℤ, again by using 2-graded consensus. Finally, we show that our edge agreement protocol enables approximate agreement in ℝ (with outputs that are at most some small parameter ε > 0 apart) in 6log₂M/(ε) + 𝒪(log log M/(ε)) rounds with 𝒪(n²) messages of size 𝒪(log log M/(ε)) per round, where M is the maximum non-byzantine input magnitude.

In this work, we study multivalued byzantine agreement (BA) in an asynchronous network of n parties where up to [EQUATION] parties are byzantine. We present a new reduction from multivalued BA to binary BA. It allows one to achieve BA on ℓ-bit inputs with one instance of binary BA, one instance of crusader agreement (CA) on ℓ-bit inputs and Θ(ℓn + n2) bits of additional 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 ℝ with the optimal resilience t < n/3 with a protocol where each party reliably broadcasts a value in every iteration. This takes Θ(n²) messages per reliable broadcast, or Θ(n³) messages per iteration. In this work, we forgo reliable broadcast to achieve asynchronous approximate agreement against t < n/3 faults with 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 𝒪(n²) messages of size 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 ℤ, again with the help of 2-graded consensus. Finally, we show that our edge agreement protocol enables ε-agreement in ℝ in 6log₂M/(ε) + 𝒪(log log M/(ε)) rounds with 𝒪(n² log M/(ε)) messages and 𝒪(n²log M/(ε)log log M/(ε)) bits of communication, where M is the maximum non-byzantine input magnitude.