Approximate Agreement (AA) is a fundamental primitive that, even in the presence of Byzantine faults, allows honest parties to obtain close (but not necessarily identical) outputs that lie within the range of their inputs. While the optimal round complexity of synchronous AA on real values is well understood, its extension to other input spaces has remained open, with fundamental questions regarding achievable resilience and round efficiency still unresolved.

Multidimensional Approximate Agreement (D-AA) considers a setting with n parties with inputs in ℝD. Out of the n parties, up to t may be byzantine (malicious). The goal is for the honest parties to obtain ϵ-close outputs that lie in the convex hull of the honest inputs.

In stable matching, one must find a matching between two sets of agents, commonly men and women, or job applicants and job positions. Each agent has a preference ordering over who they want to be matched with. Moreover a matching is said to be stable if no pair of agents prefer each other over their current matching.

Approximate Agreement (AA) is a key consensus primitive that allows honest parties to achieve close but not necessarily identical outputs, even in the presence of Byzantine faults. While optimal round complexity for synchronous AA on real values is well understood, its extension to other input spaces remains an open problem.

Byzantine Agreement (BA) allows a set of n parties to agree on a value even when up to t of the parties involved are corrupted. While previous works have shown that, for ℓ-bit inputs, BA can be achieved with the optimal communication complexity O(ℓn) for sufficiently large ℓ, BA only ensures that honest parties agree on a meaningful output when they hold the same input, rendering the primitive inadequate for many real-world applications.

Byzantine Agreement (BA) considers a setting of $n$ parties, out of which up to $t$ can exhibit byzantine (malicious) behavior. Honest parties must decide on a common value (agreement), which must belong to a set determined by the honest inputs (validity). Depending on the use case, this set can grow or shrink, leading to various possible desiderata collectively known as validity conditions. Varying the validity property requirement can affect the regime under which BA is solvable. Our work investigates how the selected validity property impacts BA solvability in the network-agnostic model, where the network can either be synchronous with up to $t_s$ byzantine parties or asynchronous with up to $t_a \leq t_s$ byzantine parties. We give necessary and sufficient conditions for a validity property to render BA solvable, both for the case with cryptographic setup and for the one without. This traces the precise boundary of solvability in the network-agnostic model for every validity property. Our proof of sufficiency provides a universal protocol, that achieves BA for a given validity property whenever the provided conditions are satisfied. We note that, for any non-trivial validity property, the condition $2 \cdot t_s + t_a < n$ is necessary for BA to be solvable, even with cryptographic setup. Specializing this claim to $t_a = 0$ gives that $t < n / 2$ is required whenever one expects a purely synchronous protocol to also work in an asynchronous network when there are no corruptions. This is especially surprising given that, for some validity properties, $t < n$ is a sufficient condition without the last stipulation.

Byzantine Agreement (BA) allows a set of n parties to agree on a value even when up to t of the parties involved are corrupted. While previous works have shown that, for ℓ-bit inputs, BA can be achieved with the optimal communication complexity Õ(ℓn) for sufficiently large ℓ, BA only ensures that honest parties agree on a meaningful output when they hold the same input, rendering the primitive inadequate for many real-world applications.

Traditional blockchain design gives miners or validators full control over transaction ordering, i.e., they can freely choose which transactions to include or exclude, as well as in which order. While not an issue initially, the emergence of decentralized finance has introduced new transaction order dependencies allowing parties in control of the ordering to make a profit by front-running others' transactions. In this work, we present the Decentralized Clock Network, a new approach for achieving fair transaction ordering. Users submit their transactions to the network's clocks, which run an agreement protocol that provides each transaction with a timestamp of receipt which is then used to define the transactions' order. By separating agreement from ordering, our protocol is efficient and has a simpler design compared to other available solutions. Moreover, our protocol brings to the blockchain world the paradigm of asynchronous fallback, where the algorithm operates with stronger fairness guarantees during periods of synchronous use, switching to an asynchronous mode only during times of increased network delay.

Multidimensional Approximate Agreement considers a setting of n parties, where each party holds a vector in ℝD as input. The honest parties are required to obtain very close outputs in ℝD that lie inside the convex hull of their inputs.

Approximate Agreement (AA) allows a set of n parties that start with real-valued inputs to obtain values that are at most within a parameter ε > 0 from each other and within the range of their inputs. Existing AA protocols, both for the synchronous network model (where any message is delivered within a known delay Δ time) and the asynchronous network model, are secure when up to t < n/3 of the parties are corrupted and require no initial setup (such as a public-key infrastructure (PKI) for signatures). We consider AA protocols where a PKI is available, and show the first AA protocol that achieves simultaneously security against ts corruptions when the network is synchronous and ta corruptions when the network is asynchronous, for any 0 ≤ ta < n/3 ≤ ts < n/2 such that ta + 2 · ts < n. We further show that our protocol is optimal by proving that achieving AA for ta +2·ts ≥ n is impossible (even with setup). Remarkably, this is also the first AA protocol that tolerates more than n/3 corruptions in the synchronous network model.