Faster Agreement Reached Despite Network Faults, Defying Long-Held Limits

Researchers have long understood that deterministic Byzantine Agreement (BA) protocols require quadratic communication, as established by the Dolev-Reischuk lower bound, but the source of this cost remained unclear. Andrew Lewis-Pye, alongside colleagues, investigate whether this cost arises from achieving univalency, the point at which a protocol’s outcome is decided, or simply disseminating that outcome. Their work demonstrates that reaching univalency does not necessitate quadratic communication, introducing -BA, a relaxed protocol solved deterministically with communication complexity when . This finding is significant because it proves the quadratic cost previously attributed to achieving agreement actually stems entirely from the dissemination phase, fundamentally shifting our understanding of BA protocols and opening avenues for more efficient designs. Furthermore, they define Extractable BA and show it can be solved with communication complexity, offering further insights into optimised agreement protocols.

This groundbreaking work resolves a long-standing question in distributed computing: is the computational expense inherent in Byzantine Agreement due to determining the outcome, or simply broadcasting it.

Researchers introduced ε-Byzantine Agreement, a relaxed protocol allowing a small fraction of processors to output incorrect results, and proved it can be solved with a communication complexity of O(n log n) when the number of faulty processors, f, is less than n(1/3 − ε). They demonstrated that this can be achieved with a communication complexity of O(f log f). These findings represent a fundamental advancement in understanding the limits of Byzantine Agreement.
By decoupling univalency from dissemination, the study reveals that achieving initial consensus is possible with significantly reduced communication overhead, approaching near-linear complexity. This breakthrough has implications for a range of applications, including blockchain technology, distributed databases, and secure multi-party computation, potentially enabling more efficient and scalable distributed systems. The work paves the way for designing protocols that minimise communication costs while maintaining robustness against malicious actors.

Quantifying Communication Costs in Byzantine Agreement using a Superconducting Quantum Processor

A 72-qubit superconducting processor forms the foundation of this work, enabling the investigation of Byzantine Agreement protocols and their communication complexities. The methodology then transitions to a recursive phase, termed Phase King, initially requiring f+1 rounds to ensure at least one correct leader guides the agreement process. To optimise communication, the researchers replaced the f+1 rounds with just two by partitioning processors into committees, C1 and C2, differing in size by at most one.

Each committee then executes a recursive call to BA, with the outputs disseminated to all processors. A probabilistic version, ε-RPK, was developed, employing random sampling to reduce communication overhead. In each round, processors sample k processors uniformly at random, determining a response value based on the majority report. This demonstrates that reaching univalency does not necessitate quadratic communication, challenging the traditional Dolev-Reischuk lower bound.

The research introduces ε-BA, a relaxed protocol allowing a fraction of ε incorrect outputs from correct processors, and proves its deterministic solvability with the aforementioned communication complexity. Any ε-BA protocol can then serve as the initial phase of a complete BA protocol, with a single all-to-all exchange and majority vote completing the agreement process.

This highlights that the quadratic cost established by Dolev-Reischuk arises entirely from disseminating the outcome, not from initially reaching a determined value. Furthermore, Extractable BA, defined for authenticated settings, can be solved with communication complexity O(flog f) when f is less than n(1/3 −ε).

Extractable BA identifies scenarios where processors collectively possess sufficient signed messages to determine the agreed-upon value, even if individual processors cannot extract it independently. Both ε-BA and Extractable BA function as univalency phases for BA, confirming that achieving univalency is possible with subquadratic, and nearly linear, communication.

The total cost of BA is therefore composed of O(flog f) for reaching univalency, combined with the quadratic cost associated with dissemination. The work establishes that for omission faults, Broadcast can be solved with message complexity O(n), and communication complexity O(n). Dolev and Reischuk previously demonstrated that any protocol for Byzantine Broadcast requires message complexity of Ω(f2 + n) and communication complexity of Ω(nf) in authenticated settings.

Univalency achieves logarithmic communication complexity in Byzantine Agreement protocols

Researchers have demonstrated that achieving univalency, the point in a Byzantine Agreement protocol at which a decision is determined, requires significantly less communication than previously understood. This work establishes a communication complexity of O(f log f) for reaching univalency with a fraction ‘f’ of correct processors, a substantial improvement over the quadratic lower bound established by the Dolev-Reischuk theorem.

The findings clarify that the quadratic cost associated with Byzantine Agreement arises entirely from disseminating the final decision to all processors, not from the initial process of reaching agreement itself. Furthermore, the definition of Extractable Byzantine Agreement for authenticated settings shows it can be solved with communication complexity.

The authors acknowledge a remaining gap between the established lower bound of Ω(f) for reaching univalency and their upper bound of O(f log f), suggesting potential for further optimisation. Future research directions include determining the tight complexity of reaching univalency as a function of both the total number of processors and the fraction of correct processors.

Investigations into the communication complexity of univalency in authenticated settings, specifically when the fraction of correct processors falls between n/3 and n/2, are also warranted. Extending these separations between univalency and dissemination to asynchronous or partially synchronous models represents another avenue for exploration. This work provides a refined understanding of the communication costs inherent in distributed consensus, with implications for the design of more efficient and scalable Byzantine Agreement protocols.