Fast-to-Finalize Nakamoto-Like Consensus

Abstract

As the fundamental component of blockchains, proof-of-work (PoW) scheme has been widely leveraged to provide consensus for maintaining a distributed public ledger. However, the long confirmation time, and hence the slow finality rate, is far from satisfactory. Alternative paradigms with performance improvement emerge. Nevertheless, there are fewer attempts in modifying the PoW mechanism itself.

We find that the slow finality rate in PoW is caused by using only one bit to measure the computational power, namely, whether the attained hash value is smaller than a given target. In this paper, we first propose Demo-of-Work (DoW), a generalized PoW which assigns the computational work with a score depending on the hash value. We also treat the bitcoin blockchain as a global “clock” to attain synchronization for ensuring that each participant takes part in DoW for roughly the same time interval for ensuring fairness. With these two tools, we construct an alternative blockchain called AB-chain which provides a significantly faster finality rate when compared with the existing PoW-based blockchains, without sacrificing communication complexity or fairness.

Part of the work was done while the first author was a research intern in CUHK. The second author is supported by General Research Funds (CUHK 14210217) of the Research Grants Council, Hong Kong. The third author is supported by the National Natural Science Foundation of China (Grant No. 61672347). A preliminary version appeared as “Fast-to-Converge PoW-like Consensus Protocol” in China Blockchain Conference 2018.

A Detailed Protocols of The Simulation Experiment

For each instantiation of the generalized model with a potential function $\mathcal {L}(\cdot )$, an experiment is performed to reveal its finality rate with a Monte Carlo method. In our experiment, $M=2^{20}$, $T=D=2^{10}$ and $q=10^{-3}$ are chosen, the algorithms listed below are executed (starting from the main function, Algorithm 7). After the execution, the main function returns the expected number of rounds required to form the safety gap.

• 1. Preparation($\alpha , T$) prepares two arrays to provide outcomes of two discrete cumulative distribution functions. Specifically, $\textsf {aCDF}[i]$ is the probability of having a hash value no greater than i found by the adversary. $\textsf {hCDF}[i]$ is similarly the probability of having a hash value no greater than i found by honest nodes.

• 2. GetH(CDF) returns a random number according to a distribution of a cumulative distribution function recorded in the array CDF.

• 3. SimAttack($\varDelta , {\mathbf {\mathsf{{aCDF}}}}, {\mathbf {\mathsf{{hCDF}}}}$) models the behaviour of the adversary attempt of forming a new chain of blocks with the total weight greater than the honest one by a certain gap $\varDelta$.

• 4. Test($\varDelta , q, {\mathbf {\mathsf{{aCDF}}}}, {\mathbf {\mathsf{{hCDF}}}}$) performs “SimAttack” for sufficiently enough times, to show (via Monte Carlo method) whether the probability of adversary in successfully performing an attack (and overrunning the honest one by a total weight of $\varDelta$) is smaller than q.

• 5. FindMinGap($q, {\mathbf {\mathsf{{aCDF}}}},{\mathbf {\mathsf{{hCDF}}}}$) utilizes a binary search to find the minimal gap $\delta$ such that the adversary can catch up with the honest chain by a total weight of $\delta$ only with a negligible probability q.

• 6. Expc(CDF) returns the expected block weight attained by either honest parties or the adversary by another Monte Carlo experiment.

• 7. Main($\alpha , T, q$) is the main function that returns (the inverse of) the finality rate $\varLambda _{\alpha }$ of the blockchain with potential function $\mathcal {L}(\cdot )$, i.e., the expected number of rounds required to form the safety gap.

In our final experiment, we execute the algorithms with $\texttt {NUM\_TEST\_SAMPLE}=100000$, $\texttt {NUM\_TEST\_BLOCK}=200$, and $\epsilon =10^{-4}$.

Table 3. Experiments on few establishments of the potential function