PoTA: A hybrid consensus protocol to avoid miners’ collusion for BaaS platform

Abstract

The emergence of the blockchain-as-a-service (BaaS) platform reduces the application barrier of blockchain technology. However, in BaaS, the transaction processing demand generated by BaaS users is continuous isolating from the transaction processing capability that comes from blockchain miner community. This results in the phenomenon that miner community spontaneously reduces the transaction processing capacity to obtain higher revenue, which is called the miners’ collusion. The BaaS platform requires a new consensus protocol that prevent the miners’ collusion while remain the security and immutability of blockchain. Based on this challenge, in this article, we propose a hybrid consensus protocol for BaaS called the Proof-of-Transaction Amount (PoTA). First, we theoretically analyze and formally define the miners’ collusion. The existence of Nash-equilibrium collusion strategy has been proven. Second, the calculation method of the miner’s effective collusion strategy space is provided to help determine whether the BaaS is facing the threat of collusion. Third, by organizing a transaction packaging committee (TPC) based on transaction amount, PoTA successfully avoids single-identity miners who have greater effective collusion strategy space obtaining the authority of transaction packaging. In the meanwhile, we redefine the reference relationship of blocks to enable cross-verification of PoW and PoS consensus. Finally, we prove our protocol can effectively avoid the occurrence of the miners’ collusion. Control experiments prove the outperforms security and performance of PoTA in the comparison with PoW, PoS and other hybrid protocols.

Appendix

We abstract the strategy transfers process as a $\mu$-round first-price sealed-bid auction which contains Bayesian Nash equilibrium solutions. Under the miners’ collusion, we assume that miner community will not accept the behavior fight against the auction, which means if the strategy transfers, users who maintain their original bids will not have their transactions processed. Therefore, the price gap of two strategies $\mathbb {Q}$ is the BaaS users’ valuation of this auction. We define there are two users $n_i$ and $n_j$ biding for transaction process with their positive bids $q_i$ and $q_i$ which obeys a uniformly distribution within $[0,\mathbb {Q}]$. If one user succeed in the auction with bid q, the revenue of this auction is $\mathbb {Q}-q$. According to the rules of auction, the higher bidder gets the commodity, i.e., gets his transactions processed. If both sides have the same bids, miner community will toss a coin to decide the winner. Therefore, the revenue function of $n_i$ is:

$$\begin{aligned} u_i={\left\{ \begin{array}{ll} \mathbb {Q}-q_i &{}, if\quad q_i > q_j;\\ \frac{1}{2}\times ( \mathbb {Q}-q_i) &{}, if\quad q_i = q_j;\\ 0 &{}, if\quad q_i < q_j. \end{array}\right. } \end{aligned}$$

(16)

Then, to achieve the optimal strategy in auction, $q_i$ needs to satisfy

$$\begin{aligned} max \quad (\mathbb {Q}-q_i)Pro\{q_i > q_j\}+\frac{1}{2}\times (\mathbb {Q}-q_i)Pro\{q_i = q_j\} \end{aligned}$$

(17)

According to [21], since users’ bids are uniformly distributed, there exists an unique linear Bayesian Nash equilibrium solution to Eq. (17). We define users final bids as $b(q_i)=Q(s)+\theta _i \times \mathbb {Q}$ and $b(q_i)=Q(s)+\theta _j \times \mathbb {Q}$, respectively. The users’ bids satisfy a continuous uniform distribution. Therefore, the probability of $q_i = q_j$ is 0. The optimal bidding strategy is the solution of the following formula:

$$\begin{aligned} max\quad (\mathbb {Q}-q_i)Pro\{b(q_i) > Q(s)+\theta _j \times \mathbb {Q}\} \end{aligned}$$

(18)

$n_i$ needs a final bid that higher than the Q(s) and lower than the highest possible bid of $n_j$, $Q(s)+\mathbb {Q}$. Therefore, we have

$$\begin{aligned} Pro\{b(q_i) > Q(s)+\theta _j \times \mathbb {Q}\}&=Pro\{\mathbb {Q}<\frac{b(q_i)-Q(s)}{\theta _i}\}\\&=\frac{b(q_i)-Q(s)}{\theta _i} \end{aligned}$$

(19)

The optimal strategy of bidding for $n_i$ is:

$$\begin{aligned} b(q_i) {\left\{ \begin{array}{ll} Q(s)+\frac{\mathbb {Q}}{2} &{}, if\quad Q(s^*) \ge Q(s);\\ q(s) &{}, if\quad Q(s^*) < Q(s). \end{array}\right. } \end{aligned}$$

(20)

Obviously, the case $Q(s^*) < Q(s)$ does not meet our assumption. Therefore, we have $\theta =0.5$ as the optimal bidding value for users in the strategy transfer process caused by a collusion. The derivation of optimal strategy for $n_j$ is the same as above. In a $\mu$-round strategy transfer process, the final valuation of strategy $s^*$ can be obtained by overlaying the user estimates from $\mu$-round auction, which is $Q(s^*)=Q(s)+\sum ^{t=1}_{\mu }\theta ^{\mu -t}\times \mathbb {Q}$. The equation for the user’s optimal bid contains only one unknown variable. Therefore, for a multi-player auction model, $\theta =0.5$ is still the only solution. Then, we successfully generalize the optimal bid case to the multi-player auction game.