The Ticket Price Matters in Sharding Blockchain

Abstract

Sharding technology has been recognized to be a promising solution for blockchain scalability problems in recent years. For safety guarantees in each shard, mainly to prevent the single-shard takeover attack, sharding requires an identity establishment protocol in which participants have to pay a certain amount of resources (i.e., ticket price) to get a node and participate in the network. However, state-of-the-art sharding protocols overlook a non-democratic state of the real-world where every participant has a different amount of resources, termed a non-democratic environment. This oversight raises combined problems of security and scalability due to the design of the identity establishment protocol.

In this paper, we examine the effects of the non-democracy of blockchain networks in terms of the security and scalability of blockchain sharding and suggest formulae to quantitatively analyze the trade-off between security and scalability. Moreover, we conduct a numerical analysis by capturing four real-world resource distributions from renowned permissionless cryptocurrency networks. We re-evaluate the well-known sharding protocols through this numerical analysis and present the changed fault tolerance bounds and damage to scalability. The results show that the ticket price plays a leading role in tuning the effect of non-democracy. The main contribution of this paper is the proposal of new metrics for accessing the degree of security and scalability with regard to the ticket price in the identity establishment phase. Our discussion suggests further research on a more delicate ticket price control algorithm when designing a new sharding model for blockchain.

Appendices

A Amount of Resource to Participate in a Certain Number of Shards

To show this, we define additional notations in Table 5. When a participant has $\alpha / t$ nodes, a probability that the participant has at least a node in one of the shards is the complement of the probability that all of the nodes are not in that shard.

$$\begin{aligned} (1 - (\frac{S - 1}{S})^{\lfloor \alpha / t\rfloor }) \end{aligned}$$

(5)

Table 5. Additional notations.

Now, we can estimate the expected number of shards that a participant with $\alpha / t$ nodes would take part in.

$$\begin{aligned} S \times (1 - (\frac{S - 1}{S})^{\lfloor \alpha / t\rfloor }) \end{aligned}$$

(6)

Equation (6) should be equal to the target number of shards.

$$\begin{aligned} \lambda S = S \times (1 - (\frac{S - 1}{S})^{\lfloor \alpha / t\rfloor }) \end{aligned}$$

(7)

Solving for $\alpha$, we can get the amount of resource with which a participant has nodes in the target number of shards (i.e., $\lambda S$) on average.

$$\begin{aligned} \alpha = \frac{t \log (1 - \lambda )}{\log (1 - 1/S)} \end{aligned}$$

(8)

B Resiliency Bound Recalculation

What we want to do is to calculate the minimum ratio of resources that is smaller than the total resiliency bound ($\theta$), yet can generate more number of nodes which goes over the bound under the resource quantization. By equating Eq. (3), we can derive the equation for this value and it will be the changed resiliency bounds ($\theta ^\prime$).

$$\begin{aligned} \frac{h}{\lfloor \frac{R}{T}\rfloor - \lfloor \frac{I\beta }{T}\rfloor } = \theta \end{aligned}$$

(9)

$$\begin{aligned} \frac{h}{(R - I \beta )/T} = \theta \end{aligned}$$

(10)

Solved for h.

$$\begin{aligned} h = \frac{\theta (R - I \beta )}{T} \end{aligned}$$

(11)

Multiplied by the ratio of ticket price.

$$\begin{aligned} \frac{T}{R} \times h = \frac{T}{R} \times \frac{\theta (R - I \beta )}{T} \end{aligned}$$

(12)

$$\begin{aligned} t \times h = \frac{\theta (R - I \beta )}{R} \end{aligned}$$

(13)

$$\begin{aligned} \theta ^\prime = \frac{\theta (R - I \beta )}{R} \end{aligned}$$

(14)

Thus, the rich adversary with the resource more than $\theta ^\prime$ but less than $\theta$ can still subvert the network.