# Satellite-Aided Consensus Protocol for Scalable Blockchains
## 4 Security Implications  
In this section, we assess the risk to the satellite-aided consensus protocol in the presence of an adversary. Denote \(\beta_{t}\) as the fraction of the total terrestrial mining power controlled by an adversary. Hence, in a round, the probability that a benign miner is chosen as the winner is \(1-\beta_{t}\), and the  
Figure 2: Possible patterns of blockchain evolution based on the satellite-aided consensus. (**a**) Regular extension. (**b**) Oracle missing. (**c**) Block missing. (**d**) Invalid blocks. (**e**) Fraud identity. (**f**) Multi-block in one round. (**g**) Alternative history attack.  
winning probability of an adversary is \(\beta_{t}\). Moreover, we have to consider that some satellites may be controlled or hacked by an adversary. Let \(\beta_{s}\) be the fraction of adversarial satellites. Undoubtedly, when an adversarial satellite generates an oracle, an adversarial miner will be selected as the winner. In short, the probability that a malicious miner is selected as the winner is \(\beta_{s}+\beta_{t}\left(1-\beta_{s}\right)\), while the probability that a benign miner is selected as the winner is \(\left(1-\beta_{s}\right)\left(1-\beta_{t}\right)\).  
More practically, we should take the possible outage or interrupt of satellite links into considerations. Denote \(p\) as the successful transmission probability of satellite links such that the outage probability is \(1-p\). Figure 2b shows that a winner will not generate a block if it does not receive the oracle (with probability \(1-p\)). Hence, the growth rate of the benign and adversarial chains can be, respectively, formulated as  
\[h =p\left(1-\beta_{s}\right)\left(1-\beta_{t}\right), \tag{1}\] \[f =p\left(\beta_{s}+\beta_{t}-\beta_{t}\beta_{s}\right). \tag{2}\]  
Therefore, the overall relative power controlled by an adversary, denoted by \(\beta\), can be expressed by  
\[\beta\triangleq\frac{f}{f+h}=\beta_{s}+\beta_{t}-\beta_{t}\beta_{s}. \tag{3}\]  
From Equation (3), the successful transmission probability \(p\) will not directly influence the system security. However, if the adversarial miners have back channels to the colluding satellites, the blockchain faces heightened risk. In the absence of adversarial satellites, the overall adversarial power only depends on the terrestrial miners \(\beta=\beta_{t}\) as \(\beta_{s}=0\). Obviously, the existence of adversarial satellites makes the chain more vulnerable.  
We use the probability of a successful double spending attack, i.e., confirmation error probability, to evaluate the resilience of the proposed protocol. According to the framework on alternative history attacks [41], given the confirmation number \(k\), the confirmation error probability is  
\[\mathsf{S}_{k}\left(\beta\right)=\begin{cases}1-\sum_{n=0}^{k}\binom{n+k-1}{n }\left(\beta^{n}\left(1-\beta\right)^{k}-\beta^{k}\left(1-\beta\right)^{n}\right) &\text{if }\beta<1/2\\ 1&\text{if }\beta\geq 1/2.\end{cases} \tag{4}\]  
**Remark 1**.: _For \(\beta\geq 1/2\), the adversary dominates the chain and always succeeds. For \(\beta<1/2\), the confirmation error probability is exponentially decreasing with the confirmation number \(k\), where the decay rate is the attacker's relative power \(\beta\). Note that a PoW-based blockchain with adversary mining power \(\beta_{PoW}=\beta\) has the exact same confirmation error probability as a satellite-aided blockchain._