We consider the time-restricted double-spending attack (TR-DSA) on the Proof-of-Work-based blockchain, where an adversary conducts a DSA within a finite timeframe and simultaneously launches multiple types of attacks on the blockchain. To be specific, the adversary can conduct attacks to isolate some honest miners and cause block propagation delays among miners to enhance the success probability of the TR-DSA. We first develop the closed-form expression for the success probability of a TR-DSA with the aid of multiple types of attacks, which is leveraged to develop the closed-form expression for the expected profit of a TR-DSA. The numerical analysis reveals that in scenarios where an adversary lacks the majority of computational power in the blockchain network, it is advisable for the adversary to refrain from indefinitely conducting a DSA, and moreover, the adversary can repeatedly launch “short-time” TR-DSAs to obtain their maximum expected profit. Notably, by leveraging the closed-form expression for the expected profit of a TR-DSA, the blockchain network designer can reduce the expected profit of a TR-DSA and therefore significantly mitigate the risk of TR-DSAs by adjusting system parameters, such as the number of blocks required for transaction confirmation, mining reward, and mining cost.

Distributed detection over a blockchain-aided Internet of Things (BIoT) network in the presence of attacks is considered, where the integrated blockchain is employed to secure data exchanges over the BIoT as well as data storage at the agents of the BIoT. We consider a general adversary model where attackers jointly exploit the vulnerability of IoT devices and that of the blockchain employed in the BIoT. The optimal attacking strategy which minimizes the Kullback-Leibler divergence is pursued. It can be shown that this optimization problem is nonconvex, and hence it is generally intractable to find the globally optimal solution to such a problem. To overcome this issue, we first propose a relaxation method that can convert the original nonconvex optimization problem into a convex optimization problem, and then the analytic expression for the optimal solution to the relaxed convex optimization problem is derived. The optimal value of the relaxed convex optimization problem provides a detection performance guarantee for the BIoT in the presence of attacks. In addition, we develop a coordinate descent algorithm which is based on a capped water-filling method to solve the relaxed convex optimization problem, and moreover, we show that the convergence of the proposed coordinate descent algorithm can be guaranteed.