  Bitcoin is the first secure decentralized electronic currency system.
However, it is known to be inefficient due to its proof-of-work (PoW) consensus
algorithm and has the potential hazard of double spending. In this paper, we
aim to reduce the probability of double spending by decreasing the probability
of consecutive winning. We first formalize a PoW-based decentralized secure
network model in order to present a quantitative analysis. Next, to resolve the
risk of double spending, we propose the personalized difficulty adjustment
(PDA) mechanism which modifies the difficulty of each participant such that
those who win more blocks in the past few rounds have a smaller probability to
win in the next round. To analyze the performance of the PDA mechanism, we
observe that the system can be modeled by a high-order Markov chain. Finally,
we show that PDA effectively decreases the probability of consecutive winning
and results in a more trustworthy PoW-based system.
