  Numerous blockchain applications are designed with tasks that naturally have
finite durations, and hence, a double-spending attack (DSA) on such blockchain
applications leans towards being conducted within a finite timeframe,
specifically before the completion of their tasks. Furthermore, existing
research suggests that practical attackers typically favor executing a DSA
within a finite timeframe due to their limited computational resources. These
observations serve as the impetus for this paper to investigate a
time-restricted DSA (TR-DSA) model on Proof-of-Work based blockchains. In this
TR-DSA model, an attacker only mines its branch within a finite timeframe, and
the TR-DSA is considered unsuccessful if the attacker's branch fails to surpass
the honest miners' branch when the honest miners' branch has grown by a
specific number of blocks. First, we developed a general closed-form expression
for the success probability of a TR-DSA. This developed probability not only
can assist in evaluating the risk of a DSA on blockchain applications with
timely tasks, but also can enable practical attackers with limited
computational resources to assess the feasibility and expected reward of
launching a TR-DSA. In addition, we provide rigorous proof that the success
probability of a TR-DSA is no greater than that of a time-unrestricted DSA
where the attacker indefinitely mines its branch. This result implies that
blockchain applications with timely tasks are less vulnerable to DSAs than
blockchain applications that provide attackers with an unlimited timeframe for
their attacks. Furthermore, we show that the success probability of a TR-DSA is
always smaller than one even though the attacker controls more than half of the
hash rate in the network. This result alerts attackers that there is still a
risk of failure in launching a TR-DSA even if they amass a majority of the hash
rate in the network.
