  Our aim in this paper is to investigate the profitability of double-spending
(DS) attacks that manipulate an a priori mined transaction in a blockchain. It
was well understood that a successful DS attack is established when the
proportion of computing power an attacker possesses is higher than that the
honest network does. What is not yet well understood is how threatening a DS
attack with less than 50% computing power used can be. Namely, DS attacks at
any proportion can be of a threat as long as the chance to making a good profit
exists. Profit is obtained when the revenue from making a successful DS attack
is greater than the cost of carrying out one. We have developed a novel
probability theory for calculating a finite time attack probability. This can
be used to size up attack resources needed to obtain the profit. The results
enable us to derive a sufficient and necessary condition on the value of a
transaction targeted by a DS attack. Our result is quite surprising: we
theoretically show that DS attacks at any proportion of computing power can be
made profitable. Given one's transaction size, the results can also be used to
assess the risk of a DS attack. An example of the attack resources is provided
for the BitcoinCash network.
