  We introduce a novel generalization of entropy and conditional entropy from
which most definitions from the literature can be derived as particular cases.
Within this general framework, we investigate the problem of designing
countermeasures for information leakage. In particular, we seek
metric-invariant solutions, i.e., they are robust against the choice of entropy
for quantifying the leakage. The problem can be modelled as an information
channel from the system to an adversary, and the countermeasures can be seen as
modifying this channel in order to minimise the amount of information that the
outputs reveal about the inputs. Our main result is to fully solve the problem
under the highly symmetrical design constraint that the number of inputs that
can produce the same output is capped. Our proof is constructive and the
optimal channels and the minimum leakage are derived in closed form.
