The main result of this paper consists of a pair of necessary and sufficient conditions for the input-output finite-time stability of impulsive linear systems. The former requires that an optimization problem, constrained by a coupled differential/difference linear matrix inequality (LMI), admits a feasible solution; the latter that the solution of a coupled differential/difference Lyapunov equation satisfies a constraint on the maximum eigenvalue. The first condition was already provided in Amato et al. (2011), where, however, only sufficiency was proven. The novel analysis condition (i.e. the one requiring the solution of the differential/difference Lyapunov equation) is shown to be more efficient from the computational point of view, while the result based on the differential/difference LMI is the starting point for the derivation of the design theorem. Some examples illustrate the benefits of the proposed technique.