  Transmission losses through optical fibers are one of the main obstacles
preventing both long-distance quantum communications and continuous-variable
quantum key distribution. Optical amplification provides a tool to obtain, at
least partially, signal restoration. In this work, we address a key
distribution protocol over a multispan link employing either phase-insensitive
or phase-sensitive amplifiers, considering Gaussian modulation of coherent
states followed by homodyne detection at the receiver's side. We perform the
security analysis under both unconditional and composable security frameworks
by assuming in the latter case only a single span of the whole communication
link to be untrusted. We compare the resulting key generation rate for both
kinds of amplified links with the no-amplifier protocol, identifying the
enhancement introduced by optical amplification.
