  We provide a security analysis for continuous variable quantum key
distribution protocols based on the transmission of squeezed vacuum states
measured via homodyne detection. We employ a version of the entropic
uncertainty relation for smooth entropies to give a lower bound on the number
of secret bits which can be extracted from a finite number of runs of the
protocol. This bound is valid under general coherent attacks, and gives rise to
keys which are composably secure. For comparison, we also give a lower bound
valid under the assumption of collective attacks. For both scenarios, we find
positive key rates using experimental parameters reachable today.
