  Quantum cryptography has attracted much recent attention due to its potential
for providing secret communications that cannot be decrypted by any amount of
computational effort. This is the first analysis of the secrecy of a practical
implementation of the BB84 protocol that simultaneously takes into account and
presents the {\it full} set of complete analytical expressions for effects due
to the presence of pulses containing multiple photons in the attenuated output
of the laser, the finite length of individual blocks of key material, losses
due to error correction, privacy amplification, continuous authentication,
errors in polarization detection, the efficiency of the detectors, and
attenuation processes in the transmission medium. The analysis addresses
eavesdropping attacks on individual photons rather than collective attacks in
general. Of particular importance is the first derivation of the necessary and
sufficient amount of privacy amplification compression to ensure secrecy
against the loss of key material which occurs when an eavesdropper makes
optimized individual attacks on pulses containing multiple photons. It is shown
that only a fraction of the information in the multiple photon pulses is
actually lost to the eavesdropper.
