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We improve the non-asymptotic key rate shown by Scarani and Renner byproposing several methods to construct tighter conservative confidenceintervals of the phase error rate than one shown by them. In addition, we showthat the accurate channel estimation method non-asymptotically increases thekey rate over the amplitude damping channel as well as the asymptotic case inthe BB84 protocol.

Secure Key Rate of the BB84 Protocol using Finite Sample Bits