Finite-size key in the Bennett 1992 quantum-key-distribution protocol for Rényi entropies

A realistic quantum-key-distribution protocol necessarily runs with finite resources. Usually, security proofs for existing quantum key distribution are asymptotic in the sense that certain parameters are exceedingly large compared to practical realistic values. In this paper, we derive bounds on the secret key rates for the Bennett 1992 protocol, which includes a preprocessing step. The derivation for a finite-size key is expressed as an optimization problem by using results from the uncertainty relations and the smooth Rényi entropies.