  In this work we present a security analysis for quantum key distribution,
establishing a rigorous tradeoff between various protocol and security
parameters for a class of entanglement-based and prepare-and-measure protocols.
The goal of this paper is twofold: 1) to review and clarify the
state-of-the-art security analysis based on entropic uncertainty relations, and
2) to provide an accessible resource for researchers interested in a security
analysis of quantum cryptographic protocols that takes into account finite
resource effects. For this purpose we collect and clarify several arguments
spread in the literature on the subject with the goal of making this treatment
largely self-contained.
  More precisely, we focus on a class of prepare-and-measure protocols based on
the Bennett-Brassard (BB84) protocol as well as a class of entanglement-based
protocols similar to the Bennett-Brassard-Mermin (BBM92) protocol. We carefully
formalize the different steps in these protocols, including randomization,
measurement, parameter estimation, error correction and privacy amplification,
allowing us to be mathematically precise throughout the security analysis. We
start from an operational definition of what it means for a quantum key
distribution protocol to be secure and derive simple conditions that serve as
sufficient condition for secrecy and correctness. We then derive and eventually
discuss tradeoff relations between the block length of the classical
computation, the noise tolerance, the secret key length and the security
parameters for our protocols. Our results significantly improve upon previously
reported tradeoffs.
