Definition: Ideal key and distance

For a normalized Eve law ${\sigma }_{ET}$, define

For a bit output $\tau$, write

A secure fixed-length key-distillation sequence has lengths ${\ell }_n\ge 0$ and comparison laws ${\sigma }_n$ such that

Its lower asymptotic rate is ${\mathrm{liminf}}_n{\ell }_n/n$. The corresponding $K_D$ is the supremum over admitted secure sequences, including the empty-key sequence.