  Security proof methods for quantum key distribution, QKD, that are based on
the numerical key rate calculation problem, are powerful in principle. However,
the practicality of the methods are limited by computational resources and the
efficiency and accuracy of the underlying algorithms for convex optimization.
We derive a stable reformulation of the convex nonlinear semidefinite
programming, SDP, model for the key rate calculation problems. We use this to
develop an efficient, accurate algorithm. The reformulation is based on novel
forms of facial reduction, FR, for both the linear constraints and nonlinear
relative entropy objective function. This allows for a Gauss-Newton type
interior-point approach that avoids the need for perturbations to obtain strict
feasibility, a technique currently used in the literature. The result is high
accuracy solutions with theoretically proven lower bounds for the original QKD
from the FR stable reformulation. This provides novel contributions for FR for
general SDP.
  We report on empirical results that dramatically improve on speed and
accuracy, as well as solving previously intractable problems.
