High-dimensional entanglement promises not only increased key rates but overcoming some of the obstacles faced by modern-day quantum communication. Typically, key rates are computed via convex optimization procedures, which inherently limits the dimensionality one can analyze through computational constraints. Recent progress in high-dimensional photonics far exceeds these limitations and brings forth a need for (semi-)analytic methods to compute key rates in the regime of large encoding dimensions. We present a flexible analytic framework facilitated by the dual of a semi-definite program, enabling the computation of key rates in high-dimensional systems. This method, whether purely analytical or semi-numerical, hinges on diagonalizing specific operators influenced by entanglement witnesses and efficiently solving an optimization problem. To facilitate the latter, we show how matrix completion techniques can be incorporated to yield effective and computable bounds on the key rate in paradigmatic high-dimensional systems of time- or frequency-bin entangled photons and beyond.