  Extending the functionality and overcoming the performance limitation under
which QKD can operate requires either quantum repeaters or new security models.
Investigating the latter option, we introduce the \textit{Quantum Computational
Hybrid} (QCH) security model, where we assume that computationally secure
encryption may only be broken after time much longer than the coherence time of
available quantum memories. We propose an explicit $d$-dimensional key
distribution protocol, that we call MUB-\textit{Quantum Computational Timelock}
(MUB-QCT) where one bit is encoded on a qudit state chosen among $d+1$ mutually
unbiased bases (MUBs). Short-term-secure encryption is used to share the basis
information with legitimate users while keeping it unknown from Eve until after
her quantum memory decoheres. This allows reducing Eve's optimal attack to an
immediate measurement followed by post-measurement decoding. \par We
demonstrate that MUB-QCT enables everlasting secure key distribution with input
states containing up to $O(\sqrt{d})$ photons. This leads to a series of
important improvements when compared to QKD: on the functional side, the
ability to operate securely between one sender and many receivers, whose
implementation can moreover be untrusted; significant performance increase,
characterized by a $O(\sqrt{d})$ multiplication of key rates and an extension
by $25 {\rm} km \times \log(d)$ of the attainable distance over fiber.
Implementable with a large number of modes with current or near-term multimode
photonics technologies, the MUB-QCT construction has the potential to provide a
radical shift to the performance and practicality of quantum key distribution.
