  We introduce an explicit construction for a key distribution protocol in the
Quantum Computational Timelock (QCT) security model, where one assumes that
computationally secure encryption may only be broken after a time much longer
than the coherence time of available quantum memories.
  Taking advantage of the QCT assumptions, we build a key distribution protocol
called HM-QCT from the Hidden Matching problem for which there exists an
exponential gap in one-way communication complexity between classical and
quantum strategies.
  We establish that the security of HM-QCT against arbitrary i.i.d. attacks can
be reduced to the difficulty of solving the underlying Hidden Matching problem
with classical information. Legitimate users, on the other hand, can use
quantum communication, which gives them the possibility of sending multiple
copies of the same quantum state while retaining an information advantage. This
leads to an everlasting secure key distribution scheme over $n$ bosonic modes.
Such a level of security is unattainable with purely classical techniques.
Remarkably, the scheme remains secure with up to $\mathcal{O}\big(
\frac{\sqrt{n}}{\log(n)}\big)$ input photons for each channel use, extending
the functionalities and potentially outperforming QKD rates by several orders
of magnitudes.
