  Functional encryption is a powerful paradigm for public-key encryption which
allows for controlled access to encrypted data. This primitive is generally
impossible in the standard setting so we investigate possibilities in the
bounded quantum storage model (BQSM) and the bounded classical storage model
(BCSM). In these models, ciphertexts potentially disappear which nullifies
impossibility results and allows us to obtain positive outcomes.
  Firstly, in the BQSM, we construct information-theoretically secure
functional encryption with $\texttt{q}=O(\sqrt{\texttt{s}/\texttt{r}})$ where
$\texttt{r}$ can be set to any value less than $\texttt{s}$. Here $\texttt{r}$
denotes the number of times that an adversary is restricted to
$\texttt{s}$--qubits of quantum memory in the protocol and $\texttt{q}$ denotes
the required quantum memory to run the protocol honestly. We then show that our
scheme is optimal by proving that it is impossible to attain
information-theoretically secure functional encryption with $\texttt{q} <
\sqrt{\texttt{s}/\texttt{r}}$. However, by assuming the existence of
post-quantum one-way functions, we can do far better and achieve functional
encryption with classical keys and with $\texttt{q}=0$ and $\texttt{r}=1$.
  Secondly, in the BCSM, we construct $(O(\texttt{n}),\texttt{n}^2)$ functional
encryption assuming the existence of $(\texttt{n},\texttt{n}^2)$ virtual weak
grey-box obfuscation. Here, the pair $(\texttt{n},\texttt{n}^2)$ indicates the
required memory to run honestly and the needed memory to break security,
respectively. This memory gap is optimal and the assumption is minimal. In
particular, we also construct $(O(\texttt{n}),\texttt{n}^2)$ virtual weak
grey-box obfuscation assuming $(\texttt{n},\texttt{n}^2)$ functional
encryption.

Cryptography and Security