  Secret sharing provides a means to distribute shares of a secret such that
any authorized subset of shares, specified by an access structure, can be
pooled together to recompute the secret. The standard secret sharing model
requires public access structures, which violates privacy and facilitates the
adversary by revealing high-value targets. In this paper, we address this
shortcoming by introducing \emph{hidden access structures}, which remain secret
until some authorized subset of parties collaborate. The central piece of this
work is the construction of a set-system $\mathcal{H}$ with strictly greater
than $\exp\left(c \dfrac{1.5 (\log h)^2}{\log \log h}\right)$ subsets of a set
of $h$ elements. Our set-system $\mathcal{H}$ is defined over $\mathbb{Z}_m$,
where $m$ is a non-prime-power, such that the size of each set in $\mathcal{H}$
is divisible by $m$ but the sizes of their pairwise intersections are not
divisible by $m$, unless one set is a subset of another. We derive a vector
family $\mathcal{V}$ from $\mathcal{H}$ such that superset-subset relationships
in $\mathcal{H}$ are represented by inner products in $\mathcal{V}$. We use
$\mathcal{V}$ to "encode" the access structures and thereby develop the first
\emph{access structure hiding} secret sharing scheme. For a setting with $\ell$
parties, our scheme supports $2^{\binom{\ell}{\ell/2+1}}$ out of the
$2^{2^{\ell - O(\log \ell)}}$ total monotone access structures, and its maximum
share size for any access structures is $(1+ o(1)) \dfrac{2^{\ell+1}}{\sqrt{\pi
\ell/2}}$. The scheme assumes semi-honest polynomial-time parties, and its
security relies on the Generalized Diffie-Hellman assumption.
