Extremal Set Theory and LWE Based Access Structure Hiding Verifiable Secret Sharing with Malicious-Majority and Free Verification

Secret sharing allows distributing a secret among several parties such that only authorized subsets, specified by an access structure, can reconstruct the secret. Sehrawat and Desmedt (COCOON 2020) introduced hidden access structures, that remain secret until some authorized subset of parties collaborate. However, their scheme assumes semi-honest parties and supports only restricted access structures. We address these shortcomings by constructing an access structure hiding verifiable secret sharing scheme that supports all monotone access structures. It is the first secret sharing scheme to support cheater identification and share verifiability in malicious-majority settings. The verification procedure of our scheme incurs no communication overhead. As the building blocks of our scheme, we introduce and construct: (i) a set-system with $> \exp\left(c\frac{2(\log h)^2}{(\log\log h)}\right)+2\exp\left(c\frac{(\log h)^2}{(\log\log h)}\right)$ 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. The size of each set in $\mathcal{H}$ is divisible by $m$ but the sizes of their pairwise intersections are not, unless one set is a subset of another, (ii) a new variant of the learning with errors (LWE) problem, called PRIM-LWE, wherein the secret matrix is sampled such that its determinant is a generator of $\mathbb{Z}_q^*$, where $q$ is the LWE modulus. The security of our scheme relies on the hardness of the LWE problem, and its share size is $$(1+ o(1)) \dfrac{2^{\ell}}{\sqrt{\pi \ell/2}}(2 q^{\varrho + 0.5} + \sqrt{q} + \mathrm{\Theta}(h)),$$ where $\varrho \leq 1$ is a constant and $\ell$ is the total number of parties. We also provide directions for future work to reduce the share size to \[\leq \dfrac{1}{3} \left( (1+ o(1)) \dfrac{2^{\ell}}{\sqrt{\pi \ell/2}}(2 q^{\varrho + 0.5} + 2\sqrt{q}) \right).\]