Secret sharing allows a dealer to distribute a secret among a set of parties such that only authorized subsets, specified by an access structure, can reconstruct the secret. Sehrawat and Desmedt (COCOON 2020 [80]) 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 a novel access structure hiding verifiable secret sharing scheme that supports all monotone access structures. Our scheme is the first secret sharing solution to support malicious behavior identification and share verifiability in malicious-majority settings. Furthermore, the verification procedure of our scheme incurs no communication overhead, and is therefore “free”. As the building blocks of our scheme, we introduce and construct the following:

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a set-system with greater than $\exp \left(c\frac{2{(\log h)}^2}{(\log \log h)}\right)+2\exp \left(c\frac{{(\log h)}^2}{(\log \log h)}\right)$ 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 while the sizes of the pairwise intersections of different sets are not divisible by m unless one set is a (proper) subset of the other,

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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.