International Association for Cryptologic Research

International Association
for Cryptologic Research


Oded Nir


Upslices, Downslices, and Secret-Sharing with Complexity of $1.5^n$ 📺
Oded Nir Benny Applebaum
A secret-sharing scheme allows to distribute a secret $s$ among $n$ parties such that only some predefined ``authorized'' sets of parties can reconstruct the secret, and all other ``unauthorized'' sets learn nothing about $s$. The collection of authorized/unauthorized sets is be captured by a monotone function $f:\{0,1\}^n\rightarrow \{0,1\}$. In this paper, we focus on monotone functions that all their min-terms are sets of size $a$, and on their duals -- monotone functions whose max-terms are of size $b$. We refer to these classes as $(a,n)$-\emph{upslices} and $(b,n)$-\emph{downslices}, and note that these natural families correspond to monotone $a$-regular DNFs and monotone $(n-b)$-regular CNFs. We derive the following results. \begin{enumerate} \item (General downslices) Every downslice can be realized with total share size of $1.5^{n+o(n)}<2^{0.585 n}$. Since every monotone function can be cheaply decomposed into $n$ downslices, we obtain a similar result for general access structures improving the previously known $2^{0.637n+o(n)}$ complexity of Applebaum, Beimel, Nir and Peter (STOC 2020). We also achieve a minor improvement in the exponent of linear secrets sharing schemes. \item (Random mixture of upslices) Following, Beimel and Farr{\`{a}}s (TCC 2020) who studied the complexity of random DNFs with constant-size terms, we consider the following general distribution $F$ over monotone DNFs: For each width value $a\in [n]$, uniformly sample $k_a$ monotone terms of size $a$, where $\vec{k}=(k_1,\ldots,k_n)$ is an arbitrary vector of non-negative integers. We show that, except with exponentially small probability, $F$ can be realized with share size of $2^{0.5 n+o(n)}$ and can be linearly realized with an exponent strictly smaller than $2/3$. Our proof also provides a candidate distribution for the ``exponentially-hard'' access structure. \end{enumerate} We use our results to explore connections between several seemingly unrelated questions about the complexity of secret-sharing schemes such as worst-case vs. average-case, linear vs. non-linear, and primal vs. dual access structures. We prove that, in at least one of these settings, there is a significant gap in secret-sharing complexity.
Secret-Sharing Schemes for General and Uniform Access Structures 📺
A secret-sharing scheme allows some authorized sets of parties to reconstruct a secret; the collection of authorized sets is called the access structure. For over 30 years, it was known that any (monotone) collection of authorized sets can be realized by a secret-sharing scheme whose shares are of size $$2^{n-o(n)}$$ and until recently no better scheme was known. In a recent breakthrough, Liu and Vaikuntanathan (STOC 2018) have reduced the share size to $$O(2^{0.994n})$$. Our first contribution is improving the exponent of secret sharing down to 0.892. For the special case of linear secret-sharing schemes, we get an exponent of 0.942 (compared to 0.999 of Liu and Vaikuntanathan).Motivated by the construction of Liu and Vaikuntanathan, we study secret-sharing schemes for uniform access structures. An access structure is k-uniform if all sets of size larger than k are authorized, all sets of size smaller than k are unauthorized, and each set of size k can be either authorized or unauthorized. The construction of Liu and Vaikuntanathan starts from protocols for conditional disclosure of secrets, constructs secret-sharing schemes for uniform access structures from them, and combines these schemes in order to obtain secret-sharing schemes for general access structures. Our second contribution in this paper is constructions of secret-sharing schemes for uniform access structures. We achieve the following results:A secret-sharing scheme for k-uniform access structures for large secrets in which the share size is $$O(k^2)$$ times the size of the secret.A linear secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is $$\tilde{O}(2^{h(k/n)n/2})$$ (where h is the binary entropy function). By counting arguments, this construction is optimal (up to polynomial factors).A secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is $$2^{\tilde{O}(\sqrt{k \log n})}$$. Our third contribution is a construction of ad-hoc PSM protocols, i.e., PSM protocols in which only a subset of the parties will compute a function on their inputs. This result is based on ideas we used in the construction of secret-sharing schemes for k-uniform access structures for a binary secret.