Upslices, Downslices, and Secret-Sharing with Complexity of 1. 5 ⁿ

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 can be captured by a monotone function f: { 0, 1 } ⁿ→ { 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)-upslices and (b, n)-downslices, and note that these natural families correspond to monotone a-regular DNFs and monotone (n- b) -regular CNFs. We derive the following results. 1.(General downslices) Every downslice can be realized with total share size of 1. 5 ^n+o(n)< 2 ^0.585n. 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.2.(Random mixture of upslices) Following Beimel and Farràs (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∈ [ n], uniformly sample k_a monotone terms of size a, where k= (k₁, …, 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 ^0.5n+o(n) and can be linearly realized with an exponent strictly smaller than 2/3. Our proof also provides a candidate distribution for “exponentially-hard” access structure. 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.