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标题：Upslices, Downslices, and Secret-Sharing with Complexity of 1.5n
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作者：Benny Applebaum ; Oded Nir
期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2021
卷号：21
语种：English
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：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:01n01 . 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 (an) -upslices and (bn) -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 15n+o(n)20585n. Since every monotone function can be cheaply decomposed into n downslices, we obtain a similar result for general access structures improving the previously known 20637n+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 Farras (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 ka monotone terms of size a, where k=(k1kn) is an arbitrary vector of non-negative integers. We show that, except with exponentially small probability, F can be realized with share size of 205n+o(n) and can be linearly realized with an exponent strictly smaller than 23 . 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.
关键词：Information Theoretic;secret sharing