论文标题
极端集理论和基于LWE的访问结构隐藏可验证的秘密共享，以恶意和自由验证
Extremal Set Theory and LWE Based Access Structure Hiding Verifiable Secret Sharing with Malicious-Majority and Free Verification
论文作者
论文摘要
秘密共享允许在几个方之间分配秘密，这样只有访问结构指定的授权子集才能重建秘密。 Sehrawat和Desmedt（Cocoon 2020）引入了隐藏的访问结构，这些结构一直秘密，直到各方的某些授权子集进行合作为止。但是，他们的计划假设半honest政党仅支持受限制的访问结构。我们通过构建一个支持所有单调访问结构的访问结构来解决这些缺点。这是支持骗子识别并在恶意差异设置中共享验证性的第一个秘密共享计划。我们计划的验证程序不会造成开销的沟通。作为我们方案的构建块，我们介绍和构造：（i）具有$> \ exp \ left的设定系统（c \ frac {2（\ log h）^2} {（\ \ log \ log h）} \ right）+2 \ right）+2 \ exp \ exp \ exp \ left（c \ log h）^（\ log h）^$ nog fog f log y nog pog for $ h $元素。我们的设定系统，$ \ mathcal {h} $，在$ \ mathbb {z} _m $上定义，其中$ m $是非prime-Power。 $ \ Mathcal {h} $中的每个集合的大小可除以$ m $，但是它们的成对相交的大小不是，除非一组是另一个子集，（ii）一种新的学习变体（LWE）问题（LWE）问题（称为Prim-lwe），称为prim-lwe，其中nectrix是$ quake $ quest的$ quand $ quand $ n n $ n n n $ n n n $ n n n; $ Q $是LWE模量。我们方案的安全依赖于LWE问题的硬度，其共享大小为$$（1+ o（1））\ dfrac {2^{\ ell}} {\ sqrt {\ sqrt {π\ ell/2}}}}}}}（2 q^{\ varrho + 0.5} + 0.5} + 0} + \ \ \ \ \ \ sqrt） $ \ varrho \ leq 1 $是常数，$ \ ell $是当事方的总数。我们还为将来的工作提供了指导，以将股份规模减少到 \ [\ leq \ dfrac {1} {3} \ left（（（1+ o（1））\ dfrac {2^{\ ell}} {\ sqrt {\ sqrt {π\ ell/2}}}}}}（2 q^{\ varrho + 0.5} + 0.5}
Secret sharing allows distributing a secret among several parties such that only authorized subsets, specified by an access structure, can reconstruct the secret. Sehrawat and Desmedt (COCOON 2020) introduced hidden access structures, that remain secret until some authorized subset of parties collaborate. However, their scheme assumes semi-honest parties and supports only restricted access structures. We address these shortcomings by constructing an access structure hiding verifiable secret sharing scheme that supports all monotone access structures. It is the first secret sharing scheme to support cheater identification and share verifiability in malicious-majority settings. The verification procedure of our scheme incurs no communication overhead. As the building blocks of our scheme, we introduce and construct: (i) a set-system with $> \exp\left(c\frac{2(\log h)^2}{(\log\log h)}\right)+2\exp\left(c\frac{(\log h)^2}{(\log\log h)}\right)$ subsets of a set of $h$ elements. Our set-system, $\mathcal{H}$, is defined over $\mathbb{Z}_m$, where $m$ is a non-prime-power. The size of each set in $\mathcal{H}$ is divisible by $m$ but the sizes of their pairwise intersections are not, unless one set is a subset of another, (ii) a new variant of the learning with errors (LWE) problem, called PRIM-LWE, wherein the secret matrix is sampled such that its determinant is a generator of $\mathbb{Z}_q^*$, where $q$ is the LWE modulus. The security of our scheme relies on the hardness of the LWE problem, and its share size is $$(1+ o(1)) \dfrac{2^{\ell}}{\sqrt{π\ell/2}}(2 q^{\varrho + 0.5} + \sqrt{q} + \mathrmΘ(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 \[\leq \dfrac{1}{3} \left( (1+ o(1)) \dfrac{2^{\ell}}{\sqrt{π\ell/2}}(2 q^{\varrho + 0.5} + 2\sqrt{q}) \right).\]
