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arXiv 2008.07969cs.CR
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一种隐藏秘密共享的访问结构

Access Structure Hiding Secret Sharing from Novel Set Systems and Vector Families

Vipin Singh Sehrawat, Yvo Desmedt

摘要:

秘密共享提供了一种分发秘密的方式,使得由访问结构指定的任何授权的份额子集可以被汇集在一起以重新计算秘密。标准的秘密共享模型需要公共访问结构,这侵犯了隐私,并通过暴露高价值目标来方便对手。在本文中,我们通过引入\emph{hidden access structures}来解决这个缺点,它保持秘密,直到一些授权的子集的各方合作。这项工作的核心部分是建设一套系统$\mathcal{H}$严格大于$\exp\left(c \dfrac{1.5 (\log h)^2}{\log \log h}\right)$子集的一组$h$元素。我们的集合系统$\mathcal{H}$是在$\mathbb{Z}_m$上定义的,其中$m$是一个非素幂,使得$\mathcal{H}$中每个集合的大小都可以被$m$整除,但是它们的成对交叉的大小不能被$m$整除,除非一个集合是另一个集合的子集。我们从$\mathcal{H}$导出一个向量族$\mathcal{V}$,使得$\mathcal{H}$中的超集-子集关系由$\mathcal{V}$中的内积表示。我们使用$\mathcal{V}$“编码”的访问结构,从而开发了第一个\emph{access structure hiding}秘密共享方案。对于$\ell$方的设置,我们的方案支持$2^{2^{\ell - O(\log \ell)}}$总单调访问结构中的$2^{\binom{\ell}{\ell/2+1}}$,并且其对于任何访问结构的最大份额大小是$(1+ o(1)) \dfrac{2^{\ell+1}}{\sqrt{π\ell/2}}$。该方案假设参与者是半诚实的多项式时间的,其安全性依赖于广义Diffie-Hellman假设。

英文摘要:

Secret sharing provides a means to distribute shares of a secret such that any authorized subset of shares, specified by an access structure, can be pooled together to recompute the secret. The standard secret sharing model requires public access structures, which violates privacy and facilitates the adversary by revealing high-value targets. In this paper, we address this shortcoming by introducing \emph{hidden access structures}, which remain secret until some authorized subset of parties collaborate. The central piece of this work is the construction of a set-system $\mathcal{H}$ with strictly greater than $\exp\left(c \dfrac{1.5 (\log h)^2}{\log \log h}\right)$ 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, such that the size of each set in $\mathcal{H}$ is divisible by $m$ but the sizes of their pairwise intersections are not divisible by $m$, unless one set is a subset of another. We derive a vector family $\mathcal{V}$ from $\mathcal{H}$ such that superset-subset relationships in $\mathcal{H}$ are represented by inner products in $\mathcal{V}$. We use $\mathcal{V}$ to "encode" the access structures and thereby develop the first \emph{access structure hiding} secret sharing scheme. For a setting with $\ell$ parties, our scheme supports $2^{\binom{\ell}{\ell/2+1}}$ out of the $2^{2^{\ell - O(\log \ell)}}$ total monotone access structures, and its maximum share size for any access structures is $(1+ o(1)) \dfrac{2^{\ell+1}}{\sqrt{π\ell/2}}$. The scheme assumes semi-honest polynomial-time parties, and its security relies on the Generalized Diffie-Hellman assumption.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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