We study the equivalence between non-perfect secret sharing (NSS) and symmetric private information retrieval (SPIR) with arbitrary response and collusion patterns. NSS and SPIR are defined with an access structure, which corresponds to the authorized/forbidden sets for NSS and the response/collusion patterns for SPIR. We prove the equivalence between NSS and SPIR in the following two senses. 1) Given any SPIR protocol with an access structure, an NSS protocol is constructed with the same access structure and the same rate. 2) Given any linear NSS protocol with an access structure, a linear SPIR protocol is constructed with the same access structure and the same rate. We prove the first relation even if the SPIR protocol has imperfect correctness and secrecy. From the first relation, we derive an upper bound of the SPIR capacity for arbitrary response and collusion patterns. For the special case of $\mathsf{n}$-server SPIR with $\mathsf{r}$ responsive and $\mathsf{t}$ colluding servers, this upper bound proves that the SPIR capacity is $(\mathsf{r}-\mathsf{t})/\mathsf{n}$. From the second relation, we prove that a SPIR protocol exists for any response and collusion patterns.
翻译:我们用任意反应和串通模式研究非完美秘密共享(NSS)和对称私人信息检索(SPIR)之间的等同性。NSS和SPIR的定义是连接结构,与NSS授权/禁止的数据集和SPIR的响应/包状模式相对应。我们证明NSS和SPIR在以下两种意义上的等同性。1鉴于任何有接入结构的SPIR协议,国家安全局协议的构建都具有相同的访问结构,2鉴于任何有访问结构的线性NSS协议,线性SPIR协议的构建与访问结构和速度相同。即使SPIR协议的准确性和保密性不完善,我们也证明第一种关系。从第一个关系中,我们从SPIR能力中得出了任意反应和串通模式的上限。对于以$mathsf{r{r}反应和以美元/maths{r{r} 或以美元xxxxxxxxxx/cloud 服务器的特例,这种上下框证明SPIR能力是来自美元\ma}/mas refr} a s relations sex sex sex sex sex sex sex sex sex sex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex exf} a.