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We study the equivalence between non-perfect secret sharing (NSS) andsymmetric private information retrieval (SPIR) with arbitrary response andcollusion patterns. NSS and SPIR are defined with an access structure, whichcorresponds to the authorized/forbidden sets for NSS and the response/collusionpatterns for SPIR. We prove the equivalence between NSS and SPIR in thefollowing two senses. 1) Given any SPIR protocol with an access structure, anNSS protocol is constructed with the same access structure and the same rate.2) Given any linear NSS protocol with an access structure, a linear SPIRprotocol is constructed with the same access structure and the same rate. Weprove the first relation even if the SPIR protocol has imperfect correctnessand secrecy. From the first relation, we derive an upper bound of the SPIRcapacity 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 aSPIR protocol exists for any response and collusion patterns.

Equivalence of Non-Perfect Secret Sharing and Symmetric Private Information Retrieval with General Access Structure