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

We study the equivalence between non-perfect secret sharing (NSS) and symmetric private information retrieval (SPIR) with colluding and unresponsive servers. We prove the equivalence between NSS and SPIR in the following two senses. 1) Given any SPIR protocol, we can construct an NSS protocol. 2) Given any linear NSS protocol, we can construct a SPIR protocol. From this equivalence relation, we prove that the capacity of $\mathsf{n}$-server SPIR with $\mathsf{r}$ responsive servers and $\mathsf{r}$ colluding servers is $(\mathsf{r}-\mathsf{t})/\mathsf{n}$.