They are called PIR with sublinear online computation or offline/online PIR.
Key idea is the client issues a query that is independent of what they really want. This is the “offline” query. This query is linear (unavoidable) and the client gets some hints as a result
Then later, the client can issue one or more queries for elements they want and those queries can be sublinear in terms of computation. The client uses hints to make that happen.
Key idea is the client issues a query that is independent of what they really want. This is the “offline” query. This query is linear (unavoidable) and the client gets some hints as a result
Then later, the client can issue one or more queries for elements they want and those queries can be sublinear in terms of computation. The client uses hints to make that happen.
Some papers on this are:
https://eprint.iacr.org/2019/1075.pdf
https://eprint.iacr.org/2021/1438
https://eprint.iacr.org/2022/081.pdf