On Arithmetic Private Information Retrieval: Why Code-Based PIR (Usually) Fails

We initiate the study of arithmetic private information retrieval (APIR) schemes, in which the database is a vector of field elements and the scheme makes a black-box use of the field. We obtain the following results.

1. Our main result is a negative one: We show that no single-server APIR scheme can achieve non-trivial download cost smaller than $n$ field elements. We observe that recent proposals for code-based PIR (Holzbaur et al., ISIT'20; Verma and Hollanti, ISIT'24) are arithmetic, and show how to break them within a few minutes on a standard workstation for all suggested parameters.

2. We complement the above by positive results in alternative models. Concretely, we show that with either two servers or a single server with secret-key preprocessing, it is possible to construct computationally secure APIR schemes based on well-studied coding assumptions. This is achieved by arithmetizing the distributed-point-function-based PIR of Boyle et al.(CCS'16), and by observing that the recent construction of secret-key single-server PIR by Chen et al.(STOC'26) also arithmetizes.

3. Finally, we characterize the existence of information-theoretic two-server APIR schemes in linear-algebraic terms, and show that communication of $O(n^{1/3})$ can be achieved in this setting based on the original approach of Chor et al.(FOCS'95). The optimality of this result remains an interesting open question.