Fault-Tolerant Private Information Retrieval via Threshold Distributed Point Functions

Multi-server private information retrieval (PIR) based on function secret sharing (FSS) has emerged as a prominent paradigm for achieving sublinear communication. However, standard FSS constructions strictly require full server participation, making them highly vulnerable to single-node fail-stop faults. Existing fault-tolerant schemes mitigate this but inevitably inflate the downlink response overhead to scale with the database size N (e.g., \( O(\sqrt{N}) \)). To overcome this limitation, we propose a (t,p)-fault-tolerant PIR (FT-PIR) protocol grounded in a newly designed generalized (t,p)-fault-tolerant distributed point function (FT-DPF). By introducing a hierarchical recursive patching mechanism, our scheme transforms rigid all-party evaluations into flexible t-out-of-p reconstructions. This architecture completely decouples the response communication from N and ensures efficient client-side reconstruction via lightweight XOR aggregations, fundamentally bypassing heavy algebraic interpolations. Formal analysis proves that our strictly stateless protocol guarantees (t-1)-computational privacy under the semi-honest model. Asymptotic evaluations demonstrate that the proposed FT-PIR achieves an optimal downlink complexity bounded to O(\( poly(t,p) \cdot \log p \)), significantly outperforming existing robust baselines for large-scale datasets.