Data security and availability for operational use are frequently seen as conflicting goals. Research on searchable encryption and homomorphic encryption are a start, but they typically build from encryption methods that, at best, provide protections based on problems assumed to be computationally hard. By contrast, data encoding methods such as secret sharing provide information-theoretic data protections. Archives that distribute data using secret sharing can provide data protections that are resilient to malicious insiders, compromised systems, and untrusted components. In this paper, we create the Serial Interpolation Filter, a method for storing and interacting with sets of data that are secured and distributed using secret sharing. We provide the ability to operate over set-oriented data distributed across multiple repositories without exposing the original data. Furthermore, we demonstrate the security of our method under various attacker models and provide protocol extensions to handle colluding attackers. The Serial Interpolation Filter provides information-theoretic protections from a single attacker and computationally hard protections from colluding attackers.