Weighted Lifted Codes: Local Correctabilities and Application to Robust Private Information Retrieval
This work addresses practical barriers in PIR protocols for secure data retrieval, offering incremental improvements over existing codes like lifted Reed-Solomon codes.
The paper tackles the problem of designing private information retrieval (PIR) protocols that resist server collusions and have optimal computation complexity, by proposing weighted Reed-Muller codes and their lifted versions, which achieve remarkable asymptotic parameters.
Low degree Reed-Muller codes are known to satisfy local decoding properties which find applications in private information retrieval (PIR) protocols, for instance. However, their practical instantiation encounters a first barrier due to their poor information rate in the low degree regime. This lead the community to design codes with similar local properties but larger dimension, namely the lifted Reed-Solomon codes. However, a second practical barrier appears when one requires that the PIR protocol resists collusions of servers. In this paper, we propose a solution to this problem by considering \emph{weighted} Reed-Muller codes. We prove that such codes allow us to build PIR protocols with optimal computation complexity and resisting to a small number of colluding servers. In order to improve the dimension of the codes, we then introduce an analogue of the lifting process for weigthed degrees. With a careful analysis of their degree sets, we notably show that the weighted lifting of Reed-Solomon codes produces families of codes with remarkable asymptotic parameters.