X-Secure T-Private Information Retrieval From MDS Coded Storage With Byzantine and Unresponsive Servers

• Published in: IEEE transactions on information theory Vol. 66; no. 12; pp. 7427 - 7438
• Main Authors: Jia, Zhuqing, Jafar, Syed Ali
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-12-2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Asymptotic properties; Convexity; Decoding; distributed storage; Encoding; Indexes; Information retrieval; Interference cancellation; Lower bounds; MDS codes; Messages; Multiplication; Privacy; Security; Servers
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2020.3013152

The problem of <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula>-secure <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula>-private information retrieval from MDS coded storage is studied in this paper, where the user wishes to privately retrieve one out of <inline-formula> <tex-math notation="LaTeX">K </tex-math></inline-formula> independent messages that are distributed over <inline-formula> <tex-math notation="LaTeX">N </tex-math></inline-formula> servers according to an MDS code. It is guaranteed that any group of up to <inline-formula> <tex-math notation="LaTeX">X </tex-math></inline-formula> colluding servers learn nothing about the messages and that any group of up to <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula> colluding servers learn nothing about the identity of desired message. A lower bound of achievable rates is proved by presenting a novel scheme based on cross-subspace alignment and a successive decoding with interference cancellation strategy. For large number of messages <inline-formula> <tex-math notation="LaTeX">(K\rightarrow \infty) </tex-math></inline-formula> the achieved rate, which we conjecture to be optimal, improves upon the best known rates previously reported in the literature by Raviv and Karpuk, and generalizes an achievable rate for MDS-TPIR previously found by Freij-Hollanti et al. that is also conjectured to be asymptotically optimal. The setting is then expanded to allow unresponsive and Byzantine servers. Finally, the scheme is applied to find a new lower convex hull of (download, upload) pairs of secure and private distributed matrix multiplication that generalizes, and in certain asymptotic settings strictly improves upon the best known previous results.