TY - JOUR
T1 - Bounds on the Length of Functional PIR and Batch Codes
AU - Zhang, Yiwei
AU - Etzion, Tuvi
AU - Yaakobi, Eitan
N1 - Funding Information: Manuscript received April 15, 2019; revised December 25, 2019; accepted February 14, 2020. Date of publication March 2, 2020; date of current version July 14, 2020. The work of Yiwei Zhang and Eitan Yaakobi was supported by the ISF Grant 1817/18. The work of Yiwei Zhang and Tuvi Etzion was supported by the BSF-NSF Grant 2016692. The work of Yiwei Zhang was supported by a Technion Fellowship. This work was supported by the Technion Hiroshi Fujiwara Cyber Security Research Center and the Israel Cyber Directorate. This article was presented in part at the 2019 IEEE International Symposium on Information Theory (ISIT). (Corresponding author: Tuvi Etzion.) Yiwei Zhang was with the Department of Computer Science, Technion— Israel Institute of Technology, Haifa 3200003, Israel. He is now with the School of Cyber Science and Technology, Shandong University, Qingdao 266237, China, and also with the Key Laboratory of Cryptologic Technology and Information Security, Ministry of Education, Shandong University, Qingdao 266237, China (e-mail: [email protected]). Publisher Copyright: © 1963-2012 IEEE.
PY - 2020/8
Y1 - 2020/8
N2 - A functional k-Private Information Retrieval (k-PIR) code of dimension s consists of n servers storing linear combinations of s linearly independent information symbols. Any linear combination of the s information symbols can be recovered by k disjoint subsets of servers. The goal is to find the minimum number of servers for given k and s. We provide lower bounds on the minimum number of servers and constructions which yield upper bounds on this number. For k ≤ 4, exact bounds on this number are proved. Furthermore, we provide some asymptotic bounds. The problem coincides with the well known PIR problem based on a coded database to reduce the storage overhead, when each linear combination contains exactly one information symbol. If any multiset of size k of linear combinations from the linearly independent information symbols can be recovered by k disjoint subset of servers, then the servers form a functional k -batch code. A functional k-batch code is a functional k-PIR code, where all the k linear combinations in the multiset are equal. We provide some bounds on the minimum number of servers for functional k-batch codes. In particular we present a random construction and a construction based on simplex codes, Write-Once Memory (WOM) codes, and Random I/O (RIO) codes.
AB - A functional k-Private Information Retrieval (k-PIR) code of dimension s consists of n servers storing linear combinations of s linearly independent information symbols. Any linear combination of the s information symbols can be recovered by k disjoint subsets of servers. The goal is to find the minimum number of servers for given k and s. We provide lower bounds on the minimum number of servers and constructions which yield upper bounds on this number. For k ≤ 4, exact bounds on this number are proved. Furthermore, we provide some asymptotic bounds. The problem coincides with the well known PIR problem based on a coded database to reduce the storage overhead, when each linear combination contains exactly one information symbol. If any multiset of size k of linear combinations from the linearly independent information symbols can be recovered by k disjoint subset of servers, then the servers form a functional k -batch code. A functional k-batch code is a functional k-PIR code, where all the k linear combinations in the multiset are equal. We provide some bounds on the minimum number of servers for functional k-batch codes. In particular we present a random construction and a construction based on simplex codes, Write-Once Memory (WOM) codes, and Random I/O (RIO) codes.
KW - Private Information Retrieval (PIR) codes
KW - batch codes
KW - distributed storage codes
UR - https://www.scopus.com/pages/publications/85088538900
U2 - 10.1109/TIT.2020.2977631
DO - 10.1109/TIT.2020.2977631
M3 - Article
SN - 0018-9448
VL - 66
SP - 4917
EP - 4934
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 8
M1 - 9020118
ER -