Private Information Retrieval is Graph Based Replication Systems

Replication is prevalent in both theory and practice as a means for obtaining robustness in distributed storage systems. A system in which every data entry is stored on two separate servers gives rise to a graph structure in a natural way, and the combinatorial properties of this graph shed light on the possible features of the system. One possible feature of interest, that has recently gained renewed attention, is private information retrieval (PIR). A PIR protocol enables a user to obtain a data entry from a storage system without revealing the identity of the requested entry to sets of colluding servers. In this paper we suggest a simple PIR protocol for graph based replication systems, which guarantees perfect secrecy against any set of colluding servers that does not induce a cycle. Furthermore, it is shown that the secrecy deteriorates gracefully with the number of cycles in the colluding set, and that the upload complexity can be reduced for graphs of certain specialized structure.

[1]  M. Murty.  Ramanujan Graphs , 1965 .

[2]  László Babai,et al.  A short proof of the nonuniform Ray-Chaudhuri-Wilson inequality , 1988, Comb..

[3]  Daniel A. Spielman,et al.  Expander codes , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[4]  Eyal Kushilevitz,et al.  Private information retrieval , 1998, JACM.

[5]  Felix Lazebnik,et al.  On Hypergraphs of Girth Five , 2003, Electron. J. Comb..

[6]  Sergey Yekhanin,et al.  Locally Decodable Codes and Private Information Retrieval Schemes , 2010, Information Security and Cryptography.

[7]  Kannan Ramchandran,et al.  Fractional repetition codes for repair in distributed storage systems , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[8]  Daniela Kühn,et al.  Loose Hamilton cycles in hypergraphs , 2008, Discret. Math..

[9]  Ervin Györi,et al.  Hypergraphs with no cycle of length 4 , 2012, Discret. Math..

[10]  Sriram Vishwanath,et al.  Explicit MBR all-symbol locality codes , 2013, 2013 IEEE International Symposium on Information Theory.

[11]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[12]  Zeev Dvir,et al.  2-Server PIR with Sub-Polynomial Communication , 2014, STOC.

[13]  David Ellis,et al.  On Regular Hypergraphs of High Girth , 2013, Electron. J. Comb..

[14]  Kannan Ramchandran,et al.  One extra bit of download ensures perfectly private information retrieval , 2014, 2014 IEEE International Symposium on Information Theory.

[15]  Eitan Yaakobi,et al.  PIR with Low Storage Overhead: Coding instead of Replication , 2015, ArXiv.

[16]  Natalia Silberstein,et al.  Optimal Fractional Repetition Codes Based on Graphs and Designs , 2014, IEEE Transactions on Information Theory.

[17]  Zeev Dvir,et al.  2-Server PIR with Subpolynomial Communication , 2016, J. ACM.

[18]  Salim El Rouayheb,et al.  Private Information Retrieval From MDS Coded Data in Distributed Storage Systems , 2016, IEEE Transactions on Information Theory.

[19]  Hua Sun,et al.  The Capacity of Private Information Retrieval , 2017, IEEE Transactions on Information Theory.

[20]  Tuvi Etzion,et al.  PIR Array Codes with Optimal PIR Rate , 2016, ArXiv.

[21]  Eitan Yaakobi,et al.  Codes for graph erasures , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[22]  PIR schemes with small download complexity and low storage requirements , 2016, 2017 IEEE International Symposium on Information Theory (ISIT).

[23]  Ravi Tandon,et al.  Private Information Retrieval from Storage Constrained Databases - Coded Caching meets PIR , 2017, ArXiv.

[24]  Eitan Yaakobi,et al.  Codes for erasures over directed graphs , 2017, 2017 IEEE Information Theory Workshop (ITW).

[25]  Camilla Hollanti,et al.  Private Information Retrieval from Coded Databases with Colluding Servers , 2016, SIAM J. Appl. Algebra Geom..

[26]  Camilla Hollanti,et al.  Private information retrieval schemes for codec data with arbitrary collusion patterns , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[27]  Tuvi Etzion,et al.  PIR array codes with optimal PIR rates , 2016, 2017 IEEE International Symposium on Information Theory (ISIT).

[28]  Mikael Skoglund,et al.  Symmetric private information retrieval for MDS coded distributed storage , 2016, 2017 IEEE International Conference on Communications (ICC).

[29]  Hua Sun,et al.  The Capacity of Private Information Retrieval with Disjoint Colluding Sets , 2017, GLOBECOM 2017 - 2017 IEEE Global Communications Conference.

[30]  Hua Sun,et al.  Multiround Private Information Retrieval: Capacity and Storage Overhead , 2016, IEEE Transactions on Information Theory.

[31]  Hua Sun,et al.  The Capacity of Robust Private Information Retrieval With Colluding Databases , 2016, IEEE Transactions on Information Theory.

[32]  Sennur Ulukus,et al.  Multi-Message Private Information Retrieval: Capacity Results and Near-Optimal Schemes , 2017, IEEE Transactions on Information Theory.

[33]  Sennur Ulukus,et al.  The Capacity of Private Information Retrieval From Coded Databases , 2016, IEEE Transactions on Information Theory.

[34]  David A. Karpuk.  Private Computation of Systematically Encoded Data with Colluding Servers , 2018, 2018 IEEE International Symposium on Information Theory (ISIT).

[35]  Sennur Ulukus,et al.  Private Information Retrieval from Non-Replicated Databases , 2018, 2019 IEEE International Symposium on Information Theory (ISIT).

[36]  Tuvi Etzion,et al.  PIR Array Codes With Optimal Virtual Server Rate , 2016, IEEE Transactions on Information Theory.

[37]  Syed Ali Jafar,et al.  On the Asymptotic Capacity of X-Secure T-Private Information Retrieval With Graph-Based Replicated Storage , 2019, IEEE Transactions on Information Theory.

[38]  Deepak Kumar,et al.  The Capacity of Private Information Retrieval From Uncoded Storage Constrained Databases , 2018, IEEE Transactions on Information Theory.