Capacity-Achieving Private Information Retrieval Codes With Optimal Message Size and Upload Cost

We propose a new capacity-achieving code for the private information retrieval (PIR) problem, and show that it has the minimum message size (being one less than the number of servers) and the minimum upload cost (being roughly linear in the number of messages) among a general class of capacity-achieving codes, and in particular, among all capacity-achieving linear codes. Different from existing code constructions, the proposed code is asymmetric, and this asymmetry appears to be the key factor leading to the optimal message size and the optimal upload cost. The converse results on the message size and the upload cost are obtained by an analysis of the information theoretic proof of the PIR capacity, from which a set of critical properties of any capacity-achieving code in the code class of interest is extracted. The symmetry structure of the PIR problem is then analyzed, which allows us to construct symmetric codes from asymmetric ones, yielding a meaningful bridge between the proposed code and existing ones in the literature.

[1]  Andris Ambainis,et al.  On Lower Bounds for the Communication Complexity of Private Information Retrieval ∗ , 2000 .

[2]  Hua Sun,et al.  Optimal Download Cost of Private Information Retrieval for Arbitrary Message Length , 2016, IEEE Transactions on Information Forensics and Security.

[3]  Hua Sun,et al.  Private Information Retrieval from MDS Coded Data With Colluding Servers: Settling a Conjecture by Freij-Hollanti et al. , 2018, IEEE Transactions on Information Theory.

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

[5]  Jie Li,et al.  A Generic Transformation to Enable Optimal Repair in MDS Codes for Distributed Storage Systems , 2016, IEEE Transactions on Information Theory.

[6]  Oliver W. Gnilke,et al.  Private Information Retrieval From MDS Coded Data in Distributed Storage Systems , 2018, IEEE Transactions on Information Theory.

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

[8]  Kenneth Zeger,et al.  Linear Network Coding Over Rings – Part I: Scalar Codes and Commutative Alphabets , 2016, IEEE Transactions on Information Theory.

[9]  Balaji Srinivasan Babu,et al.  A Tight Lower Bound on the Sub- Packetization Level of Optimal-Access MSR and MDS Codes , 2017, 2018 IEEE International Symposium on Information Theory (ISIT).

[10]  Jaime Llorca,et al.  Finite-Length Analysis of Caching-Aided Coded Multicasting , 2014, IEEE Transactions on Information Theory.

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

[12]  Yuval Ishai,et al.  Breaking the O(n/sup 1/(2k-1)/) barrier for information-theoretic Private Information Retrieval , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[13]  P. Vijay Kumar,et al.  An Explicit, Coupled-Layer Construction of a High-Rate MSR Code with Low Sub-Packetization Level, Small Field Size and All-Node Repair , 2016, ArXiv.

[14]  Xin Wang,et al.  On Private Information Retrieval Array Codes , 2016, IEEE Transactions on Information Theory.

[15]  Li Tang,et al.  Coded Caching with Low Subpacketization Levels , 2016, 2016 IEEE Globecom Workshops (GC Wkshps).

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

[17]  Sennur Ulukus,et al.  Asymmetry Hurts: Private Information Retrieval Under Asymmetric Traffic Constraints , 2018, IEEE Transactions on Information Theory.

[18]  Mikael Skoglund,et al.  Secure Private Information Retrieval from Colluding Databases with Eavesdroppers , 2017, 2018 IEEE International Symposium on Information Theory (ISIT).

[19]  Alexandre Graell i Amat,et al.  An MDS-PIR Capacity-Achieving Protocol for Distributed Storage Using Non-MDS Linear Codes , 2018, 2018 IEEE International Symposium on Information Theory (ISIT).

[20]  Sennur Ulukus,et al.  The Capacity of Private Information Retrieval from Byzantine and Colluding Databases , 2017, IEEE Transactions on Information Theory.

[21]  Ravi Tandon,et al.  The capacity of cache aided private information retrieval , 2017, 2017 55th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[22]  A. Robert Calderbank,et al.  An Improved Sub-Packetization Bound for Minimum Storage Regenerating Codes , 2013, IEEE Transactions on Information Theory.

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

[24]  Hirosuke Yamamoto,et al.  Private information retrieval for coded storage , 2014, 2015 IEEE International Symposium on Information Theory (ISIT).

[25]  Alexandre Graell i Amat,et al.  A Capacity-Achieving PIR Protocol for Distributed Storage Using an Arbitrary Linear Code , 2018, ArXiv.

[26]  Eyal Kushilevitz,et al.  Private information retrieval , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[27]  Chao Tian,et al.  A Shannon-Theoretic Approach to the Storage-Retrieval Tradeoff in PIR Systems , 2018, 2018 IEEE International Symposium on Information Theory (ISIT).

[28]  Alexandre Graell i Amat,et al.  Private information retrieval in distributed storage systems using an arbitrary linear code , 2016, 2017 IEEE International Symposium on Information Theory (ISIT).

[29]  Sergey Yekhanin,et al.  Towards 3-query locally decodable codes of subexponential length , 2008, JACM.

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

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

[32]  Chao Tian,et al.  Symmetry, Outer Bounds, and Code Constructions: A Computer-Aided Investigation on the Fundamental Limits of Caching , 2016, 2017 IEEE International Symposium on Information Theory (ISIT).

[33]  E. Kushilevitz,et al.  Barrier for Information-Theoretic Private Information Retrieval , 2002 .

[34]  Xiaohu Tang,et al.  On the Placement Delivery Array Design for Centralized Coded Caching Scheme , 2015, IEEE Transactions on Information Theory.

[35]  Hua Sun,et al.  The Capacity of Private Information Retrieval , 2016, 2016 IEEE Global Communications Conference (GLOBECOM).

[36]  Daniel J. Costello,et al.  Error Control Coding, Second Edition , 2004 .

[37]  Suhas N. Diggavi,et al.  Matched multiuser Gaussian source-channel communications via uncoded schemes , 2015, ISIT.

[38]  Yuval Ishai,et al.  On the Hardness of Information-Theoretic Multiparty Computation , 2004, EUROCRYPT.

[39]  Li Tang,et al.  Coded Caching Schemes With Reduced Subpacketization From Linear Block Codes , 2017, IEEE Transactions on Information Theory.

[40]  Alexander Barg,et al.  Explicit Constructions of Optimal-Access MDS Codes With Nearly Optimal Sub-Packetization , 2016, IEEE Transactions on Information Theory.

[41]  Chao Tian.  Characterizing the Rate Region of the (4,3,3) Exact-Repair Regenerating Codes , 2014, IEEE Journal on Selected Areas in Communications.

[42]  Eitan Yaakobi,et al.  Codes for distributed PIR with low storage overhead , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

[43]  Klim Efremenko,et al.  3-Query Locally Decodable Codes of Subexponential Length , 2008 .

[44]  Kenneth Zeger,et al.  Linear Network Coding Over Rings – Part II: Vector Codes and Non-Commutative Alphabets , 2016, IEEE Transactions on Information Theory.

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