Yekhanin Locally Decodable Codes and Private Information Retrieval Schemes

1. Auflage 2010
ISBN: 978-3-642-14358-8
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 82 Seiten

Locally decodable codes (LDCs) are codes that simultaneously provide efficient random access retrieval and high noise resilience by allowing reliable reconstruction of an arbitrary bit of a message by looking at only a small number of randomly chosen codeword bits. Local decodability comes with a certain loss in terms of efficiency – specifically, locally decodable codes require longer codeword lengths than their classical counterparts. Private information retrieval (PIR) schemes are cryptographic protocols designed to safeguard the privacy of database users. They allow clients to retrieve records from public databases while completely hiding the identity of the retrieved records from database owners. In this book the author provides a fresh algebraic look at the theory of locally decodable codes and private information retrieval schemes, obtaining new families of each which have much better parameters than those of previously known constructions, and he also proves limitations of two server PIRs in a restricted setting that covers all currently known schemes. The author's related thesis won the ACM Dissertation Award in 2007, and this book includes some expanded sections and proofs, and notes on recent developments.

Autoren/Hrsg.

Yekhanin, Sergey

Inhaltsverzeichnis

1;Preface;6
2;Acknowledgments;8
3;Contents;9
4;1 Introduction;11
4.1;1.1 Locally decodable codes;11
4.1.1;1.1.1 Hadamard code;12
4.1.2;1.1.2 A code based on polynomial interpolation;13
4.2;1.2 Private information retrieval schemes;14
4.2.1;1.2.1 A PIR scheme based on polynomial interpolation;15
4.3;1.3 The history of LDCs and PIR schemes;16
4.3.1;1.3.1 The first generation: interpolation;17
4.3.2;1.3.2 The second generation: recursion;18
4.3.3;1.3.3 The third generation: point removal;19
4.3.4;1.3.4 Lower bounds;22
4.4;1.4 Applications of LDCs and PIR schemes;23
4.4.1;1.4.1 Secure multiparty computation;23
4.4.2;1.4.2 Other models of private information retrieval;24
4.4.3;1.4.3 Average-case complexity;26
4.5;1.5 Organization of the book;26
4.6;1.6 Addendum;27
5;2 Locally decodable codes via the point removal method;28
5.1;2.1 Notation;28
5.2;2.2 Locally decodable codes;29
5.3;2.3 Binary LDCs via point removal;29
5.3.1;2.3.1 Regular intersecting families of sets;30
5.3.2;2.3.2 Basic construction;31
5.3.3;2.3.3 The main construction: point removal;33
5.4;2.4 General LDCs via point removal;35
5.5;2.5 Combinatorially nice subsets of Fp*;39
5.6;2.6 Algebraically nice subsets of Fp*;41
5.6.1;2.6.1 3-dependences between p-th roots: sufficient conditions;43
5.6.2;2.6.2 k-dependences between p-th roots: a sufficient condition;44
5.6.3;2.6.3 Summary;48
5.7;2.7 Results;48
5.7.1;2.7.1 Results for three-query binary codes;49
5.7.2;2.7.2 Results for general codes;50
5.8;2.8 Addendum;51
5.8.1;2.8.1 The code;53
6;3 Limitations of the point removal method;55
6.1;3.1 Attaining subexponential length requires a nice sequence;55
6.1.1;3.1.1 Point removal method;55
6.1.2;3.1.2 Point removal and bounds for P(rt-1);56
6.1.3;3.1.3 Our results;56
6.2;3.2 A nice sequence yields short dependences between p-th roots;57
6.2.1;3.2.1 Algebraically nice subsets of Fq*;58
6.2.2;3.2.2 Combinatorially nice subsets of Fq*;61
6.2.3;3.2.3 Summary;63
6.3;3.3 k-dependences between p-th roots: a necessary condition;64
6.4;3.4 3-dependences between p-th roots: a necessary condition;65
6.5;3.5 Summary;66
6.6;3.6 Conclusions;67
6.7;3.7 Addendum;67
7;4 Private information retrieval;68
7.1;4.1 Preliminaries;68
7.2;4.2 From LDCs to PIR schemes;69
7.2.1;4.2.1 Upper bounds for three-server binary PIR schemes;71
7.2.2;4.2.2 Upper bounds for general PIR schemes;72
7.3;4.3 A combinatorial view of two-server PIR;73
7.3.1;4.3.1 Bilinear PIR;76
7.3.2;4.3.2 Group-based PIR;76
7.4;4.4 Complexity of bilinear group-based PIR;77
7.4.1;4.4.1 Algebraic preliminaries;77
7.4.2;4.4.2 Algebraic formulation;78
7.4.3;4.4.3 Low-dimensional principal ideals in group algebras;79
7.5;4.5 Summary of lower bounds for two-server PIR;80
7.6;4.6 Addendum;81
8;References;82
9;Index;87