TY - JOUR
T1 - On locally decodable codes, self-correctable codes, and t-private PIR
AU - Barkol, Omer
AU - Ishai, Yuval
AU - Weinreb, Enav
N1 - Funding Information:
Research supported by grant 1310/06 from the Israel Science Foundation, grant 2004361 from the US–Israel Binational Science Foundation, and the Technion VPR fund. Part of this research was done while visiting IPAM.
PY - 2010/12
Y1 - 2010/12
N2 - A k-query locally decodable code (LDC) allows to probabilistically decode any bit of an encoded message by probing only k bits of its corrupted encoding. A stronger and desirable property is that of self-correction, allowing to efficiently recover not only bits of the message but also arbitrary bits of its encoding. In contrast to the initial constructions of LDCs, the recent and most efficient constructions are not known to be self-correctable. The existence of self-correctable codes of comparable efficiency remains open. A closely related problem with a very different motivation is that of private information retrieval (PIR). A k-server PIR protocol allows a user to retrieve the i-th bit of a database, which is replicated among k servers, without revealing information about i to any individual server. A natural generalization is t-private PIR, which keeps i hidden from any t colluding servers. In contrast to the initial PIR protocols, it is not known how to generalize the recent and most efficient protocols to yield t-private protocols of comparable efficiency. In this work we study both of the above questions, showing that they are in fact related. We start by presenting a general transformation of any 1-private PIR protocol (equivalently, LDC) into a t-private protocol with a similar amount of communication per server. Combined with the recent result of Yekhanin (STOC 2007), this yields an improvement over previous t-private PIR protocols. A major weakness of our transformation is that the number of servers grows exponentially with t. We show that if the underlying LDC satisfies the stronger self-correction property, then there is a similar transformation in which the number of servers grows only linearly with t, which is the best one can hope for. Finally, we explore the possibility of improving current constructions of self-correctable codes and relate this question to a conjecture of Hamada concerning the algebraic rank of combinatorial designs.
AB - A k-query locally decodable code (LDC) allows to probabilistically decode any bit of an encoded message by probing only k bits of its corrupted encoding. A stronger and desirable property is that of self-correction, allowing to efficiently recover not only bits of the message but also arbitrary bits of its encoding. In contrast to the initial constructions of LDCs, the recent and most efficient constructions are not known to be self-correctable. The existence of self-correctable codes of comparable efficiency remains open. A closely related problem with a very different motivation is that of private information retrieval (PIR). A k-server PIR protocol allows a user to retrieve the i-th bit of a database, which is replicated among k servers, without revealing information about i to any individual server. A natural generalization is t-private PIR, which keeps i hidden from any t colluding servers. In contrast to the initial PIR protocols, it is not known how to generalize the recent and most efficient protocols to yield t-private protocols of comparable efficiency. In this work we study both of the above questions, showing that they are in fact related. We start by presenting a general transformation of any 1-private PIR protocol (equivalently, LDC) into a t-private protocol with a similar amount of communication per server. Combined with the recent result of Yekhanin (STOC 2007), this yields an improvement over previous t-private PIR protocols. A major weakness of our transformation is that the number of servers grows exponentially with t. We show that if the underlying LDC satisfies the stronger self-correction property, then there is a similar transformation in which the number of servers grows only linearly with t, which is the best one can hope for. Finally, we explore the possibility of improving current constructions of self-correctable codes and relate this question to a conjecture of Hamada concerning the algebraic rank of combinatorial designs.
KW - Combinatorial designs
KW - Hamada's conjecture
KW - Locally decodable codes
KW - Private information retrieval
KW - Reed muller codes
KW - Self correctable codes
KW - T-privacy
UR - http://www.scopus.com/inward/record.url?scp=78149358428&partnerID=8YFLogxK
U2 - 10.1007/s00453-008-9272-1
DO - 10.1007/s00453-008-9272-1
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AN - SCOPUS:78149358428
SN - 0178-4617
VL - 58
SP - 831
EP - 859
JO - Algorithmica (New York)
JF - Algorithmica (New York)
IS - 4
ER -