New Locally Decodable Codes and New Locally Decodable Codes and Private Information Retrieval SchemesPrivate Information Retrieval Schemes

Sergey YekhaninSergey Yekhanin

LDCs and PIRsLDCs and PIRs

Definition: Definition: A code C encoding n bits to N bits is called q LDC if every bit of the message can be recovered (w.h.p.) by a randomized decoder reading only q bits of the encoding even after some constant fraction of the encoding has been corrupted.

Definition: A q server PIR protocol is a protocol between a user and q non-communicating servers holding an n bit database D, that allows the user to retrieve any bit Di, while leaking no information about i to any server.

LDCs: progressLDCs: progress

qq Lower boundLower bound Upper boundUpper bound

11 Do not exist [KT]Do not exist [KT]

22 Exp(n) [KdW]Exp(n) [KdW] Exp(n) [Folklore]Exp(n) [Folklore]

33 ΩΩ(n(n3/23/2) [KT]) [KT]

ΩΩ(n(n22/log/log22 n) [KdW] n) [KdW]

ΩΩ(n(n22/log n) [W]/log n) [W]

• Exp(nExp(n1/21/2) [BIK]) [BIK]

• ExpExp((nn1/32,582,6571/32,582,657)) [Y][Y]

• ExpExp((nnO(1/log log n)O(1/log log n))) [Y][Y]

• ExpExp((nnO(1/logO(1/log1-1-εε log n) log n))) [Y][Y]

PIRs: progressPIRs: progress

qq Lower boundLower bound Upper boundUpper bound

11 ΘΘ(n) [CGKS](n) [CGKS]

22 5 log n [WdW]5 log n [WdW] O(nO(n1/31/3) [CGKS]) [CGKS]

33 • O(nO(n1/31/3) [CGKS]) [CGKS]

• O(nO(n1/51/5) [A]) [A]

• O(nO(n1/5.251/5.25) [BIKR]) [BIKR]

• OO((nn1/32,582,6581/32,582,658)) [Y][Y]

• OO((nnO(1/log log n)O(1/log log n))) [Y][Y]

• OO((nnO(1/logO(1/log1-1-εε log n) log n))) [Y][Y]

Proof overviewProof overview

Goal: 3 query LDCs of length Exp(nGoal: 3 query LDCs of length Exp(n1/31/3))

• Regular Intersecting Families (RIFs)Regular Intersecting Families (RIFs)

• RIFs yield LDCsRIFs yield LDCs

• Basic linear-algebraic construction of RIFsBasic linear-algebraic construction of RIFs

• Combinatorial and algebraic niceness of setsCombinatorial and algebraic niceness of sets

• Main construction of RIFsMain construction of RIFs