Multi query computationally private information retrieval with constant communication rate
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Multi-Query Computationally-Private Information Retrieval with Constant Communication Rate. Jens Groth, University College London Aggelos Kiayias, University of Athens Helger Lipmaa, Cybernetica AS and Tallinn University. TexPoint fonts used in EMF.

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Multi query computationally private information retrieval with constant communication rate

Multi-Query Computationally-PrivateInformation Retrieval with ConstantCommunication Rate

Jens Groth, University College London

Aggelos Kiayias, University of Athens

Helger Lipmaa, Cybernetica AS and Tallinn University

TexPoint fonts used in EMF.

Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA


Multi query computationally private information retrieval with constant communication rate

Information retrieval

Client Server

xi

i x1,...,xn


Multi query computationally private information retrieval with constant communication rate

Privacy

Index i ?

Client Server

i


Multi query computationally private information retrieval with constant communication rate

Example of a trivial PIR protocol

Perfectly private:Client reveals nothing

x1,...,xn

xi

i x1,...,xn

Communication: nℓ bits with ℓ-bit records


Communication
Communication

bits

nℓ Trivial protocol

O(nk1/-1ℓ) Kushilevitz-Ostrovsky 97

O(kℓ) Cachin-Micali-Stadler 99

O(k log2n+ℓlog n) Lipmaa 05

O(k+ℓ) Gentry-Ramzan 05

Database size: nrecords Record size: ℓ bitsSecurity parameter: k bits (size of RSA modulus)


Multi query computationally private information retrieval with constant communication rate

Multi-query information retrieval

Client Server

xi1,...,xim

i1,...,im x1,...,xn


Multi query computationally private information retrieval with constant communication rate

Privacy

i1,...,im?

Client Server

i1,...,im


Our contribution
Our contribution

  • Lower bound (information theoretic):(mℓ+m log(n/m)) bits

  • Upper bound (CPIR protocol): O(mℓ+m log(n/m)+k) bits


Lower bound m m log n m bits
Lower bound (mℓ+m log(n/m)) bits

Client Server

xi1,...,xim

i1,...,im x1,...,xn

Client and server have unlimited computational power We do not require protocol to be private

We assume perfect correctnessWe assume worst case indices and records


Lower bound for 2 move cpir
Lower bound for 2-move CPIR

Client Server

xi1,...,xim

i1,...,im x1,...,xn

Query: possible indices (m log(n/m))

Response: m records (mℓ)


Lower bound for many move cpir
Lower bound for many-move CPIR

Client Server

xi1,...,xim

i1,...,im x1,...,xn

Proof overview:At loss of factor 2 assume 1-bit messages exhangedView function as tree with client at leaf choosing an outputWe will prove the tree has at least (leaf, output) pairs


Multi query computationally private information retrieval with constant communication rate

Input to the tree-function: I=(i1,...,im) and X=(x1,...,xn)

C(i1,...,im)

0 1

S(x1,...,xn,0) S(x1,...,xn,1)

0 1 0 1

C(i1,...,im,0,0) C(i1,...,im,0,1)C(i1,...,im,1,0) C(i1,...,im,1,1)

xi1,...,xim

Observation: If (I,X) and (I´,X´) lead to same leaf and output, then also (I,X´) lead to this leaf and output


Multi query computationally private information retrieval with constant communication rate

Define F = { (I,X)=(i1,...,im,x1,...,xn) | xi=1ℓ if iI and else xi=0ℓ}

If (I,X) F and (I´,X´)  F then (I,X´)  F

This means each (I,X) F leads to different (leaf,output) pair

For each (I,X) F the output is 1ℓ,...,1ℓ

There are pairs in F, so the tree must have leaves

This means the height is at least log ≥ m log(n/m)

So the client and server risk sending ½m log(n/m) bits

For the general case we then get a lower bound of max(mℓ, ½m log(n/m)) = (mℓ+m log(n/m)) bits


Four cases
Four cases

Trivial PIR (nℓ bits)

2

4

1

ℓ=log(n/m)

3

m=k2/3

m=n/9


Tool restricted cpir protocol
Tool: Restricted CPIR protocol

  • Perfect correctness

  • Constant >0 (e.g. =1/25) so CPIR with k bits of communication for parameters satisfying

  • m = poly(k), n = poly(k), ℓ = poly(k)

mℓ+m log n  k


Example gentry ramzan cpir
Example: Gentry-Ramzan CPIR

Primes: p1,…,pn |pi| = O(log n)

Prime powers: 1,…,n |i| > ℓ

  • Query: N, g i1…im | ord(g)

  • Response: c = gx mod N x = xi mod i

  • Extract: (cord(g)/i1…im) = (gord(g)/i1…im)x

    compute x mod i1…im extract xi1,…,xim


Three remaining cases
Three remaining cases

Restricted CPIR mℓ+m log n  k

ℓm/k CPIRs with record size k/m in parallel

2

4

ℓ=log(n/m)

3

m=k2/3

m=n/9


Two remaining cases
Two remaining cases

mℓ/log(n/m)-out of-n CPIR with record sizelog(n/m)

4

ℓ=log(n/m)

3

m=k2/3

m=n/9


One remaining case
One remaining case

Restricted CPIR mℓ+m log n  k

ℓ=log(n/m)

3

m=k2/3

m=n/9


Parallel extraction
Parallel extraction

Res-CPIR Res-CPIR Res-CPIR Res-CPIR


The problem
The problem

  • If ℓ = (log n) we could use parallel repetition of the restricted CPIR for mℓ+m log n  k on blocks of the database to get a constant rate

  • But if ℓ is small and m is large, we may loose a multiplicative factor (mℓ+m log n)/(mℓ+m log(n/m)) = 1+log m/(ℓ+log(n/m)) by parallel repetition of the restricted CPIR


Solution
Solution

aℓ-bit records

x1,x2,x3

(x1,x2)(x1,x3)(x2,x3)

x4,x5,x6

(x4,x5)(x4,x6)(x5,x6)

x7,x8,x9

(x7,x8)(x7,x9)(x8,x9)

ℓ’=aℓ, m’=m/a, n’= n/a

Restricted CPIR mℓ+m log n  k


Summary
Summary

Client Server

  • Lower bound: (mℓ+m log(n/m)) bits

  • CPIR protocol: O(mℓ+m log(n/m)+k) bits

xi1,...,xim

i1,...,im x1,...,xn