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Multi-Query Computationally-Private Information Retrieval with Constant Communication Rate

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### 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

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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

bits

nℓ Trivial protocol

O(nk1/-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)

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 (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

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

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

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

Define F = { (I,X)=(i1,...,im,x1,...,xn) | xi=1ℓ if iI 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

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

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

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

Parallel extraction

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

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

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

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

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