Private Information Retrieval Over Gaussian MAC Ori Shmuel Joint - - PowerPoint PPT Presentation

Private Information Retrieval PIR for Gaussian Multiple Access Channels

Private Information Retrieval Over Gaussian MAC

Ori Shmuel Joint Work with Asaf Cohen

Ben Gurion University of the Negev shmuelor@bgu.ac.il IEEE International Symposium on Information Theory

June 21, 2020

1 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

PIR - Problem Description

• N non-communicating and

non-colluding databases (servers).

• N noiseless orthogonal channels.
• All servers are identical with M

messages of length L.

• The user wants Wi privately.
• What is the most efficient way to

retrieve Wi?

• Non-efficient way: Download

all messages.

• First introduced by Chor et al.1

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User

2 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

PIR - Problem Description

• N non-communicating and

non-colluding databases (servers).

• N noiseless orthogonal channels.
• All servers are identical with M

messages of length L.

• The user wants Wi privately.
• What is the most efficient way to

retrieve Wi?

• Non-efficient way: Download

all messages.

• First introduced by Chor et al.1

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User

2 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

PIR - Problem Description

• N non-communicating and

non-colluding databases (servers).

• N noiseless orthogonal channels.
• All servers are identical with M

messages of length L.

• The user wants Wi privately.
• What is the most efficient way to

retrieve Wi?

• Non-efficient way: Download

all messages.

• First introduced by Chor et al.1

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User

2 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

PIR - Problem Description

• N non-communicating and

non-colluding databases (servers).

• N noiseless orthogonal channels.
• All servers are identical with M

messages of length L.

• The user wants Wi privately.
• What is the most efficient way to

retrieve Wi?

• Non-efficient way: Download

all messages.

• First introduced by Chor et al.1

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User

2 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

PIR - Problem Description

• N non-communicating and

non-colluding databases (servers).

• N noiseless orthogonal channels.
• All servers are identical with M

messages of length L.

• The user wants Wi privately.
• What is the most efficient way to

retrieve Wi?

• Non-efficient way: Download

all messages.

• First introduced by Chor et al.1

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User

• 1B. Chor, O. Goldreich, E. Kushilevitz, and M. Sudan, ”Private information

retrieval,” in Proceedings of IEEE 36th Annual Foundations of Computer Science. IEEE, 1995, pp. 41-50.

2 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

PIR - General Course of Action

• Queries: Q1(i), Q2(i), ..., QN(i)
• Answers: A1(i), A2(i), ..., AN(i)
• Example: N=2

− User: generates a uniform binary vector b ∈ FM

2 .

− Q1(i) = b − Q2(i) = b ⊕ ei A1(i) =

M

• m=1

bmWm mod 2, A2(i) =

M

• m=1

(bm + δ{m=i})Wm mod 2,

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User 𝑅"(𝑗) 𝑅#(𝑗) 𝑅*(𝑗)

3 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

PIR - General Course of Action

• Queries: Q1(i), Q2(i), ..., QN(i)
• Answers: A1(i), A2(i), ..., AN(i)
• Example: N=2

− User: generates a uniform binary vector b ∈ FM

2 .

− Q1(i) = b − Q2(i) = b ⊕ ei A1(i) =

M

• m=1

bmWm mod 2, A2(i) =

M

• m=1

(bm + δ{m=i})Wm mod 2,

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User 𝐵"(𝑗) 𝐵#(𝑗) 𝐵*(𝑗)

3 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

PIR - General Course of Action

• Queries: Q1(i), Q2(i), ..., QN(i)
• Answers: A1(i), A2(i), ..., AN(i)
• Example: N=2

− User: generates a uniform binary vector b ∈ FM

2 .

− Q1(i) = b − Q2(i) = b ⊕ ei A1(i) =

M

• m=1

bmWm mod 2, A2(i) =

M

• m=1

(bm + δ{m=i})Wm mod 2,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

3 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

PIR - General Course of Action

• Queries: Q1(i), Q2(i), ..., QN(i)
• Answers: A1(i), A2(i), ..., AN(i)
• Example: N=2

− User: generates a uniform binary vector b ∈ FM

2 .

− Q1(i) = b − Q2(i) = b ⊕ ei A1(i) =

M

• m=1

bmWm mod 2, A2(i) =

M

• m=1

(bm + δ{m=i})Wm mod 2,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

A1(i) + A2(i) = Wi

3 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• We define the PIR rate as,

R L D,

• D is the total bits downloaded.
• In the example:

R L 2L = 1 2

W

"

W# W$

…

W

"

W# W$

…

1 2

User

A1(i) + A2(i) = Wi

4 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• We define the PIR rate as,

R L D,

• D is the total bits downloaded.
• In the example:

R L 2L = 1 2

W

"

W# W$

…

W

"

W# W$

…

1 2

User

A1(i) + A2(i) = Wi

4 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• We define the PIR rate as,

R L D,

• D is the total bits downloaded.
• In the example:

R L 2L = 1 2

W

"

W# W$

…

W

"

W# W$

…

1 2

User

A1(i) + A2(i) = Wi

4 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• We define the PIR rate as,

R L D,

• D is the total bits downloaded.
• In the example:

R L 2L = 1 2

• Can we do better than that?

W

"

W# W$

…

W

"

W# W$

…

1 2

User

A1(i) + A2(i) = Wi

4 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• The PIR capacity is1,

CP IR =

• 1 − 1

N

• 1 −

1

N

M

• L = N M
• Example: N = 2, M = 2
• → L = 4, CP IR = 2

3.

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User

• 1H. Sun and S. A. Jafar, ”The capacity of private information retrieval,” IEEE

Transactions on Information Theory, vol. 63, no. 7, pp. 4075-4088, 2017

5 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• The PIR capacity is1,

CP IR =

• 1 − 1

N

• 1 −

1

N

M

• L = N M
• Example: N = 2, M = 2
• → L = 4, CP IR = 2

3.

5 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• The PIR capacity is1,

CP IR =

• 1 − 1

N

• 1 −

1

N

M

• L = N M
• Example: N = 2, M = 2
• → L = 4, CP IR = 2

3.

• The user generates a random

private permutations of L = 4 indices. − W1 : [a1, a2, a3, a4] − W2 : [b1, b2, b3, b4]

• Assume the desired message is W1

5 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• The PIR capacity is1,

CP IR =

• 1 − 1

N

• 1 −

1

N

M

• L = N M
• Example: N = 2, M = 2
• → L = 4, CP IR = 2

3.

• The user generates a random

private permutations of L = 4 indices. − W1 : [a1, a2, a3, a4] − W2 : [b1, b2, b3, b4]

• Assume the desired message is W1

Server 1 Server 2

5 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• The PIR capacity is1,

CP IR =

• 1 − 1

N

• 1 −

1

N

M

• L = N M
• Example: N = 2, M = 2
• → L = 4, CP IR = 2

3.

• The user generates a random

private permutations of L = 4 indices. − W1 : [a1, a2, a3, a4] − W2 : [b1, b2, b3, b4]

• Assume the desired message is W1

Server 1 Server 2 𝑏"

5 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• The PIR capacity is1,

CP IR =

• 1 − 1

N

• 1 −

1

N

M

• L = N M
• Example: N = 2, M = 2
• → L = 4, CP IR = 2

3.

• Symmetry across servers
• Symmetry in queries
• Exploiting side information
• The user generates a random

private permutations of L = 4 indices. − W1 : [a1, a2, a3, a4] − W2 : [b1, b2, b3, b4]

• Assume the desired message is W1

Server 1 Server 2 𝑏" 𝑏#

5 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• The PIR capacity is1,

CP IR =

• 1 − 1

N

• 1 −

1

N

M

• L = N M
• Example: N = 2, M = 2
• → L = 4, CP IR = 2

3.

• Symmetry across servers
• Symmetry in queries
• Exploiting side information
• The user generates a random

private permutations of L = 4 indices. − W1 : [a1, a2, a3, a4] − W2 : [b1, b2, b3, b4]

• Assume the desired message is W1

Server 1 Server 2 𝑏" 𝑐" 𝑏$

5 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• The PIR capacity is1,

CP IR =

• 1 − 1

N

• 1 −

1

N

M

• L = N M
• Example: N = 2, M = 2
• → L = 4, CP IR = 2

3.

• Symmetry across servers
• Symmetry in queries
• Exploiting side information
• The user generates a random

private permutations of L = 4 indices. − W1 : [a1, a2, a3, a4] − W2 : [b1, b2, b3, b4]

• Assume the desired message is W1

Server 1 Server 2 𝑏" 𝑐" 𝑏$ 𝑐$

5 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• The PIR capacity is1,

CP IR =

• 1 − 1

N

• 1 −

1

N

M

• L = N M
• Example: N = 2, M = 2
• → L = 4, CP IR = 2

3.

• Symmetry across servers
• Symmetry in queries
• Exploiting side information
• The user generates a random

private permutations of L = 4 indices. − W1 : [a1, a2, a3, a4] − W2 : [b1, b2, b3, b4]

• Assume the desired message is W1

Server 1 Server 2 𝑏" 𝑐" 𝑏$ + 𝑐& 𝑏& 𝑐&

5 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• The PIR capacity is1,

CP IR =

• 1 − 1

N

• 1 −

1

N

M

• L = N M
• Example: N = 2, M = 2
• → L = 4, CP IR = 2

3.

• Symmetry across servers
• Symmetry in queries
• Exploiting side information
• The user generates a random

private permutations of L = 4 indices. − W1 : [a1, a2, a3, a4] − W2 : [b1, b2, b3, b4]

• Assume the desired message is W1

Server 1 Server 2 𝑏" 𝑐" 𝑏$ + 𝑐& 𝑏& 𝑐& 𝑏' + 𝑐"

5 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Performance Metric and Known Results

• The PIR capacity is1,

CP IR =

• 1 − 1

N

• 1 −

1

N

M

• L = N M
• Example: N = 2, M = 2
• → L = 4, CP IR = 2

3.

R = 4 6 = 2 3

• The user generates a random

private permutations of L = 4 indices. − W1 : [a1, a2, a3, a4] − W2 : [b1, b2, b3, b4]

• Assume the desired message is W1

Server 1 Server 2 𝑏" 𝑐" 𝑏$ + 𝑐& 𝑏& 𝑐& 𝑏' + 𝑐"

5 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Related works

• PIR with colluding servers1.
• Byzantine servers which respond with erroneous answers2.
• Symmetric PIR - privacy on Wj, j = i from the user 3.
• Side information (user cache)4.
• PIR from coded servers5.

− Function of the coding rate Rcode, CP IR = (1 − Rcode)

• 1 − (Rcode)M
• 1H. Sun and S. A. Jafar, ”The capacity of robust private information retrieval

with colluding databases,” IEEE Transactions on Information Theory, vol. 64, no. 4, pp. 2361-2370, 2017.

6 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Related works

• PIR with colluding servers1.
• Byzantine servers which respond with erroneous answers2.
• Symmetric PIR - privacy on Wj, j = i from the user 3.
• Side information (user cache)4.
• PIR from coded servers5.

− Function of the coding rate Rcode, CP IR = (1 − Rcode)

• 1 − (Rcode)M
• 2K. Banawan and S. Ulukus, ”The capacity of private information retrieval from

byzantine and colluding databases,” IEEE Transactions on Information Theory, vol. 65,

• no. 2, pp. 1206-1219, 2018.

6 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Related works

• PIR with colluding servers1.
• Byzantine servers which respond with erroneous answers2.
• Symmetric PIR - privacy on Wj, j = i from the user 3.
• Side information (user cache)4.
• PIR from coded servers5.

− Function of the coding rate Rcode, CP IR = (1 − Rcode)

• 1 − (Rcode)M
• 3H. Sun and S. A. Jafar, ”The capacity of symmetric private information retrieval,”

IEEE Transactions on Information Theory, vol. 65, no. 1, pp. 322-329, 2018.

6 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Related works

• PIR with colluding servers1.
• Byzantine servers which respond with erroneous answers2.
• Symmetric PIR - privacy on Wj, j = i from the user 3.
• Side information (user cache)4.
• PIR from coded servers5.

− Function of the coding rate Rcode, CP IR = (1 − Rcode)

• 1 − (Rcode)M

4Y.-P. Wei, K. Banawan, and S. Ulukus, ”The capacity of private information

retrieval with partially known private side information,” IEEE Transactions on Information Theory, 2019.

6 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Related works

• PIR with colluding servers1.
• Byzantine servers which respond with erroneous answers2.
• Symmetric PIR - privacy on Wj, j = i from the user 3.
• Side information (user cache)4.
• PIR from coded servers5.

− Function of the coding rate Rcode, CP IR = (1 − Rcode)

• 1 − (Rcode)M
• 5T. H. Chan, S.-W. Ho, and H. Yamamoto, ”Private information retrieval for coded

storage,” in 2015 IEEE International Symposium on Information Theory (ISIT). IEEE, 2015, pp. 2842-2846.

• K. Banawan and S. Ulukus, ”The capacity of private information retrieval from coded

databases,” IEEE Transactions on Information Theory, vol. 64, no. 3, pp. 1945-1956, 2018.

6 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels General background

Related works

• PIR with colluding servers1.
• Byzantine servers which respond with erroneous answers2.
• Symmetric PIR - privacy on Wj, j = i from the user 3.
• Side information (user cache)4.
• PIR from coded servers5.

− Function of the coding rate Rcode, CP IR = (1 − Rcode)

• 1 − (Rcode)M
• Noisy orthogonal links and binary additive MAC6.

− Separation between the PIR and channel coding schemes.

• 6K. Banawan and S. Ulukus, ”Noisy private information retrieval: On separability
• f channel coding and information retrieval,” IEEE Transactions on Information

Theory, 2019.

6 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Channel Model

• N-server block fading AWGN MAC:

y(i) =

N

• k=1

hkxk(i) + z,

• Channel coefficients hk ∼ N(0, 1).
• z is an i.i.d., Gaussian noise, z ∼ N(0, In×n).
• Average power constraint per server on the

codewords, i.e., xk2 ≤ nP.

• First case: Gaussian MAC without fading - fix

hk = 1 for k = 1, ..., N.

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User ℎ" ℎ# ℎ' 𝑨

7 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• 2-server AWGN MAC:

y = x1 + x2 + z.

• Capacity achieving MAC code creates

”virtual” noiseless orthogonal links.

• Each server encodes his d private bits (Am(i))

into an n-length codeword.

• Achievable PIR rate using separation is,

R = L n = L

Nd RSR

= L DRSR ≤ CP IR · CSR,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

8 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• 2-server AWGN MAC:

y = x1 + x2 + z.

• Capacity achieving MAC code creates

”virtual” noiseless orthogonal links.

• Each server encodes his d private bits (Am(i))

into an n-length codeword.

• Achievable PIR rate using separation is,

R = L n = L

Nd RSR

= L DRSR ≤ CP IR · CSR,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

8 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• 2-server AWGN MAC:

y = x1 + x2 + z.

• Capacity achieving MAC code creates

”virtual” noiseless orthogonal links.

• Each server encodes his d private bits (Am(i))

into an n-length codeword.

• Achievable PIR rate using separation is,

R = L n = L

Nd RSR

= L DRSR ≤ CP IR · CSR,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

8 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• 2-server AWGN MAC:

y = x1 + x2 + z.

• Capacity achieving MAC code creates

”virtual” noiseless orthogonal links.

• Each server encodes his d private bits (Am(i))

into an n-length codeword.

• Achievable PIR rate using separation is,

R = L n = L

Nd RSR

= L DRSR ≤ CP IR · CSR,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

8 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• 2-server AWGN MAC:

y = x1 + x2 + z.

• Capacity achieving MAC code creates

”virtual” noiseless orthogonal links.

• Each server encodes his d private bits (Am(i))

into an n-length codeword.

• Achievable PIR rate using separation is,

R = L n = L

Nd RSR

= L DRSR ≤ CP IR · CSR,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

CSR = 1

2 log (1 + NP) is the sum-capacity of the AWGN MAC.

8 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• 2-server AWGN MAC:

y = x1 + x2 + z.

• Capacity achieving MAC code creates

”virtual” noiseless orthogonal links.

• Each server encodes his d private bits (Am(i))

into an n-length codeword.

• Achievable PIR rate using separation is,

R = L n = L

Nd RSR

= L DRSR ≤ CP IR · CSR,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

CSR = 1

2 log (1 + NP) is the sum-capacity of the AWGN MAC.

We neglect the additive nature of the channel.

8 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Compute and Forward

• Compute-and-Forward1 (CF) is a coding scheme which enables

receivers to decode linear combinations of transmitted messages.

• First introduced as a practical solution in relays networks.
• Useful in many other communication problems.

1Bobak Nazer and Michael Gastpar. ”Compute-and-forward: Harnessing

interference through structured codes.” IEEE Transactions on Information Theory 57.10 (2011): 6463-6486.

9 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Compute and Forward

• Compute-and-Forward1 (CF) is a coding scheme which enables

receivers to decode linear combinations of transmitted messages.

• First introduced as a practical solution in relays networks.
• Useful in many other communication problems.

1Bobak Nazer and Michael Gastpar. ”Compute-and-forward: Harnessing

interference through structured codes.” IEEE Transactions on Information Theory 57.10 (2011): 6463-6486.

9 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Compute and Forward

• Compute-and-Forward1 (CF) is a coding scheme which enables

receivers to decode linear combinations of transmitted messages.

• First introduced as a practical solution in relays networks.
• Useful in many other communication problems.

1Bobak Nazer and Michael Gastpar. ”Compute-and-forward: Harnessing

interference through structured codes.” IEEE Transactions on Information Theory 57.10 (2011): 6463-6486.

9 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Compute and Forward

• Compute-and-Forward1 (CF) is a coding scheme which enables

receivers to decode linear combinations of transmitted messages.

• First introduced as a practical solution in relays networks.
• Useful in many other communication problems.

9 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Compute and Forward

• Compute-and-Forward1 (CF) is a coding scheme which enables

receivers to decode linear combinations of transmitted messages.

• First introduced as a practical solution in relays networks.
• Useful in many other communication problems.

9 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Compute and Forward

• Compute-and-Forward1 (CF) is a coding scheme which enables

receivers to decode linear combinations of transmitted messages.

• First introduced as a practical solution in relays networks.
• Useful in many other communication problems.

αy =

3

• i=1

αhixi + αz =

3

• i=1

aixi

• required

linear combination

+

3

• i=1

(αhi − ai)xi

• ”quantization noise”

+ αz

• Gaussian

noise

• effective noise

.

9 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Compute and Forward

• Compute-and-Forward1 (CF) is a coding scheme which enables

receivers to decode linear combinations of transmitted messages.

• First introduced as a practical solution in relays networks.
• Useful in many other communication problems.

αy =

3

• i=1

αhixi + αz =

3

• i=1

aixi

• required

linear combination

+

3

• i=1

(αhi − ai)xi

• ”quantization noise”

+ αz

• Gaussian

noise

• effective noise

.

9 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Compute and Forward

• Compute-and-Forward1 (CF) is a coding scheme which enables

receivers to decode linear combinations of transmitted messages.

• First introduced as a practical solution in relays networks.
• Useful in many other communication problems.

Lattice codes

𝒚" 𝒚# 𝒛

Random codes

𝒚" 𝒚# 𝒛

• Every Linear combination of lattice points is a lattice point.
• The decoded linear combination is a legitimate codeword.

9 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Compute and Forward

• Compute-and-Forward1 (CF) is a coding scheme which enables

receivers to decode linear combinations of transmitted messages.

• First introduced as a practical solution in relays networks.
• Useful in many other communication problems.

Theorem ([Nazer-Gastpar’11]) The computation rate for decoding a linear combination with respect to a coefficient vector a ∈ Z and a channel vector h ∈ R is R(h, a) = 1 2 log+

• 1 + Ph2

a2 + P (a2h2 − (hT a)2)

• .
• This is the transmission rate or the rate for decoding the linear

combination.

9 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC - Achievable PIR Rate

• Using a lattice code and the

additive nature of the MAC we have,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

Theorem For the 2 servers AWGN MAC, the following PIR rate is achievable, RJ

P IR = 1

2 log+ 1 2 + P

• .
• Simple task of computation.
• Single-user decoding.

10 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC - Achievable PIR Rate

• Using a lattice code and the

additive nature of the MAC we have,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

Theorem For the 2 servers AWGN MAC, the following PIR rate is achievable, RJ

P IR = 1

2 log+ 1 2 + P

• .
• Simple task of computation.
• Single-user decoding.

10 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• The capacity region of the

Gaussian MAC R1 ≤ 1 2 log (1 + P) R2 ≤ 1 2 log (1 + P) R1 + R2 ≤ 1 2 log (1 + 2P)

• Every point inside is achievable.
• Servers transmit with equal rate.
• Represents the axis of the PIR

rate.

• With no privacy constraints.

11 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• The capacity region of the

Gaussian MAC R1 ≤ 1 2 log (1 + P) R2 ≤ 1 2 log (1 + P) R1 + R2 ≤ 1 2 log (1 + 2P)

• Every point inside is achievable.
• Servers transmit with equal rate.
• Represents the axis of the PIR

rate.

• With no privacy constraints.

11 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• The capacity region of the

Gaussian MAC R1 ≤ 1 2 log (1 + P) R2 ≤ 1 2 log (1 + P) R1 + R2 ≤ 1 2 log (1 + 2P)

• Every point inside is achievable.
• Servers transmit with equal rate.
• Represents the axis of the PIR

rate.

• With no privacy constraints.

11 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• The capacity region of the

Gaussian MAC R1 ≤ 1 2 log (1 + P) R2 ≤ 1 2 log (1 + P) R1 + R2 ≤ 1 2 log (1 + 2P)

• Every point inside is achievable.
• Servers transmit with equal rate.
• Represents the axis of the PIR

rate.

• With no privacy constraints.

11 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• Privacy with separation.
• Privacy using lattice codes.

Rs

P IR ≤ CP IR·CSR =

• 1 − 1

N

• 1 −

1

N

M·1 2 log (1 + NP) {N=2

M=2}

≤ 1 3 log (1 + 2P).

11 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

• Privacy with separation.
• Privacy using lattice codes.

Rs

P IR ≤ CP IR·CSR =

• 1 − 1

N

• 1 −

1

N

M·1 2 log (1 + NP) {N=2

M=2}

≤ 1 3 log (1 + 2P).

11 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC - Scheme Details

• Recall the first example:

− User: generates a random binary vector b ∈ FM

2 .

− Q1(i) = b − Q2(i) = b + ei mod 2 A1(i) =

M

• m=1

bmWm mod 2, A2(i) =

M

• m=1

(bm + δ{m=i})Wm mod 2,

• In our scheme each server encode his answer

using the lattice code.

W

"

W# W$

…

W

"

W# W$

…

1 2

User

12 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC - Scheme Details

• Recall the first example:

− User: generates a random binary vector b ∈ FM

2 .

− Q1(i) = b − Q2(i) = b + ei mod 2 A1(i) =

M

• m=1

bmWm mod 2, A2(i) =

M

• m=1

(bm + δ{m=i})Wm mod 2,

• In our scheme each server encode his answer

using the lattice code.

W

"

W# W$

…

W

"

W# W$

…

1 2

User

12 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC- Scheme Details

• The received input at the user is thus,

y = x1 + x2 + z = x∗ + z,

• x∗ is a legitimate codeword.
• Each server knows only his codeword → x∗ is

private.

• The linear lattice code preserves the

associativity.

• Wi can be recovered reliably from x∗ if,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

13 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC- Scheme Details

• The received input at the user is thus,

y = x1 + x2 + z = x∗ + z,

• x∗ is a legitimate codeword.
• Each server knows only his codeword → x∗ is

private.

• The linear lattice code preserves the

associativity.

• Wi can be recovered reliably from x∗ if,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

13 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC- Scheme Details

• The received input at the user is thus,

y = x1 + x2 + z = x∗ + z,

• x∗ is a legitimate codeword.
• Each server knows only his codeword → x∗ is

private.

• The linear lattice code preserves the

associativity.

• Wi can be recovered reliably from x∗ if,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

13 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC- Scheme Details

• The received input at the user is thus,

y = x1 + x2 + z = x∗ + z,

• x∗ is a legitimate codeword.
• Each server knows only his codeword → x∗ is

private.

• The linear lattice code preserves the

associativity.

• Wi can be recovered reliably from x∗ if,

W

"

W# W$

…

W

"

W# W$

…

1 2

User

R = 1 2 log+

• 1 + Ph2

a2 + P (a2h2 − (hT a)2)

• = 1

2 log+ 1 2 + P

• .

13 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

Theorem For the N servers AWGN MAC the following PIR rate is achievable, RJ

P IR = 1

2 log

• 1

2 + N 2 2 P

• .
• With N servers we have a power gain of

N

2

2.

• As N grows, the joint scheme is twice as good as separation.

lim

N→∞

CJ

P IR

Cs

P IR

≥ 2.

• Finite gap from the capacity without privacy constraints.

14 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

Theorem For the N servers AWGN MAC the following PIR rate is achievable, RJ

P IR = 1

2 log

• 1

2 + N 2 2 P

• .
• With N servers we have a power gain of

N

2

2.

• As N grows, the joint scheme is twice as good as separation.

lim

N→∞

CJ

P IR

Cs

P IR

≥ 2.

• Finite gap from the capacity without privacy constraints.

14 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

Theorem For the N servers AWGN MAC the following PIR rate is achievable, RJ

P IR = 1

2 log

• 1

2 + N 2 2 P

• .
• With N servers we have a power gain of

N

2

2.

• As N grows, the joint scheme is twice as good as separation.

lim

N→∞

CJ

P IR

Cs

P IR

≥ 2.

• Finite gap from the capacity without privacy constraints.

14 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

Theorem For the N servers AWGN MAC the following PIR rate is achievable, RJ

P IR = 1

2 log

• 1

2 + N 2 2 P

• .
• With N servers we have a power gain of

N

2

2.

• As N grows, the joint scheme is twice as good as separation.

lim

N→∞

CJ

P IR

Cs

P IR

≥ 2.

• Finite gap from the capacity without privacy constraints.

CMISO

SR

= 1 2 log

• 1 + N 2P
• .

14 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC

Theorem For the N servers AWGN MAC the following PIR rate is achievable, RJ

P IR = 1

2 log

• 1

2 + N 2 2 P

• .
• With N servers we have a power gain of

N

2

2.

• As N grows, the joint scheme is twice as good as separation.

lim

N→∞

CJ

P IR

Cs

P IR

≥ 2.

• Finite gap from the capacity without privacy constraints.

Lemma CMISO

SR

− RJ

P IR ≤ 2.

14 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC N servers

Lattice Based Schemes Separeation Based Schemes N=8 N=4 N=2

• 5

5 10 15 20 SNR [dB] 1 2 3 4 5 6 R

The PIR rate as a function of the SNR for N = {2, 4, 8}. The dashed lines represent the achievable PIR rate when the separation scheme is

• applied. The solid lines are the achievable PIR rate of the lattice

based scheme.

15 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

PIR for AWGN MAC N servers

P=1[dB] P=5[dB] P=10[dB] Separeation Based Schemes Lattice Based Schemes 20 40 60 80 100 N 1 2 3 4 5 6 7 R

The PIR rate as a function of N for P = {1dB, 5dB, 8dB}. The dashed lines represent the achievable PIR rate when the separation scheme is applied. The solid lines are the achievable PIR rate of the lattice based scheme.

16 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Block Fading AWGN MAC

• With fading we have asymmetry between the

channels. y(i) =

N

• k=1

hkxk(i) + z,

• In general, asymmetry hurts1.
• Fading - CSI known/unknown.

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User ℎ" ℎ# ℎ' 𝑨

17 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Block Fading AWGN MAC

• With fading we have asymmetry between the

channels. y(i) =

N

• k=1

hkxk(i) + z,

• In general, asymmetry hurts1.
• Fading - CSI known/unknown.

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User ℎ" ℎ# ℎ' 𝑨

• 1K. Banawan and S. Ulukus, ”Asymmetry hurts: Private information retrieval

under asymmetric traffic constraints,” IEEE Transactions on Information Theory,

• vol. 65, no. 11, pp. 7628?7645, 2019.

17 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Block Fading AWGN MAC

• With fading we have asymmetry between the

channels. y(i) =

N

• k=1

hkxk(i) + z,

• In general, asymmetry hurts1.
• Fading - CSI known/unknown.

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User ℎ" ℎ# ℎ' 𝑨

17 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

Block Fading AWGN MAC

• With fading we have asymmetry between the

channels. y(i) =

N

• k=1

hkxk(i) + z,

• In general, asymmetry hurts1.
• Fading - CSI known/unknown.

W

"

W# W$

…

W

"

W# W$

…

W

"

W# W$

… …

1 2 𝑂

User ℎ" ℎ# ℎ' 𝑨

Results for fading channels

• We suggest a joint scheme which is based on CF.
• The scheme depends on a complex optimization problem.
• We provide sub-optimal solutions which achieve the scaling laws
• f the channel capacity with either N or P.

17 / 18

Private Information Retrieval PIR for Gaussian Multiple Access Channels Channel Model Achievable PIR Rate - Separation Compute and Forward Achievable PIR Rate - Joint

The End

18 / 18