Capacity Region of the Gaussian Arbitrarily-Varying Broadcast - - PowerPoint PPT Presentation

Capacity Region of the Gaussian Arbitrarily-Varying Broadcast Channel

Fatemeh Hosseinigoki and Oliver Kosut

ARIZONA STATE UNIVERSITY ISIT 2020

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Gaussian Arbitrarily-Varying Broadcast Channel

Transmitter power constraint X2 ≤ nP

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Gaussian Arbitrarily-Varying Broadcast Channel

Transmitter power constraint X2 ≤ nP Jammer power constraints Sj2 ≤ nΛj, j = 1, 2

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Gaussian Arbitrarily-Varying Broadcast Channel

Transmitter power constraint X2 ≤ nP Jammer power constraints Sj2 ≤ nΛj, j = 1, 2 We want to decode messages no matter what the jammers do

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Gaussian Arbitrarily-Varying Broadcast Channel

Transmitter power constraint X2 ≤ nP Jammer power constraints Sj2 ≤ nΛj, j = 1, 2 We want to decode messages no matter what the jammers do

Assumptions: Oblivious adversary, but knows the code

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Gaussian Arbitrarily-Varying Channel

Transmitter power constraint X2 ≤ nP Jammer power constraint S2 ≤ nΛ

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Gaussian Arbitrarily-Varying Channel

Transmitter power constraint X2 ≤ nP Jammer power constraint S2 ≤ nΛ [Csisz´ ar-Narayan 1991]: Capacity is C =    C

• P

Λ + N

• ,

Λ < P 0, Λ ≥ P where C(x) = 1 2 log(1 + x)

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Gaussian Arbitrarily-Varying Channel

Transmitter power constraint X2 ≤ nP Jammer power constraint S2 ≤ nΛ [Csisz´ ar-Narayan 1991]: Capacity is C =    C

• P

Λ + N

• ,

Λ < P 0, Λ ≥ P where C(x) = 1 2 log(1 + x) As long as jammer’s power is less than legitimate transmitter’s, then the worst the jammer can do is send noise

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Symmetrizability

If Λ ≥ P, then the jammer can choose a false message m′ and transmit S = X(m′)

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Symmetrizability

If Λ ≥ P, then the jammer can choose a false message m′ and transmit S = X(m′) Receiver gets Y = X(m) + X(m′) + V Cannot tell which message is correct

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Gaussian AVC with Common Randomness

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Gaussian AVC with Common Randomness

[Hughes-Narayan 1982]: If transmitter/receiver have common randomness (unknown to adversary) of at least O(log n) bits, then C = C

• P

N + Λ

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Gaussian AVC with Common Randomness

[Hughes-Narayan 1982]: If transmitter/receiver have common randomness (unknown to adversary) of at least O(log n) bits, then C = C

• P

N + Λ

• Symmetrizability is not a problem because two possibilities can be

distinguished via the common randomness

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Gaussian AVBC Capacity Region

Let Cno-adv(S1, S2) be the capacity region for the no-adversary BC with SNRs S1, S2

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Gaussian AVBC Capacity Region

Let Cno-adv(S1, S2) be the capacity region for the no-adversary BC with SNRs S1, S2 i.e., if S1 ≥ S2, then Cno-adv(S1, S1) =   (R1, R2) : R1 ≤ C(αS1), R2 ≤ C

• ¯

αS2 αS2 + 1 for some α ∈ [0, 1]   

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Gaussian AVBC Capacity Region

Let Cno-adv(S1, S2) be the capacity region for the no-adversary BC with SNRs S1, S2 i.e., if S1 ≥ S2, then Cno-adv(S1, S1) =   (R1, R2) : R1 ≤ C(αS1), R2 ≤ C

• ¯

αS2 αS2 + 1 for some α ∈ [0, 1]   

Theorem

(R1, R2) ∈ CGAV BC if and only if (R1, R2) ∈ Cno-adv

• P

N1 + Λ1 , P N2 + Λ2

• If Λ1 ≥ P, then R1 = 0

If Λ2 ≥ P, then R2 = 0

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Gaussian AVBC Capacity Region

Let Cno-adv(S1, S2) be the capacity region for the no-adversary BC with SNRs S1, S2 i.e., if S1 ≥ S2, then Cno-adv(S1, S1) =   (R1, R2) : R1 ≤ C(αS1), R2 ≤ C

• ¯

αS2 αS2 + 1 for some α ∈ [0, 1]   

Theorem

(R1, R2) ∈ CGAV BC if and only if (R1, R2) ∈ Cno-adv

• P

N1 + Λ1 , P N2 + Λ2

• If Λ1 ≥ P, then R1 = 0

If Λ2 ≥ P, then R2 = 0 Converse is straightforward: (a) adversaries send Gaussian noise, (b) symmetrizability

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Gaussian AVBC with Common Randomness

Easy to show that, with common randomness, the capacity region is Cno-adv

• P

N1 + Λ1 , P N2 + Λ2

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Gaussian AVBC with Common Randomness

Problem: How to send common randomness. . .

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Gaussian AVBC with Common Randomness

Problem: How to send common randomness. . .

1 without the adversary preventing it,

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Gaussian AVBC with Common Randomness

Problem: How to send common randomness. . .

1 without the adversary preventing it, 2 without conflicting with superposition coding

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Idea that doesn’t work #1

[Ahlswede 1978]

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Idea that doesn’t work #1

[Ahlswede 1978] Works for unconstrained discrete channels, e.g. [Jahn 1981]

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Idea that doesn’t work #1

[Ahlswede 1978] Works for unconstrained discrete channels, e.g. [Jahn 1981] Doesn’t work in the power-constrained setting, because the adversary can inject more power during the short common randomness transmission

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Idea that doesn’t work #2

Use the common message as common randomness

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Idea that doesn’t work #2

Use the common message as common randomness Codebooks X1(m1, m2) ∼ N(0, αP), X2(m2) ∼ N(0, ¯ αP)

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Idea that doesn’t work #2

Use the common message as common randomness Codebooks X1(m1, m2) ∼ N(0, αP), X2(m2) ∼ N(0, ¯ αP) X = X1(m1, m2) + X2(m2)

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Idea that doesn’t work #2

Use the common message as common randomness Codebooks X1(m1, m2) ∼ N(0, αP), X2(m2) ∼ N(0, ¯ αP) X = X1(m1, m2) + X2(m2) Decoder 2 decodes m2, treats X1 as noise

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Idea that doesn’t work #2

Use the common message as common randomness Codebooks X1(m1, m2) ∼ N(0, αP), X2(m2) ∼ N(0, ¯ αP) X = X1(m1, m2) + X2(m2) Decoder 2 decodes m2, treats X1 as noise Decoder 1 decodes m2, then decodes m1 using m2 as common randomness

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Idea that doesn’t work #2

Use the common message as common randomness Codebooks X1(m1, m2) ∼ N(0, αP), X2(m2) ∼ N(0, ¯ αP) X = X1(m1, m2) + X2(m2) Decoder 2 decodes m2, treats X1 as noise Decoder 1 decodes m2, then decodes m1 using m2 as common randomness Problem: To avoid symmetrizability, we need Λ1 < ¯ αP, Λ2 < ¯ αP

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Idea that doesn’t work #2

Use the common message as common randomness Codebooks X1(m1, m2) ∼ N(0, αP), X2(m2) ∼ N(0, ¯ αP) X = X1(m1, m2) + X2(m2) Decoder 2 decodes m2, treats X1 as noise Decoder 1 decodes m2, then decodes m1 using m2 as common randomness Problem: To avoid symmetrizability, we need Λ1 < ¯ αP, Λ2 < ¯ αP This idea does work for the interference channel1

• 1F. Hosseinigoki and O. Kosut, “The Gaussian Interference Channel in the

Presence of a Malicious Jammer,” Allerton, 2016.

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Idea that works

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Idea that works

Divide length-n block into √n segments of length √n

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Idea that works

Divide length-n block into √n segments of length √n In each segment, transmit at either full power or 0

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Idea that works

Divide length-n block into √n segments of length √n In each segment, transmit at either full power or 0 The main BC code is contained in the full power segments

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Idea that works

Divide length-n block into √n segments of length √n In each segment, transmit at either full power or 0 The main BC code is contained in the full power segments The on/off sequence contains the common randomness

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Idea that works

Divide length-n block into √n segments of length √n In each segment, transmit at either full power or 0 The main BC code is contained in the full power segments The on/off sequence contains the common randomness

• Wait. . . can’t the adversary mess up the on/off sequence?

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Common randomness coding

At decoders, determine if the received power in each segment is high or low

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Common randomness coding

At decoders, determine if the received power in each segment is high or low Transmitter power Adversary power Received power P anything high >P high <P low

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Common randomness coding

At decoders, determine if the received power in each segment is high or low Transmitter power Adversary power Received power P anything high >P high <P low This is essentially a binary “or” channel

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Common randomness coding

At decoders, determine if the received power in each segment is high or low Transmitter power Adversary power Received power P anything high >P high <P low This is essentially a binary “or” channel If Λ < P, the adversary can only send power >P so often

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Common randomness coding

At decoders, determine if the received power in each segment is high or low Transmitter power Adversary power Received power P anything high >P high <P low This is essentially a binary “or” channel If Λ < P, the adversary can only send power >P so often Looks like a discrete AVC with cost-constrained adversary

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Common randomness coding

At decoders, determine if the received power in each segment is high or low Transmitter power Adversary power Received power P anything high >P high <P low This is essentially a binary “or” channel If Λ < P, the adversary can only send power >P so often Looks like a discrete AVC with cost-constrained adversary [Csiszar-Narayan 1988]: The capacity of this channel is positive, even if >1 − ǫ fraction of segments have high transmit power

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Conclusion

Main idea: On/off power code to transmit common randomness Applied to Gaussian arbitrarily-varying broadcast channel, but should generalize to many other settings Question? → fhossei1@asu.edu

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