Stealth Communication with Vanishing Power over Binary Symmetric - - PowerPoint PPT Presentation

Stealth Communication with Vanishing Power over Binary Symmetric Channels

Diego Lentner, Gerhard Kramer Technical University of Munich Institute for Communications Engineering ISIT 2020

Covert Communication

Alice Bob Warren I s A l i c e c

• m

m u n i c a t i n g w i t h B

• b

? ( y e s / n

• )

Covert Communication vs. Secrecy:

• In secrecy, we want to hide the content of a

message, i.e., we minimize the mutual information between Alice’s message and Eve’s channel output.

• In covert communication/stealth, we want to hide

the presence of communication! Other names:

• Low probability of detection (LPD),
• Hiding information in noise.

Diego Lentner, Gerhard Kramer (TUM) 2

Hypothesis testing

• Binary hypothesis test:

◮ Null hypothesis H0: zn ∼ P0, ◮ Alternative hypothesis H1: zn ∼ P1.

• Optimal test (e.g., Neyman-Pearson) minimizes

PFA + PMD where PFA and PMD are the probabilities of false alarm and missed detection, respectively.

• Bounds on the performance of an optimal test:

◮ Variational distance:

PFA + PMD = 1 − VT(P0, P1).

◮ KL divergence (Pinsker’s inequality): VT(P0, P1) ≤

• 1

2 D(P0P1).

Diego Lentner, Gerhard Kramer (TUM) 3

Measuring Covertness

Alice PY|X X n Bob Y n PZ|X Warren Z n

• Alice sends ”0” when she is not communicating
• Warren observes his channel output zn and performs a

binary hypothesis test:

◮ H0: Alice is not communicating ⇔ zn ∼ Pn

Z |X(·|0),

◮ H1: Alice is communicating ⇔ zn ∼ PZ n.

• Alice can bound Warren’s detection performance when she is communicating to Bob by ensuring

D(PZ nPn

Z |X(·|0)) ≤ δ

for a small δ > 0.

Diego Lentner, Gerhard Kramer (TUM) 4

Covert Communication: Main Results

Square root law (Bash et al. ’13, Wang et al. ’16, Bloch ’16)

• Let n be the total number of channel uses.
• Warren observes Z n and compares its statistics to the null hypothesis.
• Alice can reliably transmit O(√

n) bits to Bob without being detected by Warren. This means that the rate is zero.

= ⇒ limn→∞ O(√

n)/n = 0 !

Secret key lengths (Bash et al. ’13, Bloch ’16)

• Alice and Bob pre-share a secret key of length K.
• In general, O(√

n log n) pre-shared key bits are necessary. If Alice knows the statistics of her channel to Warren, O(√ n) key bits suffice.

• If Warren’s channel is noisier than Bob’s, then covert communication can be achieved without key.
• If Warren has uncertainty, e.g., about the transmission time or his noise model, then these results

can be improved.

Diego Lentner, Gerhard Kramer (TUM) 5

Stealth Communication

Hou and Kramer ’14

• Alice does not have to remain silent when not communicating information to Bob.
• She is free to send any other symbols than the ”zero-symbol”.

Main idea: Alice confuses Warren by sending obfuscation symbols i.i.d. ∼ PXo,n

• Warren tests his observations against PZo,n = PZ |X · PXo,n.
• Alice must ensure that

D(PZ nPn

Zo) ≤ δ

for a small constant δ > 0. Then: Positive-rate stealth communication possible!

Diego Lentner, Gerhard Kramer (TUM) 6

Vanishing Power Communication Model

We are interested in stealth communication

• with vanishing power (VP) as without obfuscation
• with energy that scales as nα, 0 ≤ α < 1, with blocklenght n.

= ⇒ Average block power:

1 nE

• n
• i=1

X 2

i

• ≤ anα

n , 0 ≤ α < 1, a > 0. Assume

• ¯

p = 1 − p, 0 ≤ α ≤ 1, and 0 < a < 1.

• n is fixed (one-shot analysis!).

How much information can we transmit with VP over n channel uses?

1 1 1 − p p p 1 − p X Y 1 − anα

n anα n

PX,n

• 1 − anα

n

¯

p + anα

n p

• 1 − anα

n

• p + anα

n ¯

p PY,n

Diego Lentner, Gerhard Kramer (TUM) 7

VP Gallager Exponents

• Rate R:

M = enR.

• For Mρ = enRρ, DMS PX, and DMC PY |X:

P(E) ≤ e−nEG(R,PX) with Gallager exponent (Gallager ’68) EG(R, PX) = max

0≤ρ≤1 [E0(ρ, PX) − ρR]

and E0(ρ, PX) = − log

• y
• x

P(x)P(y|x)

1 1+ρ

1+ρ

• Scaling constant Rα for a fixed α:

M = enαRα.

• For Mρ = enαRαρ, DMS PX,n, and DMC PY |X:

P(E) ≤ e−nα ˆ

Eα

G(Rα,PX,n)

with VP Gallager exponent

ˆ

Eα

G(Rα, PX,n) = max 0≤ρ≤1

• ˆ

Eα

0 (ρ, PX,n) − ρRα

• and

ˆ

Eα

0 (ρ, PX,n) = lim n→∞

n nαE0(ρ, PX,n)

Diego Lentner, Gerhard Kramer (TUM) 8

VP Gallager Exponent

ˆ

Eα

0 (ρ, PX,n)

= lim

n→∞ n nα

• − log
• 1 − anα

n

¯

p

1 1+ρ + anα

n p

1 1+ρ

1+ρ +

• 1 − anα

n

• p

1 1+ρ + anα

n ¯

p

1 1+ρ

1+ρ =

• (1 + ρ)a
• ¯

p

1 1+ρ − p 1 1+ρ

¯

p

ρ 1+ρ − p ρ 1+ρ

• ,

α < 1

E0(ρ, PX,n),

α = 1 .

Example: a = 0.2, α = 0.5, p = 0.1. 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Rα

ˆ

Eα

G(Rα, PX,n)

Rα,max(PX,n)

Maximum VP scaling constant for BSCs

Rα,max(PX,n) = ∂ ˆ Eα

0 (ρ, PX,n)

∂ρ

• ρ=0

=

• a(1 − 2p) log ¯

p p,

α < 1

I(PX,n; PY |X),

α = 1 .

Diego Lentner, Gerhard Kramer (TUM) 9

Stealth Communication with VP

Alice PY|X X n Bob Y n PZ|X Warren Z n

• PY |X = BSC(p) and PZ |X = BSC(q).
• Alice can transmit obfuscation symbols with VP

1 n E

• n
• i=1

X 2

i

• = bnβ

n , 0 ≤ β < 1, 0 < b .

• When transmitting information, she must ensure that

◮ Bob can decode her message (with high probability) ◮ Warren cannot distinguish the message from obfuscation: D(PZ nPn

Zo)

!

≤ δ.

How large can Alice choose (α, a) for the given vanishing obfuscation power?

Diego Lentner, Gerhard Kramer (TUM) 10

Uncoded Stealth Communication

• Assume all Zi are i.i.d.:

D(PZ nPn

Zo) = n D(PZ,nPZo,n)

!

≤ δ.

• For α, β < 1 and sufficiently large n, we find via Taylor approximation

D(PZ,nPZo,n) ≈ 1

2

(¯

q − q)2 q¯ q

anα

n

− bnβ

n

2

Uncoded Stealth Communication

For sufficiently large n, uncoded stealth communication with α, β < 1 is possible if

• anα − bnβ

≤ k √

n with k =

√2q¯

q

¯

q − q

√ δ.

Diego Lentner, Gerhard Kramer (TUM) 11

Uncoded Stealth Communication

For α, β < 1:

• anα − bnβ

≤ k √

n

• Case bnβ

n = 0 (covert communication):

αmax = 0.5,

a ≤ k.

• Case β = 0.5:

αmax = 0.5,

a ≤ k + b.

• Case β = 1 (Hou and Kramer ’14):

αmax = 1. = ⇒ Positive-rate communication!

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

β α D(PZ,nPZo,n) = 0 possible ⇔

• anα − bnβ

= 0

n D(PZ,nPZo,n) ≤ δ possible

⇔

• anα − bnβ

≤ k√

n

Diego Lentner, Gerhard Kramer (TUM) 12

Coded Stealth Communication

Coded Stealth Communication

We can bound

log M

nα

<

• a(1 − 2p) log ¯

p p,

α < 1

I(PX,n; PY |X),

α = 1 log K

nα

>   

• a(1 − 2q) log ¯

q q − a(1 − 2p) log ¯ p p

+ , α < 1

• I(PX,n; PZ |X) − I(PX,n; PY |X)

+ , α = 1

where

• [x]+ = max(x, 0),
• (α, a) satisfy
• anα − bnβ

≤ k√

n for specified (β, b). Note:

• In the covert communication case bnβ

n = 0, we recover the bounds from Bloch ’16 evaluated for BSCs.

• For log K = 0, we obtain the bounds for key-less stealth / covert communication.

Diego Lentner, Gerhard Kramer (TUM) 13

Proof Sketch

• Random coding with binning
• Pre-shared key indexes a subcodebook
• Warren has to test against the entire codebook
• Reliability: proof via VP Gallager exponents as

for noisy channel coding theorem

• Stealth: bound (a),(b),(c) separately.

(a) Proof with VP resolvability exponents (following Hayashi ’06, Hou and Kramer ’13). (b) Uncoded stealth scenario, reuse result! (c) Bound with Pinsker’s and Jensen’s inequalities.

E

• D(PZ n| ˜

CPn

Zo)

• = E
• D(PZ n| ˜

CPn

Z)

• (a)

+ D(Pn

ZPn Zo)

• (b)

+ E

• zn

PZ n| ˜

C(zn| ˜

C) − Pn

Z(zn)

• log Pn

Z(zn)

Pn

Zo(zn)

• (c)

.

Diego Lentner, Gerhard Kramer (TUM) 14

Conclusion

Main results:

• Unified stealth communication framework that includes covert communication and positive-rate stealth

communication as extreme cases.

• Simple achievability proof based on suitably modified Gallager exponents.

Outlook

• Generalization to arbitrary DMCs (done!) and AWGN channels
• Practical coding scheme!
• Stealth with feedback
• ...

Diego Lentner, Gerhard Kramer (TUM) 15

References

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Diego Lentner, Gerhard Kramer (TUM) 16