Treating Interference as Noise is Optimal for Covert Communication - - PowerPoint PPT Presentation

Treating Interference as Noise is Optimal for Covert Communication over Interference Channels

Kang-Hee Cho and Si-Hyeon Lee

School of Electrical Engineering KAIST

2020 ISIT

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Covert Communication

Encoder Decoder Warden P ×n

Y;ZjX

W Xn Y n Zn ^ W H0 : Q×n H1 : ^ QZn Reliable with low probability of detection by an adversary (warden) Optimal hypothesis testing by the warden H0: no communication (output dist. is Q×n

0 ) or H1: active (output dist. is ˆ

QZn) π1|0 + π0|1 ≥ 1 −

• D( ˆ

QZnQ×n

0 )

Square-root law: Throughput over n channel uses ∝ √n in AWGN [Bash et al. 2013] and many other discrete cases [Bloch 2016; Wang et al. 2016; Tan-Lee 2019] Low transmit power compared to the noise level (AWGN)

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Main Result for Discrete Memoryless Channels (DMCs) with a Warden

Literature [Bloch 2016] Encoder Decoder Warden P ×n

Y;ZjX

W Xn Y n Zn ^ W H0 : Q×n H1 : ^ QZn Binary input DMC to the decoder (X, PY |X, Y), to the warden (X, PZ|X, Z)

Off input symbol 0: Send when no communication occurs Q1 = PZ|X (·|1) and Q0 = PZ|X (·|0), P1 = PY |X (·|1) and P0 = PY |X (·|0) Assume Q1 ≪ Q0, (i.e., supp(Q1) ⊆ supp(Q0)). Otherwise, covert comm. is impossible

Covertness requirement ⇒ The number of symbol 1 is restricted by c√n

Code rate: log M ≈ c√n × D(P1P0) Channel resolvability: Sufficient number of codewords for nearly IID dist. at the warden Message × key rate: log MK ≈ c√n × D(Q1Q0)

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Extensions

MIMO AWGN channels [Abdelaziz and Koksal 2017 ]

Square root law on the blocklength still holds Scales exponentially with the number of transmitting antennas in massive MIMO limit

Non-coherent Rayleigh-fading channels [Tahmasbi et al. 2020]

Amplitude-constrained input distribution with finite number of input points is optimal

Multiple-access channels [Arumugam and Bloch 2019 ]

No sum-rate bound

Broadcast channels [Tan and Lee 2019 ]

Time-division is optimal over a broad class of channels

Some results are quite different or simpler from the results without the covertness

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Our Work - Interference Channels

Encoder 1 Encoder 2 Decoder 1 Decoder 2 Y n

1

Y n

2

^ W1 Xn

1

Xn

2

W1 W2 P ×n

Y1;Y2jX1;X2

^ W2

Consider discrete memoryless interference channels (DM-IC) with a warden The capacity region of interference channels without warden is not known in general, except some special cases e.g.,

Strong interference channels [Sato 1978] Injective deterministic interference channels [El Gamal and Costa 1982]

Complicated coding scheme for the best known inner bound [Han-Kobayashi 1981] Our result: Treating interference as noise is optimal for covert communication over interference channels

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Channel Model & Assumptions

Rx 1 Rx K Warden Tx 1 Tx K X1 XK Y1 YK Z (W1; S1) (WK; SK) ^ W1 ^ WK H0 : Q×n H1 : ^ Qn W ×n

YKjXK

V ×n

ZjXK

SK SK

K-user-pair binary-input DM-ICs (XK, WYK|XK, YK)

XK := (X1, . . . , XK ) = {0, 1}K , where K := {1, 2, . . . . , K} Symbol 0: Off symbol that is sent when no communication occurs

The warden monitors the channel outputs of the DM-MAC (XK, VZ|XK, Z) QU ≪ Q0 for all U ⊆ K

QU: The output distribution at the warden when only Txs i, i ∈ U send symbol 1 Otherwise, covert communication is restricted to some kinds of symbol combination

Q0 cannot be represented as any convex combination of QU for some U ⊆ K

Otherwise, positive rate is achievable (we do not focus on)

W (k)

U

≪ W (k) for all k ∈ K and for all U ⊆ K

W (k)

U : The output distribution at Rx k when only Txs i, i ∈ U send symbol 1

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Definitions of a Code & Covert Capacity Region

An (MK, JK, n) code for the K-user-pair DM-IC with a warden consists of

K message sets [1 : Mk] for k ∈ K; K secret key sets [1 : Jk] for k ∈ K; K Txs xk(wk, sk) : [1 : Mk] × [1 : Jk] → X n for k ∈ K, (uniformly distributed); K Rxs ˆ wk(yk, sK) : Yn

k × (×k∈K[1 : Jk]) → [1 : Mk] for k ∈ K.

Probability of error: Pn

e := Pr

K

k=1{ ˆ

Wk = Wk}

• ,

covertness measure: D( ˆ QnQ×n

0 )

A tuple (RK, LK) ∈ R2K

+ is achievable if there exists a sequence of codes satisfying

lim inf

n→∞

log Mk

• nD( ˆ

QnQ×n

0 )

≥ Rk, ∀k ∈ K, lim sup

n→∞

log Jk

• nD( ˆ

QnQ×n

0 )

≤ Lk, ∀k ∈ K, lim

n→∞ Pn e = 0,

lim

n→∞ D( ˆ

QnQ×n

0 ) = 0.

Covert capacity region: Closure of {RK ∈ RK

+ : (RK, LK) is achievable for some LK}

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Main Results (1) - The Number of Symbol 1

Covertness requirement ⇒ The total number of symbol 1 of the Txs, N is O(√n) Fraction vector α = (α1, . . . , αK) ∈ [0, 1]K such that

k∈K αk = 1: The allocation

• f the ratio of total symbol 1 at each Tx

Total # of symbol 1, N depends on α and channels to the warden, especially the common factor χ2(α) :=

z

(

• k∈K αk Qk (z)−Q0(z))

2

Q0(z)

in the way N ∝ 1/

• χ2(α)

This chi-square distance is related to the detectability of the warden Tx 1 Tx 2 Tx K Warden's output dist. Q1 QK Q2 α1N Total # of symbol 1: N α2N αKN

N /

1

p

χ2(α)

) χ2(α)

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Main Results (2) - Covert Capacity Region

Theorem 1

The covert capacity region is the set of the rate tuple RK satisfying Rk ≤ αkD(W (k)

k

W (k) )

• χ2(α)/2

, ∀k ∈ K for some α ∈ [0, 1]K such that

k∈K αk = 1, where χ2(α) := z

(

• k∈K αk Qk (z)−Q0(z))

2

Q0(z)

. Sparse interference signals are negligible compared to inherent channel uncertainty

(Remark) W (k)

k

: output dist. at Rx k when only Tx k send symbol 1 (P2P nature) Given # of symbol 1, each user can transmit the maximal number of reliable bits

Tx 1 Tx 2 Tx K Warden's output dist. Q1 QK Q2 α1N Total # of symbol 1: N α2N αKN

N /

1

p

χ2(α)

log M1 ≈ α1ND(W (1)

1

kW (1) ) log M2 ≈ α2ND(W (2)

2

kW (2) ) log MK ≈ αKND(W (K)

K

kW (K) )

) χ2(α)

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Main Results (2) - Covert Capacity Region

Theorem 1

The covert capacity region is the set of the rate tuple RK satisfying Rk ≤ αkD(W (k)

k

W (k) )

• χ2(α)/2

, ∀k ∈ K for some α ∈ [0, 1]K such that

k∈K αk = 1, where χ2(α) := z

(

• k∈K αk Qk (z)−Q0(z))

2

Q0(z)

. If Qk(z) = Q(z), ∀k, z (symmetric to the warden), χ2(α) and total # of 1 is fixed

Time-division scheme is optimal (α plays a role of time fraction)

Tx 1 Tx 2 Tx K Warden's output dist. Q1 QK Q2 α1N Total # of symbol 1: N α2N αKN

N /

1

p

χ2(α)

log M1 ≈ α1ND(W (1)

1

kW (1) ) log M2 ≈ α2ND(W (2)

2

kW (2) ) log MK ≈ αKND(W (K)

K

kW (K) )

) χ2(α)

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Main Results (3) - Secret Key Length

Theorem 2

Given α, for Rk =

αk D(W (k)

k

W (k) )

√

χ2(α)/2

, ∀k ∈ K, a tuple (RK, LK) is achievable if and only if Lk ≥ αk[D(QkQ0) − D(W (k)

k

W (k) )]+

• χ2(α)/2

, ∀k ∈ K. Using channel resolvability approach for covertness analysis requiring that Rk + Lk ≥ αkD(QkQ0)

• χ2(α)/2

, ∀k ∈ K

Sufficient number of codewords ⇒ Output distribution at the warden is nearly IID

Treating interference as noise ⇒ Key sharing between only each Tx-Rx pair If D(QkQ0) ≤ D(W (k)

k

W (k) ) (i.e. channel from Tx k to the warden is worse than the channel from Tx k to Rx k), a secret key between user pair k is unnecessary.

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Achievability Sketch - Reliability

Fix α. Random coding and joint typicality decoding with treating interference as noise

At transmitter k, the probability of symbol 1 is αkγn

Upper bound on the probability of error at receiver 1 over codebook ensemble

1 Equivalent point-to-point channel

W (1)

Y1jXK

Rx 1 Tx 1 Tx K Tx 2 ¯ W (1)

Y1jX1

Rx 1 Tx 1

=

2 Approximate marginal channel without interference

≈

¯ W (1)

Y1jX1

Rx 1 Tx 1 W (1)

Y1jXK

Rx 1 Tx 1 Tx K Tx 2

log Mk = (1 − ǫ)nαkγnD(W (k)

k

W (k) ) is achievable with E[Pn

e ] ≤ e−cnγn, ∀k

for arbitrarily small ǫ > 0 and a constant c > 0.

Code rate ∝ the number of symbol 1 × P2P channel quality We can choose a proper γn

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Achievability Sketch - Covertness

Channel resolvability approach for covertness analysis log MkJk = (1 + ǫ)nαkγnD(QkQ0) is achievable ∀k and arbitrarily small ǫ > 0 with

Required codebook size ∝ the number of symbol 1 × channel resolution at the warden E

• D( ˆ

QnQ×n

α,γn)

• ≤ e−cnγn: Output distribution at the warden is nearly IID

Thus,

• nD( ˆ

QnQ×n ) ≈ nγn

• χ2(α)/2

Achievable rate lim

n→∞

log Mk

• nD( ˆ

QnQ×n

0 )

= (1 − ǫ)αkD(W (k)

k

W (k) )

• χ2(α)/2

, ∀k, lim

n→∞

log MkJk

• nD( ˆ

QnQ×n

0 )

= (1 + ǫ)αkD(QkQ0)

• χ2(α)/2

, ∀k.

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Converse Sketch

Converse for reliability analysis: Similar to the achievability step

Given code, calculate the number of symbol 1 at each user and factor α ∈ [0, 1]K Upper bound each individual rate by the approximate point-to-point channel

Converse for covertness analysis

DM-MAC to the warden is similar to that in work on DM-MAC with a warden [Arumugam and Bloch 2019] Proof is similar

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Conclusion

Covert communication over K-user-pair interference channels Treating interference as noise is optimal under mild assumption on absolute continuity Time-division is optimal for the symmetric channels to the warden Secret key between each pair and unnecessary if channel between Tx and Rx is better than the channel between the Tx and the warden Refer arXiv [2003.04531] (same title) for details and extensions such as the covert capacity regions of

Non-binary input DM-IC with a warden Gaussian IC with a warden Extensions to multiple wardens

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