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Imperfect InformationTheoreticSecrecyforMultimodeFiber

Eva C. Song, Emina Soljanin, Kyle C. Guan, Peter J. Winzercsong@princeton.edu, emina@research.bell-labs.com, {kyle.guan, winzer}@alcatel-lucent.com

MotivationSpace-division multiplexing (SDM) isneeded for growing demand of opticalcommunication because single mode fiberhas been shown to reach its capacity limit(see Fig. 1). SDM can be realized viamultimode fiber (MMF). Optical networkis vulnerable to physical layer attack (seeFig. 2). The secrecy capacity under equiv-ocation for perfect secrecy was studied in[1] and [2]. In this work, we apply jointsource-channel coding and use distortionas the metric for secrecy.

Fig. 1 Fig. 2

System ModelThe communication through an M -modeoptical SDM system exposed to eavesdrop-ping is modeled as a memoryless complexgaussian MIMO broadcast channel. Eaves-dropper suffers from mode dependent loss(MDL).

Legitimate: Y = HX +N

Eavesdropper: Y e = HeX +Ne

H =√E0LU , He =

√E0Le

√V Ue

MMF input have the following per modepower constraint averaged over n uses ofchannel

1

n

n∑i=1

|X(m)i |2 ≤ 1,∀m ∈ [1 :M ]

• Sk: i.i.d. source sequence sent by Alice

• Sk: Bob’s reconstruction of source se-quence based on his channel output Y n

• T k: Eve’s estimate of source sequencebased on her channel output Zn andpossibly other side information

Definition 1. For a given distortion functiond(s, t), (R,D) is achievable if there exists a se-quence of fk,n and gk,n such that

k

n= R, lim

n→∞ P[Sk 6= Sk] = 0, (Bob)

andlim infn→∞ min

tk(zn)E[d(Sk

, tk(Z

n))] ≥ D. (Eve)

Alternatively, we also consider the case

lim infn→∞ min

{tk(zn,sj−1)}kj=1

E[1

k

k∑j=1

d(Sj, tj(Zn, S

j−1)] ≥ D.

• R: rate between Alice and Bob for reli-able transmission

• D: distortion between Alice and Eve forsecurity

Main Results

H

He

+

+

Source-channel Encoder

Source-channel Decoder

Source-channel Decoder

Sk XnY n

Zn

Sk

T k

N

Ne

tj(zn, sj�1)

Fk,n

gk,n

causal information

Theorem 1. For i.i.d. source sequence Sk and memoryless broadcast channel PY Z|X , ifthere exists V− −W− −X− −Y Z such that I(W ;Y |V ) − I(W ;Z|V ) > 0, then (R,D)is achievable if and only if

R <maxX I(X;Y )

H(S)

D ≤ Dmax

where Dmax = mint E[d(S, t)].

Theorem 2. If maxK∈HM×M ,0�K�I|SNRK+I|

|SNReUeKUe†V+I| > 0, then the following rate distor-tion pair (R,D) is achievable with no causal information:

R < M log(SNR + 1)/H(S)

D ≤ Dmax

Theorem 3. For i.i.d. source sequence Sk and Hamming distortion, the following distor-tion rate curve D(R) is in the achievable region with causal information[3]:

D =

{d(H(S)), if R ≤ R∗s

H(S)

α(K)Dmax + (1− α(K))d(Rs(K)R ), if R∗s

H(S) < R ≤ R∗pH(S)

where d(R′s) , min(f(R′s), 1 − maxs PS(s)) and f(R′s) is the linear interpolation of the points(logn, n−1

n), n = 1, 2, 3, ...; K , {K ∈ HM×M , 0 � K � I}

R∗s = maxK′∈K

log|SNRK′ + I|

|SNRe√V UeK′Ue†

√V + I|

, R∗p =M log(SNR + 1),

Rs(K) = log|SNRK + I|

|SNRe√V UeKUe†

√V + I|

,

α(K) =β(K)− γ(K)

β(K), β(K) = log

|(SNR + 1)I||SNRK + I| , γ(K) = log

|SNReV + I||SNRe

√V UeKUe†

√V + I|

.

Some Numerical ResultsAchievable rate-distortion curves of MMF with Bern(0.3) and Bern(0.5) i.i.d. sources [4]:

SNR=20dB SNRe=10dB MDL=20dB

References[1] K. Guan, P. J. Winzer, and E. Soljanin, “Information-theoretic security in space-division multiplexed fiber optic networks",

ECOC 2012

[2] T. Liu and Y. Liang, “Multiple-input multiple-output gaussian broadcast channels with common and confidential messages",IEEE Trans. Inf Theory, Vol 56, pp. 5477-5487, 2010

[3] C. Schieler, E. C. Song, P. Cuff, and H. V. Poor, “Source-channel secrecy with causal disclosure", Allerton 2012

[4] K. Guan, E. C. Song, E. Soljanin, and P. J. Winzer, “Physical layer security in space-division multiplexed fiber optic commu-nications", Asilomar 2012