Secure Identification for Gaussian Channels

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Secure Identification for Gaussian Channels

Wafa Labidi, Christian Deppe and Holger Boche

Technische Universität München Lehrstuhl für Theoretische Informationstechnik

ICASSP 2020

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Outline

1 Identification: Overview

2 Identification for the Gaussian Wiretap Channel

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Outline

3 Conclusions

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What is Identification?

m Enc noisy channelWn

Dec m0_{transmitted or not?}

Alice Bob

Ahlswede/Dueck Picture 1989

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What is Identification?

Shannon Picture 1948

• Receiver’s goal:Whatis the message sent?

• Sender chooses and sends the

messagem∈M={1, . . . ,M=

2nC_}

Ahlswede/Dueck Picture 1989

• Receiver’s goal:Ism0the message sent?

• Sender chooses and sends the

identitym∈N={1, . . . ,N=22nC

}

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Randomized Identification (ID)-Code

Randomized ID-code

A randomized(n,N,λ1,λ2)ID-code for a discrete memoryless channel (DMC)

Wis a family of pairs{(Qi,Di)|i=1, . . . ,N}withλ1,λ26λ <1₂and

∀i∈{1, . . . ,N}: • Qi∈P(Xn), Di⊆Yn • P_xn_∈_XnQi(xn)Wn(Dci|x n₎ 6λ1⇐=channel noise • P_xn_∈_XnQj(xn)Wn(Di|xn)6λ2⇐=ID-code

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Randomized ID-Code

Yn₌_Xn ∝λ(i,j) 2 Di Dj ∝λ(j,l) 2 Dl ∝λ(2i,k) Dk .. . . . . Dm

ID-code for a binary noisless channel

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Randomized ID-Code

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Randomized ID-Code

ID-code for a binary noisless channel

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Randomized ID-Code

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Identification Coding Theorem

1 2

Theorem

LetWbe afinite DMCandN(n,λ)the maximal number s.t. an(n,N,λ1,λ2)

ID-code forW(f,P)exists withλ1,λ26λthen:

CID(W) =C(W), ∀λ∈(0, 1 2) C(W)is the Shannon transmission capacity ofW,

CID(W),limn→∞ n1 log logN(n,λ)

1

R. Ahlswede and G. Dueck, "Identification via channels," in IEEE Transactions on Information Theory, vol. 35, no. 1, pp. 15-29, Jan. 1989 2

T. S. Han and S. Verdu, "New results in the theory of identification via channels," in IEEE Transactions on Information Theory, vol. 38, no. 1, pp. 14-25, Jan. 1992.

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Direct Proof: Coding Scheme

1 To send a messagei, we prepare a set of coloring functionsTiknown by

the sender and the receiver(s)

Ti:{1, . . . ,M0}−→{1, . . . ,M00} : _|{z}j coloring 7→Ti(j) |{z} color 1 2 .. . j .. . M0 1 2 . . . k . . . M00 T1 1 2 .. . j .. . M0 1 2 . . . k . . . M00 TN . . . . T1(1) =TN(1) =2

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Direct Proof: Coding Scheme

Ti:{1, . . . ,M0}−→{1, . . . ,M00} : _|{z}j coloring 7→Ti(j) |{z} color 1 2 .. . j .. . M0 1 2 . . . k . . . M00 T1 1 2 .. . j .. . M0 1 2 . . . k . . . M00 TN . . . . T1(1) =TN(1) =2

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Direct Proof: Coding Scheme

2 The sender chooses a coloringjrandomly and calculates the color of the

messageiunder coloringjdenoted byTi(j)

3 Send(j,Ti(j))over the channel

4 The receiver, interested ini0, calculatesTi0(^j)

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Direct Proof: Coding Scheme

5 IfTi(^j) =Ti0(^j), theni=i0

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Direct Proof: Coding Scheme

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Direct Proof: Coding Scheme

5 IfTi(^j) =Ti0(^j), theni=i0

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Direct Proof: Code Construction

u0 _u00 n d √ ne m u01 u0 2 .. . u0 M0 C0₌_{₍_u0 j,D0j),j∈{1, . . . ,M0}} (n,M0_{, 2}−nγ₎ |C0_|₌_d₂n(C−)_e u001 u00 2 . . . u00 M00 C00₌_{₍_u00 k,D00k),k∈{1, . . . ,M00}} (d√ne,M00_{, 2}−√nγ₎ |C00_|₌_d₂√n_e ∈C={(Qi,Di),i∈{1, . . . ,N}}

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Outline

3 Conclusions

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Wiretap Channel

Requirements:

• Secrecy (here strong): I(M;Zn₎

6ξ1, ξ1>0 • Reliability:Pe(n),Pr[ ^M6=M]6ξ2, ξ2>0 M Encoder Alice Channel(W,V) Decoder Bob Eavesdropper Eve ^ M M X Y Z

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Dichotomy Theorem

5

Theorem

We denote byC(W)the capacity of the channelWand byCSID(W,V)the

identification capacity of the wiretap channel(W,V)then:

CSID(W,V) =

C(W) ifCS(W,V)>0

0 ifCS(W,V) =0

5

R. Ahlswede and Z. Zhang, "New directions in the theory of identification via channels," in IEEE Transactions on Information Theory, vol. 41, no. 4, pp. 1040-1050, July 1995.

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Proof: Code Construction

u0 _u00 n d √ ne m u01 u0 2 .. . u0 M0 C0₌_{₍_u0 j,D0j),j∈{1, . . . ,M0}} transmission code |C0_|₌_d₂n(C−)_e u001 u00 2 . . . u00 M00 C00₌_{₍_u00 k,D00k),k∈{1, . . . ,M00}} wiretap code |C00_|₌_d₂√n_e ∈C={(Qi,Di),i∈{1, . . . ,N}}

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Secure Codes for the Gaussian Channel

Wiretap transmission codes

An(n,M,λ)wiretap code for(W,V, g, g0,P)is a family of pairs

{(Q(·|i),Di), i=1, . . . ,M}such that for allin∈{1, . . . ,M}:

• Q(·|i)∈P(Xn₎_, _D i⊂Yn

• Di∩Dj=∅, ∀i6=j

• R_xn_∈_XnQ(xn|i)Wn(Dc_i|xn)dnxn6λ

• I(U;Zn)₆λ

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Secure Codes for the Gaussian Channel

Wiretap ID-codes

A randomized(n,N,λ1,λ2)wiretap ID-code for(V,W, g, g0,P)is a family of pairs

{(Q(·|i),Di),i=1, . . . ,N}such that forλ1,λ26λ < 1₂,∀E⊂Zn,∀i:

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Extension for the Gaussian Case

• W: yi=xi+ni, ni∼ N(0,N),g, 16i6n

• V: zi=xi+n0i, ni0 ∼ N(0,N0),g0, 16i6n

• Average power constraint: _n1Pn_i₌₁x2

i6P

• Y=Z=_R

=⇒ We call this channel(W,V, g, g0,P)

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Extension for the Gaussian Case: Dichotomy

Theorem

Theorem (

Secure identification capacity)

LetCSID(g,g0,P)be the identification capacity of the wiretap channel

(W,V,g,g0,P)then:

CSID(g,g0,P) =

C(g,P) ifCS(g,g0,P)>0

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Outline

3 Conclusions

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Conclusions and Outlook

• We provided a coding scheme for the Gaussian wiretap channel and

calculated the corresponding secure identification capacity. _,

• Future:

• Explore identification and secure identification for the single-user MIMO channel