physical layer security over wiretap channels with random...

Physical Layer Security over Wiretap Channelswith Random Parameters

Ziv Goldfeld, Paul Cuff and Haim Permuter

Ben Gurion University and Princeton University

2017 International Symposium on Cyber Security Cryptography andMachine Learning Conference

June 29-30, 2017

Physical Layer Security

Ziv Goldfeld, Paul Cuff and Haim Permuter Ben Gurion University and Princeton University

Physical Layer Security over Wiretap Channels with Random Parameters 2


Rooted in Information Theory.





Alternative approach to cryptography:






◮ Exploit the noisy channel for secrecy (no shared key).







◮ Computationally unlimited eavesdroppers.







◮ Computationally unlimited eavesdroppers.

Appropriate for securing low complexity devices such as IoT.



Some Background



Basic Information Measures

(X, Y ) ∼ PX,Y discrete RVs





Entropy: H(X) = H(PX) = −∑

x∈XPX(x) log PX(x).







Conditional Entropy: H(X|Y ) =∑

y∈YPY (y)H(PX|Y =y).







Conditional Entropy: H(X|Y ) =∑

y∈YPY (y)H(PX|Y =y).

Mutual Information: I(X; Y ) = H(X) − H(X|Y )= H(Y ) − H(Y |X).



Physical Layer Communication - IT Perspective

A mathematical model for a physical communication channel.





Channel Examples:





Channel Examples:

◮ Gaussian Channel:



X

Z ∼ N (0, σ2)

Y = X + Z



Channel Examples:




X

Z ∼ N (0, σ2)

Y = X + Z

◮ Binary Symmetric Channel:

0

1

0

1

X Y

α

1 − α

α

1 − α



Channel Examples:


Questions we ask:



X

Z ∼ N (0, σ2)

Y = X + Z


0

1

0

1

X Y

α

1 − α

α

1 − α



Channel Examples:


Questions we ask:

◮ What is the maximal rate [bits/ch. use] of reliable communication?



X

Z ∼ N (0, σ2)

Y = X + Z


0

1

0

1

X Y

α

1 − α

α

1 − α



Channel Examples:


Questions we ask:

◮ What is the maximal rate [bits/ch. use] of reliable communication?

◮ How to design codes at that rate?



X

Z ∼ N (0, σ2)

Y = X + Z


0

1

0

1

X Y

α

1 − α

α

1 − α

Physical Layer Communication - Formal Definition

MEnc

XnPY |X

Y nDec

M̂




MEnc

XnPY |X

Y nDec

M̂

Message: nR information bits.




MEnc

XnPY |X

Y nDec

M̂


(n,R)-Code: Block Encoding and Decoding functions.




MEnc

XnPY |X

Y nDec

M̂



Channel: P(

Y n = yn∣

∣Xn = xn)

=n∏

i=1PY |X(yi|xi).




MEnc

XnPY |X

Y nDec

M̂



Channel: P(

Y n = yn∣

∣Xn = xn)

=n∏

i=1PY |X(yi|xi).

Capacity: C , sup{

R∣

∣

∣∃(n, R) − codes s.t. P(

M 6= M̂)

→n

0}

.




MEnc

XnPY |X

Y nDec

M̂



Channel: P(

Y n = yn∣

∣Xn = xn)

=n∏

i=1PY |X(yi|xi).

Capacity: C , sup{

R∣

∣

∣∃(n, R) − codes s.t. P(

M 6= M̂)

→n

0}

.

Reminder: I(X; Y ) = H(Y ) − H(Y |X)Ziv Goldfeld, Paul Cuff and Haim Permuter Ben Gurion University and Princeton University


Theorem (Shannon 1948)

The capacity of a channel PY |X is C = maxPX

I(X; Y ).

State-Dependent Wiretap Channels



State-Dependent Channels[Gelfand-Pinsker 1980]



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂


State-Dependent Channel:



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂



State: S ∼ Bernoulli(

12

)

, S = {0, 1}.



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂




12

)

, S = {0, 1}.

AGN Channel: Y = X + ZS



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂




12

)

, S = {0, 1}.


Z0 ∼ N (0, σ20), Z1 ∼ N (0, σ

21), σ0 ≪ σ1.



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂




12

)

, S = {0, 1}.


Z0 ∼ N (0, σ20), Z1 ∼ N (0, σ

21), σ0 ≪ σ1.

Time t 1 2 3 4 5 6 . . .

State StSNR at time t



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂




12

)

, S = {0, 1}.


Z0 ∼ N (0, σ20), Z1 ∼ N (0, σ

21), σ0 ≪ σ1.

Time t 1 2 3 4 5 6 . . .

State St 1 1 0 1 0 0 . . .

SNR at time t



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂




12

)

, S = {0, 1}.


Z0 ∼ N (0, σ20), Z1 ∼ N (0, σ

21), σ0 ≪ σ1.

Time t 1 2 3 4 5 6 . . .

State St 1 1 0 1 0 0 . . .

SNR at time t Low



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂




12

)

, S = {0, 1}.


Z0 ∼ N (0, σ20), Z1 ∼ N (0, σ

21), σ0 ≪ σ1.

Time t 1 2 3 4 5 6 . . .

State St 1 1 0 1 0 0 . . .

SNR at time t Low Low



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂




12

)

, S = {0, 1}.


Z0 ∼ N (0, σ20), Z1 ∼ N (0, σ

21), σ0 ≪ σ1.

Time t 1 2 3 4 5 6 . . .

State St 1 1 0 1 0 0 . . .

SNR at time t Low Low High



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂




12

)

, S = {0, 1}.


Z0 ∼ N (0, σ20), Z1 ∼ N (0, σ

21), σ0 ≪ σ1.

Time t 1 2 3 4 5 6 . . .

State St 1 1 0 1 0 0 . . .

SNR at time t Low Low High Low



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂




12

)

, S = {0, 1}.


Z0 ∼ N (0, σ20), Z1 ∼ N (0, σ

21), σ0 ≪ σ1.

Time t 1 2 3 4 5 6 . . .

State St 1 1 0 1 0 0 . . .

SNR at time t Low Low High Low High



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂




12

)

, S = {0, 1}.


Z0 ∼ N (0, σ20), Z1 ∼ N (0, σ

21), σ0 ≪ σ1.

Time t 1 2 3 4 5 6 . . .

State St 1 1 0 1 0 0 . . .

SNR at time t Low Low High Low High High . . .



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂


Encoder knows the state sequence non-causally.



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂



Enhance communication rates via coherent transmission.



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂



Enhance communication rates via coherent transmission.



M

(nR bits)Encoder

XnPY |X,S

Sn ∼ P nS

Y n

DecoderM̂

Theorem (Gelfand-Pinsker 1980)

CGP = maxPU,X|S

[

I(U ; Y ) − I(U ; S)]

Joint distribution: PU,X|SPY |X,S

The Gelfand-Pinsker Channel - Encoding

Pad nR message bits with nR̃ redundancy bits.





00101101000110100010101100 01001011101010

Message Padding

Transmitted together in one block





00101101000110100010101100 01001011101010

Message Padding


Codebook: (Message ,Padding) → Codeword Un.





00101101000110100010101100 01001011101010

Message Padding



Correlating Un with Sn: R̃ > I(U ;S).





00101101000110100010101100 01001011101010

Message Padding



Correlating Un with Sn: R̃ > I(U ;S).

Reliability: R+ R̃ < I(U ;Y ).



Wiretap ChannelsDegraded [Wyner 1975], General [Csiszár-Körner 1978]

M

(nR bits)Encoder

XnPY,Z|X

Y n

Zn

Decoder

Eve

M̂

M




M

(nR bits)Encoder

XnPY,Z|X

Y n

Zn

Decoder

Eve

M̂

M

Secrecy-Capacity:




M

(nR bits)Encoder

XnPY,Z|X

Y n

Zn

Decoder

Eve

M̂

M

Secrecy-Capacity:



Reliable Communication.


M

(nR bits)Encoder

XnPY,Z|X

Y n

Zn

Decoder

Eve

M̂

M

Secrecy-Capacity:




Zn contains no information about M .


M

(nR bits)Encoder

XnPY,Z|X

Y n

Zn

Decoder

Eve

M̂

M

Secrecy-Capacity:




Zn contains no information about M .

Theorem (Csiszár-Körner 1978)

CWTC = maxPU,X

[

I(U ; Y ) − I(U ; Z)]

Joint distribution: PU,XPY,Z|X

Wiretap Channels - Encoding






00101101000110100010101100 01001011101010

Message Padding






00101101000110100010101100 01001011101010

Message Padding



Security: R̃ > I(U ;Z).





00101101000110100010101100 01001011101010

Message Padding



Security: R̃ > I(U ;Z).

Reliability: R+ R̃ < I(U ;Y ).



The Gelfand-Pinsker Wiretap Channel

M

(nR bits)Encoder

Xn

PY,Z|X,S

Eve

Sn

Y n

Zn

DecoderM̂

M




M

(nR bits)Encoder

Xn

PY,Z|X,S

Eve

Sn

Y n

Zn

DecoderM̂

M

Secrecy Capacity: Reliable and Secure Communication.




M

(nR bits)Encoder

Xn

PY,Z|X,S

Eve

Sn

Y n

Zn

DecoderM̂

M


Tension: Utilize state knowledge to simultaneously




M

(nR bits)Encoder

Xn

PY,Z|X,S

Eve

Sn

Y n

Zn

DecoderM̂

M



◮ Enhance reliable communication rate (coherent transmission).




M

(nR bits)Encoder

Xn

PY,Z|X,S

Eve

Sn

Y n

Zn

DecoderM̂

M



◮ Enhance reliable communication rate (coherent transmission).

◮ Boost security performance (e.g., via secret key agreement).




M

(nR bits)Encoder

Xn

PY,Z|X,S

Eve

Sn

Y n

Zn

DecoderM̂

M

Naive Approach:




M

(nR bits)Encoder

Xn

PY,Z|X,S

Eve

Sn

Y n

Zn

DecoderM̂

M

Naive Approach: Combine wiretap coding with GP coding.




M

(nR bits)Encoder

Xn

PY,Z|X,S

Eve

Sn

Y n

Zn

DecoderM̂

M


00101101000110100010101100 01001011101010




M

(nR bits)Encoder

Xn

PY,Z|X,S

Eve

Sn

Y n

Zn

DecoderM̂

M


00101101000110100010101100 01001011101010



Theorem (Chen-Han Vinck 2006)

CGP−WTC ≥ maxPU,X|S

[

I(U ; Y )−max{

I(U ;Z), I(U ;S)}

]

Joint distribution: PSPU,X|SPY,Z|X,S


M

(nR bits)Encoder

Xn

PY,Z|X,S

Eve

Sn

Y n

Zn

DecoderM̂

M


00101101000110100010101100 01001011101010

⋆ Always preforms wiretap codingZiv Goldfeld, Paul Cuff and Haim Permuter Ben Gurion University and Princeton University




[

I(U ; Y )−max{

I(U ;Z), I(U ;S)}

]



M

(nR bits)Encoder

Xn

PY,Z|X,S

Eve

Sn

Y n

Zn

DecoderM̂

M


00101101000110100010101100 01001011101010

⋆ Always preforms wiretap coding =⇒ Suboptimal vs. strong EveZiv Goldfeld, Paul Cuff and Haim Permuter Ben Gurion University and Princeton University




[

I(U ; Y )−max{

I(U ;Z), I(U ;S)}

]


Improved Two-Layered Coding Scheme

Theorem (ZG-Cuff-Permuter 2016)

CGP−WTC ≥ maxPU,V,X|S :

I(U ;Y )≥I(U ;S)

min

{

I(V ; Y |U) − I(V ; Z|U),I(U, V ; Y ) − I(U, V ; S)

}

Joint distribution PSPU,V,X|SPY,Z|X,S .






I(U ;Y )≥I(U ;S)

min

{


}


Two-Layered Coding Scheme (Superposition Code):






I(U ;Y )≥I(U ;S)

min

{


}



Inner Layer: No wiretap coding - Supports key-agreement strategies.






I(U ;Y )≥I(U ;S)

min

{


}




Outer Layer: Wiretap coding (when channel favors legit users).






I(U ;Y )≥I(U ;S)

min

{


}





Generalization: Captures all past results as special cases.






I(U ;Y )≥I(U ;S)

min

{


}





Generalization: Captures all past results as special cases.

Strict Improvement: Explicit example & Analytic derivation.



Summary

The Gelfand-Pinsker wiretap channel



Summary


◮ Combination of two fundamental IT setups.



Summary



◮ Simultaneously exploit state for reliability and security.



Summary




Novel superposition coding scheme



Summary





◮ Recovers all past results.



Summary






◮ Strictly outperforms previous benchmark.



Summary







Available on ArXiV: https://arxiv.org/abs/1608.00743v1.



Summary







Available on ArXiV: https://arxiv.org/abs/1608.00743v1.

Thank you!



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physical layer security over wiretap channels with random...

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