Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20....
Transcript of Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20....
![Page 1: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/1.jpg)
Secret-Key Generation from ChannelReciprocity: A Separation Approach
Ashish Khisti
Department of Electrical and Computer EngineeringUniversity of Toronto
Feb 11, 2013
![Page 2: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/2.jpg)
Security at PHY-Layer
Use PHY Resources for designing security mechanisms.
A p p l i c a t i o n L a y e r( S e m a n t i c s o f I n f o r m a t i o n )
T r a n s p o r t L a y e r( E n d t o E n d C o n n e c t i v i t y )
( R o u t i n g a n d P a t h D i s c o v e r y )
D a t a L i n k L a y e r( E r r o r C o r r e c t i o n C o d e s )
N e t w o r k L a y e r
P h y s i c a l L a y e r
( S i g n a l s , R F h a r d w a r e )
E n c r y p t i o n ,A u t h e n t i c a t i o n
S e c u r e S o c k e tL a y e r
V i r t u a l P r i v a t e N e t w o r k sA n o n y m o u s R o u t i n g
D e v i c e l e v e lA u t h e n t i c a t i o n
A n t i - J a m m i n g
Wireless Systems
Applications:
Secret-Key Generation
Secure Message Transmission
Physical Layer Authentication
Jamming ResistanceFeb 11, 2013 2/ 20
![Page 3: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/3.jpg)
Motivation
Secret-Key Generation in Wireless Fading Channels
A B
KAKB
Forward Link
Reverse Link
mA mBABAABB nxhy +=
BABBAA nxhy +=
time
Channel Gain
Forward ChannelReverse Channel
Fading:
yB(t) = hAB(t)xA(t)+nB(t)
Reciprocity:
yB(t) = hAB(t)xA(t) + nB(t)
yA(t) = hBA(t)xB(t) + nA(t)
Feb 11, 2013 3/ 20
![Page 4: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/4.jpg)
Motivation
Secret-Key Generation in Wireless Fading Channels
A B
E
KAKB
Forward Link
Reverse Link
mA mBABAABB nxhy +=
BABBAA nxhy +=
AEAAEAE nxgz += BEBBEBE nxgz +=
time
Channel Gain
Forward ChannelReverse Channel
Eavesdropper Link
Spatial Decorrelation:
yB(t)= hAB(t)xA(t) + nB(t)
yA(t)= hBA(t)xB(t) + nA(t)
zA(t)= gA(t)xA(t) + nAE(t)
zB(t)= gB(t)xB(t) + nBE(t)
Feb 11, 2013 3/ 20
![Page 5: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/5.jpg)
Secret-Key Generation - A Systems Approach
Key Generation in Wireless Systems
UWB Systems: Wilson-Tse-Scholz (’07), M. Ko (’07),Madiseh-Neville-McGuire(’12)Narrowband Systems: Azimi Sadjadi- Kiayias-Mercado-Yener (’07),Mathur-Trappe-Mandayam -Ye-Reznick (’10), Patware and Kasera (’07)OFDM reciprocity: Haile (’09), Tsouri and Wulich (’09)
Implementations
Experimental UWB: Measurements for Key Generation Madiseh (’12)Software Radio Implementations: Jana et. al. (’09)MIMO systems: Wallace and Sharma (’10), Shimizu et al. Zeng-Wu-Mohapatra
Signal Processing for Secret-Key Generation
Quantization Techniques: Ye-Reznik-Shah (’07), Hamida-Pierrot-Castelluccia(’09), Sun-Zhu-Jiang-Zhao (’11)Adaptive Channel Probing: Wei-Zheng-Mohapatra (’10)Mobility Assisted Key Generation: Gungor-Chen-Koksal (’11)
Attacks
Active Eavesdroppers: Ebrez et. al (’11) Zafer-Agrawal-Srivatsa (’11),Unauthenticated Channels: Mathur et al. (’10),Xiao-Greenstein-Mandayam-Trappe (’07).
Feb 11, 2013 4/ 20
![Page 6: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/6.jpg)
Secret-Key Generation: A Systems Approach II
A B
E
KAKB
Forward Link
Reverse Link
mA mBABAABB nxhy
BABBAA nxhy
AEAAEAE nxgz BEBBEBE nxgz
Start
Channel Probing
NABh N
BAh
Shared Key
Key Extraction
Two Phase Approach:
Phase I: Channel Probing andEstimation: (hNAB, h
NBA)
Phase 2: Source Reconciliationand Key Extraction
Secret-Key Generation: CapacityLimits
Feb 11, 2013 5/ 20
![Page 7: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/7.jpg)
Secret-Key Generation - Source ModelMaurer (’93), Ahlswede-Csiszar (’93)
Terminal A Terminal BPublic Discussion Channel
Eavesdropper
NAX N
BX
KA KB
DMMS Model: (xNA , xNB ) ∼
∏Ni=1 pxA,xB (xA(i), xB(i))
Interactive Public Communication: F
Key Generation: ki = Fi(xNi ,F), i ∈ {A,B}.
Reliability: Pr(kA 6= kB) ≤ εN ,
Secrecy: 1N I(kA;F) ≤ εN
Secret-Key Rate: R = 1NH(kA)
Feb 11, 2013 6/ 20
![Page 8: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/8.jpg)
Secret-Key Generation - Source ModelMaurer (’93), Csiszar-Ahlswede (’93)
Slepian-WolfEncoder
Slepian-WolfDecoder
Eavesdropper
NAX N
BX
KAKB
FA
NAX
FA
FA
Key-DistillationKey-Distillation
Capacity: C = I(xA; xB)
One-Round of Communication
Capacity Unknown when Eavesdropper also observes a sourcesequence
Feb 11, 2013 7/ 20
![Page 9: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/9.jpg)
Problem Setup
A B
E
KAKB
Forward Link
Reverse Link
mA mBABAABB nxhy +=
BABBAA nxhy +=
AEAAEAE nxgz += BEBBEBE nxgz +=
Two-Way Reciprocal Fading Channel
yB(i) = hAB(i)xA(i) + nAB(i), yA(i) = hBA(i)xB(i) + nBA(i)
zA(i) = gA(i)xA(i) + nAE(i), zB(i) = gB(i)xB(i) + nBE(i)
Feb 11, 2013 8/ 20
![Page 10: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/10.jpg)
Problem Setup
A B
E
KAKB
Forward Link
Reverse Link
mA mBABAABB nxhy +=
BABBAA nxhy +=
AEAAEAE nxgz += BEBBEBE nxgz +=
Two-Way Reciprocal Fading Channel
yB(i) = hAB(i)xA(i) + nAB(i), yA(i) = hBA(i)xB(i) + nBA(i)
zA(i) = gA(i)xA(i) + nAE(i), zB(i) = gB(i)xB(i) + nBE(i)
Channel Model Assumptions:Non-Coherent Model: hAB(i) and hBA(i)
Perfect Eavesdropper CSI: gA(i) & gB(i) known to EveBlock-Fading Channel with Coherence Period: T .Approximate Reciprocity: (hAB, hBA) ∼ phAB ,hBA
(·, ·)Independence: (gA, gB) ⊥ (hAB, hBA)
Feb 11, 2013 8/ 20
![Page 11: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/11.jpg)
Problem Setup
A B
E
KAKB
Forward Link
Reverse Link
mA mBABAABB nxhy +=
BABBAA nxhy +=
AEAAEAE nxgz += BEBBEBE nxgz +=
Two-Way Reciprocal Fading Channel
yB(i) = hAB(i)xA(i) + nAB(i), yA(i) = hBA(i)xB(i) + nBA(i)
zA(i) = gA(i)xA(i) + nAE(i), zB(i) = gB(i)xB(i) + nBE(i)
Secret-Key Agreement Protocols:Interactive: xA(i) = fA(mA, y
i−1A ), xB(i) = fB(mB, y
i−1B )
Average Power Constraints E[|xA|2] ≤ P, E[|xB|2] ≤ P .kA = KA(yNA ,mA), kB = KB(yNB ,mB)
Reliability and Secrecy Constraint.Secret-Key Capacity
Feb 11, 2013 8/ 20
![Page 12: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/12.jpg)
Outline
Upper Bound
Lower Bound — With Public Discussion
Lower Bound — No Public Discussion
Asymptotic Regimes and Numerical Results
Feb 11, 2013 9/ 20
![Page 13: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/13.jpg)
Secret-Key Capacity — Upper BoundKhisti’12
Theorem
An upper bound on the secret-key capacity is C ≤ R+:
R+=1
TI(hAB; hBA) + max
P (hAB)∈PE
[log
(1 +
P (hAB)|hAB|2)1 + P (hAB)|gA|2
)]+ max
P (hBA)∈PE
[log
(1 +
P (hBA)|hBA|2
1 + P (hBA)|gB|2)
)]where P (hAB) and P (hBA) are power allocation function acrossthe fading states.
Feb 11, 2013 10/ 20
![Page 14: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/14.jpg)
Secret-Key Capacity — Upper Bound
Genie-Aided Channel:
A B
hAB
hBA
xA yBxByA
Feb 11, 2013 11/ 20
![Page 15: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/15.jpg)
Secret-Key Capacity — Upper Bound
Genie-Aided Channel:
A B
hAB
hBA
xA yBxByA
NTR ≤ I(mA, hNBA, y
NTA ;mB, h
NAB, y
NTB |zNT , gN )
Feb 11, 2013 11/ 20
![Page 16: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/16.jpg)
Secret-Key Capacity — Upper Bound
Genie-Aided Channel:
A B
hAB
hBA
xA yBxByA
NTR ≤ I(mA, hNBA, y
NTA ;mB, h
NAB, y
NTB |zNT , gN )
≤ I(xA(NT ); yB(NT )|hAB(N), zA(NT ), gA(N))
+ I(xB(NT ); yA(NT )|hBA(N), zB(NT ), gB(N))
+ I(mA, hNBA, y
NT−1A ;mB, h
NAB, y
NT−1B |zNT−1, gN )
Feb 11, 2013 11/ 20
![Page 17: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/17.jpg)
Secret-Key Capacity — Upper Bound
Genie-Aided Channel:
A B
hAB
hBA
xA yBxByA
NTR ≤ I(mA, hNBA, y
NTA ;mB, h
NAB, y
NTB |zNT , gN )
≤NT∑n=1
I(xA(n); yB(n)|hAB(n), zA(n), gA(n))
+
NT∑n=1
I(xB(n); yA(n)|hBA(n), zB(n), gB(n))
+NI(hAB; hBA)Feb 11, 2013 11/ 20
![Page 18: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/18.jpg)
Secret-Key Capacity — Upper Bound
Genie-Aided Channel:
A B
hAB
hBA
xA yBxByA
Interpretation of the Upper Bound:
Channel Reciprocity: 1T I(hAB; hBA)
Forward Channel: I(yB; xA|hAB, zA, gA)
Reverse Channel: I(yA; xB|hBA, zB, gB)
Feb 11, 2013 11/ 20
![Page 19: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/19.jpg)
Secret-Key Capacity — Upper Bound
Genie-Aided Channel:
A B
hAB
hBA
xA yBxByA
Public Discussion Channel
Interpretation of the Upper Bound:
Channel Reciprocity: 1T I(hAB; hBA)
Forward Channel: I(yB; xA|hAB, zA, gA)
Reverse Channel: I(yA; xB|hBA, zB, gB)
Upper Bound also holds if a public discussion channel is available.
Feb 11, 2013 11/ 20
![Page 20: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/20.jpg)
Lower Bound: Separation Based SchemeKhisti ’12
N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 )1P 1P 1P
T T
Randomness Sharing Randomness Sharing
Training: xA(i, 1) =√P1
Randomness Sharing: xA(i, t) ∼ CN (0, P2) for t = 2, . . . , TxA(i) = [xA(i, 2), . . . , xA(i, T )] ∈ CT−1.
Training: hAB(i) and hBA(i)
Correlated Sources:Forward Channel: yB(i) = hAB(i)xA(i) + nB(i) ∈ CT−1,Reverse Channel: yA(i) = hBA(i)xB(i) + nA(i) ∈ CT−1.
Feb 11, 2013 12/ 20
![Page 21: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/21.jpg)
Lower Bound: Separation Based SchemeKhisti ’12
N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 )1P 1P 1P
T T
Randomness Sharing Randomness Sharing
A B E
Channel State hKBA hKAB (gKA , g
KB )
Forward Channel xKA yKB zKAReverse Channel yKA xKB zKB
Feb 11, 2013 12/ 20
![Page 22: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/22.jpg)
Lower Bound: Separation Based SchemeKhisti ’12
N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 ) N(0,P2 )1P 1P 1P
T T
Randomness Sharing Randomness Sharing
A B E
Channel State hKBA hKAB (gKA , g
KB )
Forward Channel xKA yKB zKAReverse Channel yKA xKB zKB
Generate a secret-key from these sequences.
Feb 11, 2013 12/ 20
![Page 23: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/23.jpg)
Lower Bound — Overview
Start
Channel Probing KABh K
BAh
Channel Reconciliation
Randomness SharingKT
AY KTBY
Source Reconciliation
Secret-KeyExtraction
Shared Key
KABh K
BAh
KABh K
BAh KTAY KT
BY
Feb 11, 2013 13/ 20
![Page 24: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/24.jpg)
Achievable Rate with Public Discussion
Theorem (Public Discussion)
An achievable rate when a public discussion channel is available is
Rkey =
{1
TI(hAB; hBA)︸ ︷︷ ︸
Training
+T − 1
T
[I(yB; xA, hAB)− I(yB; zA, gA, hAB)
]︸ ︷︷ ︸
Forward Channel
+T − 1
T
[I(yA; xB, hBA))− I(yA; zB, gB, hBA)
]︸ ︷︷ ︸
Reverse Channel
}
Feb 11, 2013 14/ 20
![Page 25: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/25.jpg)
Achievable Rate with Public Discussion
Theorem (Public Discussion)
An achievable rate when a public discussion channel is available is
Rkey =
{1
TI(hAB; hBA)︸ ︷︷ ︸
Training
+T − 1
T
[I(yB; xA, hAB)− I(yB; zA, gA, hAB)
]︸ ︷︷ ︸
Forward Channel
+T − 1
T
[I(yA; xB, hBA))− I(yA; zB, gB, hBA)
]︸ ︷︷ ︸
Reverse Channel
}
Feb 11, 2013 14/ 20
![Page 26: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/26.jpg)
High SNR Regime
Theorem
In the high SNR regime our upper and lower bound (with publicdiscussion) coincide:
limP→∞
{R+(P )−R−PD(P )
}≤ c
T
where
c =E
[log
(1 +|hAB|2
|gA|2
)]+E
[log
(1 +|hBA|2
|gB|2
)]
Feb 11, 2013 15/ 20
![Page 27: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/27.jpg)
Lower BoundWithout Public Discussion
Training Communication
(T-1)
K Coherence Blocks 1 K
Public Discussion
2 K
Transmission
Phase Coherence BlocksProbing + Randomness Sharing K
Channel-Sequence Reconciliation ε1 ·KSource-Sequence Reconciliation ε2 ·K
Feb 11, 2013 16/ 20
![Page 28: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/28.jpg)
Numerical Plot
SNR =35 dB, h1, h2 ∼ CN (0, 1), ρ = 0.99.
0 5 10 15 200
0.5
1
1.5
2
2.5
3
3.5
4
Coherence Period (T)
Rat
e (n
ats/
sym
bol)
Lower BoundUpper BoundPublic DiscussionTraining
Feb 11, 2013 17/ 20
![Page 29: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/29.jpg)
Symmetric MIMO ExtensionM. Andersson, A. Khisti and M. Skoglund, 2012
yB = HABxA + nAB, zA = GAExA + nAE
yA = HBAxB + nBA, zB = GBExB + nBE
HA,HB ∈ CM×M , GAE ,GBE ∈ CNE×M
Independent Rayleigh Fading, Approximate Reciprocity
Block Fading with Coherence Period T
T ≥M ≥ NE
Training + Source Emulation achieves degrees of freedom given by:
d = maxM?∈[1,M ]
2(T −M?)(M? −NE)
T
Feb 11, 2013 18/ 20
![Page 30: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/30.jpg)
Conclusions
Secret-Key Agreement in Two-Way fading channels
Upper and Lower Bounds on Capacity
Asymptotic Optimality
Significant Gains over Training Based Schemes
Future Work:
Upper Bounds with Perfect Reciprocity (See AlsoLai-Liang-Poor ’12)
Stationary Fading Channels
Low SNR Regime
Stronger Eavesdropper Channels
Feb 11, 2013 19/ 20
![Page 31: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/31.jpg)
Error ReconciliationPublic Discussion Channel, Discrete-Valued Sequences
Reconciliation of Channel-Estimate Sequences (hKAB, hKBA)
Discussion ChannelA B
KBAh K
ABh
AKBAh
Discussion ChannelA B
KBAh K
ABh
BKABh
)ˆ,ˆ(ˆ KBA
KAB
K hhh
1
KH(φA) ≈ H(hAB|hBA),
1
KH(φB) ≈ H(hBA|hAB)
Feb 11, 2013 20/ 20
![Page 32: Secret-Key Generation from Channel Reciprocity: A ...akhisti/ita2013.pdfFeb 11, 2013 4/ 20. Secret-Key Generation: A Systems Approach II A. B. E. K. A. K. B. Forward Link. Reverse](https://reader033.fdocuments.us/reader033/viewer/2022042415/5f3098edef454a7a175ef3c0/html5/thumbnails/32.jpg)
Error ReconciliationPublic Discussion Channel, Discrete-Valued Sequences
Reconciliation of Channel-Estimate Sequences (hKAB, hKBA)
Discussion ChannelA B
KBAh K
ABh
AKBAh
Discussion ChannelA B
KBAh K
ABh
BKABh
)ˆ,ˆ(ˆ KBA
KAB
K hhh
Reconciliation of (yKA , yKB )
Discussion ChannelA B
KAy KK
B hx ˆ,
AKAy
Discussion ChannelA B
KKA hx ˆ, K
By
BKBy
)ˆ,,( KKB
KA hyyCommon Sequence:
1
TKH(ψA) ≈ H(yA|xB, hAB, hBA),
1
TKH(ψB) ≈ H(yB|xA, hAB, hBA)
Feb 11, 2013 20/ 20