The Sender-Excited Secret Key Agreement Model: Capacity and Error - - PowerPoint PPT Presentation

▶
the sender excited secret key agreement model capacity
SMART_READER_LITE
LIVE PREVIEW

The Sender-Excited Secret Key Agreement Model: Capacity and Error - - PowerPoint PPT Presentation

Nov 08, 2023 180 likes •920 views

The Sender-Excited Secret Key Agreement Model: Capacity and Error Exponents Tzu-Han Chou, Vincent Y. F. Tan , Stark C. Draper Department of Electrical and Computer Engineering, University of Wisconsin-Madison Allerton (Sep 2011) Vincent Tan

slide-1
SLIDE 1

The Sender-Excited Secret Key Agreement Model: Capacity and Error Exponents

Tzu-Han Chou, Vincent Y. F. Tan, Stark C. Draper

Department of Electrical and Computer Engineering, University of Wisconsin-Madison

Allerton (Sep 2011)

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 1 / 18

slide-2
SLIDE 2

Joint work with

Tzu-Han Chou Qualcomm Stark C. Draper UW-Madison

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 2 / 18

slide-3
SLIDE 3

Introduction

Consider the fundamental limits of the secret key generation problem

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 3 / 18

slide-4
SLIDE 4

Introduction

Consider the fundamental limits of the secret key generation problem There is a noiseless public discussion channel

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 3 / 18

slide-5
SLIDE 5

Introduction

Consider the fundamental limits of the secret key generation problem There is a noiseless public discussion channel Source is randomly excited by the sender

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 3 / 18

slide-6
SLIDE 6

Introduction

Consider the fundamental limits of the secret key generation problem There is a noiseless public discussion channel Source is randomly excited by the sender Motivated by

Key generation [Maurer, Ahlswede and Csiszár] Key generation with external excitation [Chou, Draper and Sayeed]

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 3 / 18

slide-7
SLIDE 7

Introduction

Consider the fundamental limits of the secret key generation problem There is a noiseless public discussion channel Source is randomly excited by the sender Motivated by

Key generation [Maurer, Ahlswede and Csiszár] Key generation with external excitation [Chou, Draper and Sayeed] Channels with action-dependent states [Weissman]

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 3 / 18

slide-8
SLIDE 8

Introduction

Consider the fundamental limits of the secret key generation problem There is a noiseless public discussion channel Source is randomly excited by the sender Motivated by

Key generation [Maurer, Ahlswede and Csiszár] Key generation with external excitation [Chou, Draper and Sayeed] Channels with action-dependent states [Weissman]

Main contributions:

Secret key capacity Inner bound for rate-reliability-secrecy-exponent region

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 3 / 18

slide-9
SLIDE 9

Wiretap Channel [Wyner, Csiszár and Körner]

✲ ✲ ✲ ✲

p(y, z|s) Dec Enc M ∈ [2nR] Alice ˆ M Bob Y n Sn

✲

Z n Eve Want to transmit message reliably to Bob but keep Eve ignorant P( ˆ M = M) → 0 and 1

nI(M; Z n) → 0

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 4 / 18

slide-10
SLIDE 10

Wiretap Channel [Wyner, Csiszár and Körner]

✲ ✲ ✲ ✲

p(y, z|s) Dec Enc M ∈ [2nR] Alice ˆ M Bob Y n Sn

✲

Z n Eve Want to transmit message reliably to Bob but keep Eve ignorant P( ˆ M = M) → 0 and 1

nI(M; Z n) → 0

Wiretap channel capacity Cwiretap = max

U−S−(Y,Z) {I(U; Y) − I(U; Z)}

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 4 / 18

slide-11
SLIDE 11

Wiretap Channel [Wyner, Csiszár and Körner]

✲ ✲ ✲ ✲

p(y, z|s) Dec Enc M ∈ [2nR] Alice ˆ M Bob Y n Sn

✲

Z n Eve Want to transmit message reliably to Bob but keep Eve ignorant P( ˆ M = M) → 0 and 1

nI(M; Z n) → 0

Wiretap channel capacity Cwiretap = max

U−S−(Y,Z) {I(U; Y) − I(U; Z)}

Channel-type model

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 4 / 18

slide-12
SLIDE 12

Secret Key Generation [Maurer, Ahlswede & Csiszár]

Public Channel

Alice Bob Eve p(x, y, z)

❄

Y n

❄

X n

❄

Z n

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 5 / 18

slide-13
SLIDE 13

Secret Key Generation [Maurer, Ahlswede & Csiszár]

Public Channel

Alice Bob Eve p(x, y, z)

❄

Y n

❄

X n

❄

Z n

❄

Φ

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 5 / 18

slide-14
SLIDE 14

Secret Key Generation [Maurer, Ahlswede & Csiszár]

Public Channel

Alice Bob Eve p(x, y, z)

❄

Y n

❄

X n

❄

Z n

❄

Φ Φ ✻ Φ ✻

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 5 / 18

slide-15
SLIDE 15

Secret Key Generation [Maurer, Ahlswede & Csiszár]

Public Channel

Alice Bob Eve p(x, y, z)

❄

Y n

❄

X n

❄

Z n

❄

Φ Φ ✻ Φ ✻

❄

KA

❄

KB

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 5 / 18

slide-16
SLIDE 16

Secret Key Generation [Maurer, Ahlswede & Csiszár]

Public Channel

Alice Bob Eve p(x, y, z)

❄

Y n

❄

X n

❄

Z n

❄

Φ Φ ✻ Φ ✻

❄

KA

❄

KB Secret keys are generated from dependent sources X, Y, Z

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 5 / 18

slide-17
SLIDE 17

Secret Key Generation [Maurer, Ahlswede & Csiszár]

Public Channel

Alice Bob Eve p(x, y, z)

❄

Y n

❄

X n

❄

Z n

❄

Φ Φ ✻ Φ ✻

❄

KA

❄

KB Secret keys are generated from dependent sources X, Y, Z P(KA = KB) → 0 and 1

nI(KA; Z n, Φ) → 0

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 5 / 18

slide-18
SLIDE 18

Secret Key Generation [Maurer, Ahlswede & Csiszár]

Public Channel

Alice Bob Eve p(x, y, z)

❄

Y n

❄

X n

❄

Z n

❄

Φ Φ ✻ Φ ✻

❄

KA

❄

KB Secret keys are generated from dependent sources X, Y, Z P(KA = KB) → 0 and 1

nI(KA; Z n, Φ) → 0

Secret key capacity CSK = max

W−U−X−(Y,Z){I(U; Y|W) − I(U; Z|W)}

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 5 / 18

slide-19
SLIDE 19

Secret Key Generation [Maurer, Ahlswede & Csiszár]

Public Channel

Alice Bob Eve p(x, y, z)

❄

Y n

❄

X n

❄

Z n

❄

Φ Φ ✻ Φ ✻

❄

KA

❄

KB Secret keys are generated from dependent sources X, Y, Z P(KA = KB) → 0 and 1

nI(KA; Z n, Φ) → 0

Secret key capacity CSK = max

W−U−X−(Y,Z){I(U; Y|W) − I(U; Z|W)}

Source-type model

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 5 / 18

slide-20
SLIDE 20

Key Generation with External Excitation [Chou et al.]

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

Φ

❄

KA

❄

KB Φ ✻ Φ ✻

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 6 / 18

slide-21
SLIDE 21

Key Generation with External Excitation [Chou et al.]

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

Φ

❄

KA

❄

KB Φ ✻ Φ ✻

❄

sn

Wireless channels ⇒ auxiliary randomness Due to multipath fading Transmissions are bi-directional ⇒ X, Y, Z generated by transmitting prearranged sounding signals.

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 6 / 18

slide-22
SLIDE 22

Key Generation with External Excitation [Chou et al.]

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

Φ

❄

KA

❄

KB Φ ✻ Φ ✻

❄

sn

Wireless channels ⇒ auxiliary randomness Due to multipath fading Transmissions are bi-directional ⇒ X, Y, Z generated by transmitting prearranged sounding signals.

External excitation via a deterministic sounding signal sn

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 6 / 18

slide-23
SLIDE 23

Key Generation with External Excitation [Chou et al.]

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

Φ

❄

KA

❄

KB Φ ✻ Φ ✻

❄

sn

Wireless channels ⇒ auxiliary randomness Due to multipath fading Transmissions are bi-directional ⇒ X, Y, Z generated by transmitting prearranged sounding signals.

External excitation via a deterministic sounding signal sn Secret key capacity CSK = max

p(w,u|s),p(x|u,s),p(s){I(U; Y|W, S) − I(U; Z|W, S)}

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 6 / 18

slide-24
SLIDE 24

Our model: Sender-Excited Secret Key Agreement

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

KA

❄

KB

❄ ✻

Φ Φ Φ

✻

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 7 / 18

slide-25
SLIDE 25

Our model: Sender-Excited Secret Key Agreement

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

KA

❄

KB

❄ ✻

Φ Φ Φ

✻ ❄

M

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 7 / 18

slide-26
SLIDE 26

Our model: Sender-Excited Secret Key Agreement

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

KA

❄

KB

❄ ✻

Φ Φ Φ

✻ ❄

M

❄

Sn(M)

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 7 / 18

slide-27
SLIDE 27

Our model: Sender-Excited Secret Key Agreement

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

KA

❄

KB

❄ ✻

Φ Φ Φ

✻ ❄

M

❄

Sn(M) A (2nRM, 2nRΦ, n, Γ) code consists of a uniform M ∈ [2nRM] and

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 7 / 18

slide-28
SLIDE 28

Our model: Sender-Excited Secret Key Agreement

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

KA

❄

KB

❄ ✻

Φ Φ Φ

✻ ❄

M

❄

Sn(M) A (2nRM, 2nRΦ, n, Γ) code consists of a uniform M ∈ [2nRM] and Channel Excitation: sn = sn(m) such that 1

n

n

i=1 Λ(si(m)) ≤ Γ

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 7 / 18

slide-29
SLIDE 29

Our model: Sender-Excited Secret Key Agreement

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

KA

❄

KB

❄ ✻

Φ Φ Φ

✻ ❄

M

❄

Sn(M) A (2nRM, 2nRΦ, n, Γ) code consists of a uniform M ∈ [2nRM] and Channel Excitation: sn = sn(m) such that 1

n

n

i=1 Λ(si(m)) ≤ Γ

One-way Public Discussion: Alice generates a public message φ = φ(m, xn) ∈ [2nRΦ] and transmits it over a noiseless channel

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 7 / 18

slide-30
SLIDE 30

Our model: Sender-Excited Secret Key Agreement

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

KA

❄

KB

❄ ✻

Φ Φ Φ

✻ ❄

M

❄

Sn(M) A (2nRM, 2nRΦ, n, Γ) code consists of a uniform M ∈ [2nRM] and Channel Excitation: sn = sn(m) such that 1

n

n

i=1 Λ(si(m)) ≤ Γ

One-way Public Discussion: Alice generates a public message φ = φ(m, xn) ∈ [2nRΦ] and transmits it over a noiseless channel Key Generation: kA = kA(m, xn) ∈ N and kB = kB(φ, yn) ∈ N

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 7 / 18

slide-31
SLIDE 31

Motivation for our model

Combine the source-type model with the wiretap channel model to extract higher SK rate?

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 8 / 18

slide-32
SLIDE 32

Motivation for our model

Combine the source-type model with the wiretap channel model to extract higher SK rate? Can parties (or sender) excite the source with private source of randomness M?

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 8 / 18

slide-33
SLIDE 33

Motivation for our model

Combine the source-type model with the wiretap channel model to extract higher SK rate? Can parties (or sender) excite the source with private source of randomness M? Our model is also inspired by

Channels with action-dependent states [Weissman 2010] Probing capacity [Asnani et al. 2010]

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 8 / 18

slide-34
SLIDE 34

Motivation for our model

Combine the source-type model with the wiretap channel model to extract higher SK rate? Can parties (or sender) excite the source with private source of randomness M? Our model is also inspired by

Channels with action-dependent states [Weissman 2010] Probing capacity [Asnani et al. 2010] Key generation when encoder and decoder have state information [Khisti, Diggavi, Wornell 2011]

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 8 / 18

slide-35
SLIDE 35

Weak Achievability

The rate RSK is weakly-achievable if there exists a sequence of (2nRM, 2nRΦ, n, Γ) codes such that lim

n→∞

P(KA = KB) = 0 lim

n→∞

1 nI(KA; Z n, Φ) = 0 lim inf

n→∞

1 nH(KA) ≥ RSK

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 9 / 18

slide-36
SLIDE 36

Weak Achievability

The rate RSK is weakly-achievable if there exists a sequence of (2nRM, 2nRΦ, n, Γ) codes such that lim

n→∞

P(KA = KB) = 0 lim

n→∞

1 nI(KA; Z n, Φ) = 0 lim inf

n→∞

1 nH(KA) ≥ RSK Definition ((Weak)-Secret key capacity) CSK(Γ) := sup{RSK : RSK weakly-achievable}

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 9 / 18

slide-37
SLIDE 37

Weak Achievability

The rate RSK is weakly-achievable if there exists a sequence of (2nRM, 2nRΦ, n, Γ) codes such that lim

n→∞

P(KA = KB) = 0 lim

n→∞

1 nI(KA; Z n, Φ) = 0 lim inf

n→∞

1 nH(KA) ≥ RSK Definition ((Weak)-Secret key capacity) CSK(Γ) := sup{RSK : RSK weakly-achievable} But weak secrecy 1

nI(KA; Z n, Φ) → 0 is usually not good enough

[Maurer & Wolf 2000], [Watanabe et al. 2009], [Bloch & Barros 2011], [Bloch & Laneman 2011],

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 9 / 18

slide-38
SLIDE 38

Strong Achievability

The rate-exponent triple (RSK, E, F) is achievable if there exists a sequence of (2nRM, 2nRΦ, n, Γ) codes such that lim inf

n→∞

−1 n log P(KA = KB) ≥ E, ⇔ P(KA = KB)

.

≤ 2−nE lim inf

n→∞

−1 n log I(KA; Z n, Φ) ≥ F, ⇔ I(KA; Z n, Φ)

.

≤ 2−nF lim inf

n→∞

1 nH(KA) ≥ RSK

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 10 / 18

slide-39
SLIDE 39

Strong Achievability

The rate-exponent triple (RSK, E, F) is achievable if there exists a sequence of (2nRM, 2nRΦ, n, Γ) codes such that lim inf

n→∞

−1 n log P(KA = KB) ≥ E, ⇔ P(KA = KB)

.

≤ 2−nE lim inf

n→∞

−1 n log I(KA; Z n, Φ) ≥ F, ⇔ I(KA; Z n, Φ)

.

≤ 2−nF lim inf

n→∞

1 nH(KA) ≥ RSK Definition (Capacity-reliability-secrecy region) R∗(p(x, y, z|s)) :=

  • (RSK, E, F) ∈ R3

+ : (RSK, E, F) achievable

  • Vincent Tan (UW-Madison)

Sender-Excited Secret Key Agreement Model Allerton 2011 10 / 18

slide-40
SLIDE 40

Strong Achievability

The rate-exponent triple (RSK, E, F) is achievable if there exists a sequence of (2nRM, 2nRΦ, n, Γ) codes such that lim inf

n→∞

−1 n log P(KA = KB) ≥ E, ⇔ P(KA = KB)

.

≤ 2−nE lim inf

n→∞

−1 n log I(KA; Z n, Φ) ≥ F, ⇔ I(KA; Z n, Φ)

.

≤ 2−nF lim inf

n→∞

1 nH(KA) ≥ RSK Definition (Capacity-reliability-secrecy region) R∗(p(x, y, z|s)) :=

  • (RSK, E, F) ∈ R3

+ : (RSK, E, F) achievable

  • Definition (Strong-achievability)

RSK is strongly-achievable if (RSK,E,F) is achievable for some E,F >0

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 10 / 18

slide-41
SLIDE 41

Capacity Result

Theorem (Secret Key Capacity for Sender-Excited Model) The secret key capacity is CSK(Γ) = max {I(U, V; Y|W) − I(U, V; Z|W)} where the max is over all joints p(w, u, v, s, x, y, z) = p(w, u)p(s|u)p(v|w, u, x)p(x, y, z|s) such that EΛ(S) ≤ Γ.

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 11 / 18

slide-42
SLIDE 42

Remarks on Capacity Result

CSK(Γ) = max{I(U, V; Y|W) − I(U, V; Z|W)}

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 12 / 18

slide-43
SLIDE 43

Remarks on Capacity Result

CSK(Γ) = max{I(U, V; Y|W) − I(U, V; Z|W)} Rate can be written as Rch + Rsrc where Rch = I(U; Y|W) − I(U; Z|W), Rsrc = I(V; Y|W, U) − I(V; Y|W, U) Rch = Confidential message rate of wiretap channel p(y, z|s) Rsrc = Secret key rate of excited source p(x, y, z|s) [Chou et al.]

Sounding signal sn deterministic Roughly, p(s) chosen to max I(V; Y|W, S) − I(V; Z|W, S)

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

Φ

❄

KA

❄

KB Φ ✻ Φ ✻

❄

sn

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 12 / 18

slide-44
SLIDE 44

Remarks on Capacity Result

CSK(Γ) = max{I(U, V; Y|W) − I(U, V; Z|W)} Rate can be written as Rch + Rsrc where Rch = I(U; Y|W) − I(U; Z|W), Rsrc = I(V; Y|W, U) − I(V; Y|W, U) Rch = Confidential message rate of wiretap channel p(y, z|s) Rsrc = Secret key rate of excited source p(x, y, z|s) [Chou et al.]

Sounding signal sn deterministic Roughly, p(s) chosen to max I(V; Y|W, S) − I(V; Z|W, S)

Public Channel

Alice Bob Eve p(x, y, z|s)

❄

Y n

❄

X n

❄

Z n

❄

Φ

❄

KA

❄

KB Φ ✻ Φ ✻

❄

sn

Capacity: Find optimal sum rate Rch + Rsrc

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 12 / 18

slide-45
SLIDE 45

Degradedness

We say that the DM-BC p(x, y, z|s) is degraded if (X, S) − Y − Z

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 13 / 18

slide-46
SLIDE 46

Degradedness

We say that the DM-BC p(x, y, z|s) is degraded if (X, S) − Y − Z Theorem (Secret Key Capacity for Degraded Sender-Excited Model) If the DM-BC p(x, y, z|s) is degraded the secret key capacity is CSK(Γ) = C(Weak)

SK

(Γ) = max

p(s):EΛ(S)≤Γ {I(X, S; Y) − I(X, S; Z)}

Also, C(Weak)

SK

(Γ) = C(Strong)

SK

(Γ)

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 13 / 18

slide-47
SLIDE 47

Degradedness

We say that the DM-BC p(x, y, z|s) is degraded if (X, S) − Y − Z Theorem (Secret Key Capacity for Degraded Sender-Excited Model) If the DM-BC p(x, y, z|s) is degraded the secret key capacity is CSK(Γ) = C(Weak)

SK

(Γ) = max

p(s):EΛ(S)≤Γ {I(X, S; Y) − I(X, S; Z)}

Also, C(Weak)

SK

(Γ) = C(Strong)

SK

(Γ) Rch = I(S; Y) − I(S; Z), Rsrc = I(X; Y|S) − I(X; Z|S)

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 13 / 18

slide-48
SLIDE 48

Binary Example

Consider the case where S, X, Y, Z ∈ F2: X = (H · S) ⊕ N1, Y = (H · S) ⊕ N2, Z = ( ˜ H · H · S) ⊕ N3

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 14 / 18

slide-49
SLIDE 49

Binary Example

Consider the case where S, X, Y, Z ∈ F2: X = (H · S) ⊕ N1, Y = (H · S) ⊕ N2, Z = ( ˜ H · H · S) ⊕ N3

0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 β RSK(β) Rch Rsrc RSK

Noises Ni indep H and ˜ H indep S ∼ Bern(β) Rch = I(S; Y) − I(S; Z) Rsrc = I(X; Y|S) − I(X; Z|S)

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 14 / 18

slide-50
SLIDE 50

Binary Example

Consider the case where S, X, Y, Z ∈ F2: X = (H · S) ⊕ N1, Y = (H · S) ⊕ N2, Z = ( ˜ H · H · S) ⊕ N3

0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 β RSK(β) Rch Rsrc RSK

Noises Ni indep H and ˜ H indep S ∼ Bern(β) Rch = I(S; Y) − I(S; Z) Rsrc = I(X; Y|S) − I(X; Z|S) Interplay between common randomness and wiretap rate

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 14 / 18

slide-51
SLIDE 51

Error Exponents: Setup

Recall that (RSK, E, F) is achievable if H(KA) ≥ n(RSK − ǫ), P(KA = KB)

.

≤ 2−nE, I(KA; Z n, Φ)

.

≤ 2−nF

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 15 / 18

slide-52
SLIDE 52

Error Exponents: Setup

Recall that (RSK, E, F) is achievable if H(KA) ≥ n(RSK − ǫ), P(KA = KB)

.

≤ 2−nE, I(KA; Z n, Φ)

.

≤ 2−nF Define reliability exponent given p(s), RΦ, RM Eo(p(s), RΦ, RM) := max

0≤ρ≤1 ρ(−RM + RΦ) − log

  • y
  • s,x

p(s)p(x, y|s)

1 1+ρ

1+ρ

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 15 / 18

slide-53
SLIDE 53

Error Exponents: Setup

Recall that (RSK, E, F) is achievable if H(KA) ≥ n(RSK − ǫ), P(KA = KB)

.

≤ 2−nE, I(KA; Z n, Φ)

.

≤ 2−nF Define reliability exponent given p(s), RΦ, RM Eo(p(s), RΦ, RM) := max

0≤ρ≤1 ρ(−RM + RΦ) − log

  • y
  • s,x

p(s)p(x, y|s)

1 1+ρ

1+ρ := max

0≤ρ≤1 −ρRM + ρRΦ − log

  • y
  • s,x

p(s)p(y|s)

1 1+ρ p(x|y, s) 1 1+ρ

1+ρ

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 15 / 18

slide-54
SLIDE 54

Error Exponents: Setup

Recall that (RSK, E, F) is achievable if H(KA) ≥ n(RSK − ǫ), P(KA = KB)

.

≤ 2−nE, I(KA; Z n, Φ)

.

≤ 2−nF Define reliability exponent given p(s), RΦ, RM Eo(p(s), RΦ, RM) := max

0≤ρ≤1 ρ(−RM + RΦ) − log

  • y
  • s,x

p(s)p(x, y|s)

1 1+ρ

1+ρ := max

0≤ρ≤1 −ρRM + ρRΦ − log

  • y
  • s,x

p(s)p(y|s)

1 1+ρ p(x|y, s) 1 1+ρ

1+ρ Gallager’s channel coding exponent [Gallager’s Book Ch. 5]

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 15 / 18

slide-55
SLIDE 55

Error Exponents: Setup

Recall that (RSK, E, F) is achievable if H(KA) ≥ n(RSK − ǫ), P(KA = KB)

.

≤ 2−nE, I(KA; Z n, Φ)

.

≤ 2−nF Define reliability exponent given p(s), RΦ, RM Eo(p(s), RΦ, RM) := max

0≤ρ≤1 ρ(−RM + RΦ) − log

  • y
  • s,x

p(s)p(x, y|s)

1 1+ρ

1+ρ := max

0≤ρ≤1 −ρRM + ρRΦ − log

  • y
  • s,x

p(s)p(y|s)

1 1+ρ p(x|y, s) 1 1+ρ

1+ρ Gallager’s source coding with side information exponent [1976]

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 15 / 18

slide-56
SLIDE 56

Error Exponents: Setup

Recall that (RSK, E, F) is achievable if H(KA) ≥ n(RSK − ǫ), P(KA = KB)

.

≤ 2−nE, I(KA; Z n, Φ)

.

≤ 2−nF Define reliability exponent given p(s), RΦ, RM Eo(p(s), RΦ, RM) := max

0≤ρ≤1 ρ(−RM + RΦ) − log

  • y
  • s,x

p(s)p(x, y|s)

1 1+ρ

1+ρ

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 15 / 18

slide-57
SLIDE 57

Error Exponents: Setup

Recall that (RSK, E, F) is achievable if H(KA) ≥ n(RSK − ǫ), P(KA = KB)

.

≤ 2−nE, I(KA; Z n, Φ)

.

≤ 2−nF Define reliability exponent given p(s), RΦ, RM Eo(p(s), RΦ, RM) := max

0≤ρ≤1 ρ(RΦ − RM) − log

  • y
  • s,x

p(s)p(x, y|s)

1 1+ρ

1+ρ Define secrecy exponent given p(s), RSK, RΦ, RM Fo(p(s), RSK, RΦ, RM) := sup

0<α≤1

−α(RSK + RΦ − RM) − log

  • x,z,s

p(x, z, s) p(x, z|s) p(z) α

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 15 / 18

slide-58
SLIDE 58

Error Exponents: Setup

Recall that (RSK, E, F) is achievable if H(KA) ≥ n(RSK − ǫ), P(KA = KB)

.

≤ 2−nE, I(KA; Z n, Φ)

.

≤ 2−nF Define reliability exponent given p(s), RΦ, RM Eo(p(s), RΦ, RM) := max

0≤ρ≤1 ρ(RΦ − RM) − log

  • y
  • s,x

p(s)p(x, y|s)

1 1+ρ

1+ρ Define secrecy exponent given p(s), RSK, RΦ, RM Fo(p(s), RSK, RΦ, RM) := sup

0<α≤1

−α(RSK + RΦ−RM)−log

  • x,z,s

p(x, z, s)

  • p(x|z, s)p(z|s)

p(z) α

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 15 / 18

slide-59
SLIDE 59

Error Exponents: Setup

Recall that (RSK, E, F) is achievable if H(KA) ≥ n(RSK − ǫ), P(KA = KB)

.

≤ 2−nE, I(KA; Z n, Φ)

.

≤ 2−nF Define reliability exponent given p(s), RΦ, RM Eo(p(s), RΦ, RM) := max

0≤ρ≤1 ρ(RΦ − RM) − log

  • y
  • s,x

p(s)p(x, y|s)

1 1+ρ

1+ρ Define secrecy exponent given p(s), RSK, RΦ, RM Fo(p(s), RSK, RΦ, RM) := sup

0<α≤1

−α(RSK + RΦ − RM)−log

  • x,z,s

p(x, z, s)

  • p(x|z, s)p(z|s)

p(z) α

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 15 / 18

slide-60
SLIDE 60

Error Exponents: Setup

Recall that (RSK, E, F) is achievable if H(KA) ≥ n(RSK − ǫ), P(KA = KB)

.

≤ 2−nE, I(KA; Z n, Φ)

.

≤ 2−nF Define reliability exponent given p(s), RΦ, RM Eo(p(s), RΦ, RM) := max

0≤ρ≤1 ρ(RΦ − RM) − log

  • y
  • s,x

p(s)p(x, y|s)

1 1+ρ

1+ρ Define secrecy exponent given p(s), RSK, RΦ, RM Fo(p(s), RSK, RΦ, RM) := sup

0<α≤1

−α(RSK + RΦ − RM) − log

  • x,z,s

p(x, z, s) p(x, z|s) p(z) α

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 15 / 18

slide-61
SLIDE 61

Error Exponents: Result

Theorem (Inner bound to Capacity-Reliability-Secrecy Region) Let R(p(s), RΦ, RM) :=

  • (RSK, E, F) ∈ R3

+ :

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 16 / 18

slide-62
SLIDE 62

Error Exponents: Result

Theorem (Inner bound to Capacity-Reliability-Secrecy Region) Let R(p(s), RΦ, RM) :=

  • (RSK, E, F) ∈ R3

+ : E ≤ Eo(p(s), RΦ, RM),

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 16 / 18

slide-63
SLIDE 63

Error Exponents: Result

Theorem (Inner bound to Capacity-Reliability-Secrecy Region) Let R(p(s), RΦ, RM) :=

  • (RSK, E, F) ∈ R3

+ : E ≤ Eo(p(s), RΦ, RM),

F ≤ Fo(p(s), RSK, RΦ, RM)

  • .

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 16 / 18

slide-64
SLIDE 64

Error Exponents: Result

Theorem (Inner bound to Capacity-Reliability-Secrecy Region) Let R(p(s), RΦ, RM) :=

  • (RSK, E, F) ∈ R3

+ : E ≤ Eo(p(s), RΦ, RM),

F ≤ Fo(p(s), RSK, RΦ, RM)

  • . Then,
  • p(s),RΦ,RM

R(p(s), RΦ, RM) ⊂ R∗(p(x, y, z|s))

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 16 / 18

slide-65
SLIDE 65

Error Exponents: Result

Theorem (Inner bound to Capacity-Reliability-Secrecy Region) Let R(p(s), RΦ, RM) :=

  • (RSK, E, F) ∈ R3

+ : E ≤ Eo(p(s), RΦ, RM),

F ≤ Fo(p(s), RSK, RΦ, RM)

  • . Then,
  • p(s),RΦ,RM

R(p(s), RΦ, RM) ⊂ R∗(p(x, y, z|s)) Reliability exponent E:

Gallager’s channel coding exponent [1968] Gallager’s source coding with side information exponent [1976]

Secrecy exponent F:

Hayashi’s wiretap channel exponents [2006, 2011] Chou’s key agreement model with external excitation [In Press]

Strongly-achievable rates for degraded case [Preprint]

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 16 / 18

slide-66
SLIDE 66

Error Exponents: Binary Example

RM: Rate of Alice’s Private mess. RΦ: Rate of Public mess. RSK: Secret key rate

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 17 / 18

slide-67
SLIDE 67

Error Exponents: Binary Example

RM: Rate of Alice’s Private mess. RΦ: Rate of Public mess. RSK: Secret key rate

0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 RΦ Er(RΦ, RM) RM = 0.1 RM = 0.2 RM = 0.3

When RM ↑ rel. exp. ↓ When RΦ ↑ rel. exp. ↑

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 17 / 18

slide-68
SLIDE 68

Error Exponents: Binary Example

RM: Rate of Alice’s Private mess. RΦ: Rate of Public mess. RSK: Secret key rate

0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 RΦ Er(RΦ, RM) RM = 0.1 RM = 0.2 RM = 0.3

When RM ↑ rel. exp. ↓ When RΦ ↑ rel. exp. ↑

0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 RΦ Fr(RSK, RΦ, RM)

RM=0.1,RSK=0.01 RM=0.2,RSK=0.01 RM=0.3,RSK=0.01 RM=0.1,RSK=0.05 RM=0.2,RSK=0.05 RM=0.3,RSK=0.05

When RM ↑ sec. exp. ↑ When RΦ ↑ sec. exp. ↓ When RSK ↑ sec. exp. ↓

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 17 / 18

slide-69
SLIDE 69

Conclusions and Open Problems

Proposed the sender-excited model for key agreement

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 18 / 18

slide-70
SLIDE 70

Conclusions and Open Problems

Proposed the sender-excited model for key agreement Derived the capacity and an inner bound to the capacity-reliability-secrecy region

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 18 / 18

slide-71
SLIDE 71

Conclusions and Open Problems

Proposed the sender-excited model for key agreement Derived the capacity and an inner bound to the capacity-reliability-secrecy region Inner bound for multi-way discussion? Strictly better?

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 18 / 18

slide-72
SLIDE 72

Conclusions and Open Problems

Proposed the sender-excited model for key agreement Derived the capacity and an inner bound to the capacity-reliability-secrecy region Inner bound for multi-way discussion? Strictly better? Outer bound to capacity-reliability-secrecy region?

Public Channel

Alice Bob Eve p(x, y, z|s) ❄ Y n ❄ Xn ❄ Zn ❄ KA ❄ KB ❄ ✻ Φ Φ Φ ✻ ❄ M ❄ Sn(M)

Vincent Tan (UW-Madison) Sender-Excited Secret Key Agreement Model Allerton 2011 18 / 18