On the Discussion Rate Region for the PIN Model Qiaoqiao Zhou - - PowerPoint PPT Presentation

On the Discussion Rate Region for the PIN Model

Qiaoqiao Zhou (zq115@ie.cuhk.edu.hk) Institute of Network Coding The Chinese University of Hong Kong

1

Joint work with Chung Chan and Raymond Yeung

Pairwise Independent Network (PIN) model

[Ye and Reznik 07] [Nitinawarat et al. 10a]

Secret Key Generation by Public Discussion

2

3 1 2 Public discussion Secret key

F1 := Xa ⊕ Xb

K := Xa

Recoverability: F1 ⊕ Xb = Xa

Secrecy:

I(K ∧ F1) = 0

[Maurer 93] [Ahlswede and Csiszár 93]

Multiterminal Secret Key Generation

[Csiszár and Narayan 04]

3

A DMMS ZV := (Zi | i ∈ V )

• Private observation:
• Public discussion:
• Key generation:

Fit := fit(Zn

i , ˜

Fit) for i ∈ V t ∈ [r]

Fi := (Fit | t ∈ [r]) and F := (Fi | i ∈ V )

Ki := θi(Zn

i , F)

for i ∈ V

lim

n→∞ Pr(9i 2 V, K 6= Ki) = 0

The secret key is required to satisfy

lim

n→∞

1 n[log |K| − H(K|F)] = 0

(recoverability) (secrecy)

Zn

i

for i ∈ V

Note that does not depend on and can be any finite positive integer.

r

n

4

We say a secret key rate and discussion rate tuple is achievable if there exists a sequence of in 𝑜 satisfying

(F, K)

Achievable rate region:

rK

rV := (ri | i ∈ V )

0 ≤ rK ≤ lim

n→∞

1 n log |K|, ri ≥ lim

n→∞

1 n log |Fi|, ∀i ∈ V

CS(ZV ) := max{rK | (rK, rV ) ∈ R}

Secrecy capacity: Discussion rate region: RS := {rV | (CS(ZV ), rV ) ∈ R}

The discussion rate region RS

Communication complexity: RS := min{

:=P

i∈V ri

z}|{ r(V ) | rV ∈ RS}

Related Works

5

[Csiszar and Narayan 04]

: Partial characterizations in [Tyagi 13] [Muhkerjee et al. 16] [Chan et al. 18] [Chan et al. 19] CS(ZV ) = H(ZV ) − RCO

RCO := min{r(V ) |

Slepian-Wolf constraints

z }| { r(B) ≥ H(ZB|ZV \B) ∀B ( V }

RS

R

[Liu et al. 17] [Z. and Chan 20]

: Multi-letter characterization for two-terminal sources Minimally connected hypergraphical sources

The Pairwise Independent Network (PIN) Model

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is said to be a pairwise independent network (PIN) model if there is a graph and some mutually independent edge variables such that

ZV

Z1 := (Xa, Xc) Z2 := (Xa, Xb ) Z3 := ( Xb, Xc)

1 2 3

a b c

G = (V, E)

Xe for e ∈ E

Zi := (Xe | e ∈ E, i ∈ e), ∀i ∈ V.

[Ye and Reznik 07] [Nitinawarat et al. 10a]

Tree Packing Protocol [Nitinawarat et al. 10b]

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Spanning trees , are a fractional spanning tree packing of the associated graph of a PIN model if

T`, ` ∈ [k], with weight µ` > 0, ` ∈ [k] respectively

X

`∈[k]:{i,j}∈T`

µ` ≤ w({i, j}), ∀ i, j

1 2 3

0.5 0.5

1 2 3

0.5 0.5

1 2 3

0.5 0.5

For a PIN model, we have Furthermore,

(rK, rV ) ∈ R with

1 2 3

1 1 1

rK := X

`∈[k]

µ` and ri := X

`∈[k]

[dT`(i) − 1]µ`, i ∈ V

CS(ZV ) = max X

`∈[k]

µ`

Multivariate Mutual Information (MMI)

8

CS(ZV ) = I(ZV ) := min

P∈Π0(V )

D ✓ PZV

• Q

C∈P

PZC ◆ |P| − 1 | {z }

IP(ZV ):=

[Csiszar and Narayan 04] [Chan and Zheng 10]

1 2 3

a b c

𝐽 Z{",$,%} = min 𝐽 Z" ∧ Z $,% , 𝐽 Z ",$ ∧ Z% , 𝐽 Z$ ∧ Z{",%} , 𝐼 Z" + 𝐼 Z$ + 𝐼 Z% − 𝐼 Z ",$,% 2 = 1.5 Z1 := (Xa, Xc) Z2 := (Xa, Xb ) Z3 := ( Xb, Xc)

Theorem [Chan et al. 15] There is a fundamental partition , i.e., the optimal partition with maximal cardinality.

P∗(ZV )

P∗(ZV ) = {{1}, {2}, {3}}

𝑊

C1

C2

C3

P = {C1, C2, C3}

PIN model with a unique tree packing discussion rate vector

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We say a PIN model admits a unique tree packing discussion rate vector, if all the maximal fractional spanning tree packings of the associated graph has the same discussion rate vector

(r1, r2, r3) = (0.5, 0.5, 0.5)

1 2 3

1 1 1

(r1, r2, r3, r4) = (2, 2, 0.5, 0.5)

2 1 4 3

3 1 1 1 1 1

1 2 3

0.5 0.5

1 2 3

0.5 0.5

1 2 3

0.5 0.5

for PIN model with a unique tree packing discussion rate vector

10

RS

RS = 8 < :rV ∈ R|V |

+

| ri ≥ X

`∈[k]

[dT`(i) − 1] µ` 9 = ;

where {(T`, µ`) | ` ∈ [k]} is a maximal fractional spanning tree packing

• f the graph

associated with the PIN model, dT`(i) is the degree of node i in tree T`.

RS = 8 < : (r1, r2, r3) ∈ R3

+

• r1 ≥ 0.5,

r2 ≥ 0.5, r3 ≥ 0.5. 9 = ;

Z1 := (Xa, Xc) Z2 := (Xa, Xb ) Z3 := ( Xb, Xc)

1 2 3

a b c

Proof Sketch

11

For each i ∈ V , choose proper P ∈ Π0(V \ {i})

ri ≥(|P| − 1)[rK − IP(ZV \{i})]

= (|P| − 1) X

`∈[k]

µ` − (|P| − 1)IP(ZV \{i})

≥

=

= (|P| − 1) X

`∈[k]

µ` − crossing bits in G/{i}

(|P| − 1) X

`∈[k]

µ` − X

`∈[k]

|crossing edges in T /{i}|µ`

| {z }

=|P|−dT`(i)

= X

`∈[k]

[dT`(i) − 1]µ`

For any achievable (rK, rV ) ∈ R, r(B) ≥ (|P| − 1)[rK − IP(ZV \B)] for any B ⊆ V : |V \ B| ≥ 2 , P ∈ Π0(V \ B).

Helper-Set Bound [Chan et al. 17]

For a PIN model with , it admits a unique tree packing discussion rate vector Consequently,

Special Case

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P∗(ZV ) = {{i} | i ∈ V }

(ri | i ∈ V ) with ri = H(Zi) − I(ZV )

RS = n rV ∈ R|V |

+

| ri ≥ H(Zi) − I(ZV )

• <latexit 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</latexit>

Z1 := (Xa, Xd) Z2 := (Xa, Xb ) Z3 := ( Xb, Xc ) Z4 := ( Xc, Xd)

2 1 4 3

a b c d

RS = 8 > > < > > : (r1, r2, r3, r4) ∈ R4

+

• r1 ≥ 2

3,

r2 ≥ 2

3,

r3 ≥ 2

3,

r4 ≥ 2

3.

9 > > = > > ;

because H(Zi) − I(ZV ) = 2 − 4

3 = 2 3, i ∈ {1, 2, 3, 4}.

I(ZV ) = 4 3 with P∗(ZV ) = {{1}, {2}, {3}, {4}}

Extensions & Future Works

• Identify another classes of sources that it is tight under Helper-Set

bound

• Improve Helper-Set bound to characterize the achievable region of

PIN model

13

2 1 4 3 5

References

[Maurer 93] U.M. Maurer, "Secrecy key agreement by public discussion from common information." IEEE TIT, 1993. [Ahlswede and Csiszár 93] R. Ahlswede and I. Csiszár, "Common randomness in information theory and cryptography—Part I: Secret sharing." IEEE TIT, 1993. [Csiszár and Narayan 04] I. Csiszár and P. Narayan, "Secrecy capacities for multiple terminals." IEEE TIT, 2004. [Ye and Reznik 07] C. Ye and A. Reznik, ”Group secret key generation algorithms." IEEE ISIT, 2007. [Chan and Zheng 10] C. Chan and L. Zheng, "Mutual dependence for secret key agreement." CISS2010. [Nitinawarat et al. 10a] S. Nitinawarat and P. Narayan, “Perfect omniscience, perfect secrecy, and steiner tree packing.” IEEE TIT, 2010. [Nitinawarat et al. 10b] S. Nitinawarat, C. Ye, A. Barg, P. Narayan, and A. Reznik, “Secret key generation for a pairwise independent network model.” IEEE TIT, 2010. [Tyagi 13] H. Tyagi, “Common information and secret key capacity,” IEEE TIT, 2013. [Chan et al. 15] C. Chan, A. Al-Bashabsheh, J. B. Ebrahimi, T. Kaced, and T. Liu, “Multivariate mutual information inspired by secret-key agreement.” Proc.

• f the IEEE, 2015.

[Chan et al. 16] C. Chan, A. Al-Bashabsheh, Q. Zhou, T. Kaced, and T. Liu, “Info-clustering: A mathematical theory of data clustering,” TMBMC, 2016. [Mukherjee et al. 16] M. Mukherjee, N. Kashyap, and Y. Sankarasubramaniam, “On the public communication needed to achieve sk capacity in the multiterminal source model.” IEEE TIT, 2016. [Liu et al. 17] J. Liu, P. W. Cuff, and S. Verdú, “Secret key generation with limited interaction,” IEEE TIT, 2017. [Chan et al. 17] C. Chan, M. Mukherjee, N. Kashyap, and Q. Zhou, “Secret key agreement under discussion rate constraints,” IEEE ISIT, 2017. [Chan et al. 18] C. Chan, M. Mukherjee, N. Kashyap, and Q. Zhou, “On the optimality of secret key agreement via omniscience,” IEEE TIT, 2018. [Chan et al. 19] C. Chan, M. Mukherjee, N. Kashyap, and Q. Zhou, “Upper bound via lamination on the constrained secrecy capacity of hypergraphical sources,” IEEE TIT, 2019. [Z. and Chan 20] Q. Zhou and C. Chan, “Secret key generation for minimally connected hypergraphical sources,” IEEE TIT, 2020.

14

Thank You!