Secure Information Exchange for Secure Information Exchange for - - PowerPoint PPT Presentation

Secure Information Exchange for Secure Information Exchange for Omniscience Omniscience

Chung Chan (CityU) Joint work with Navin Kashyap (IISc), Praneeth Kumar Vippathalla, (IISc) and Qiaoqiao Zhou (CUHK)

Audio slides: https://www.cs.cityu.edu.hk/~ccha23/isit2020

1

Network Nodes

V Z0

Private info.

Z1 ZV Z :

n = (Z

, Z , … , Z ) i.i.d. as Z

(1) (2) (n)

Z1

n

Target info.

Y0 Y1 YV X0 X1

Censored info.

XV

Public info.

F0 F1 F

Secure information exchange Secure information exchange

Formulation Formulation

Interactive Public discussion

I(F ∧

n 1

Y ∣Z ) →

n n

u (utility) I(F ∧

n 1

Y ∣Z ) →

1 n 1 n

u1 I(F ∧

n 1

X ∣Z ) →

n n

l (leakage) I(F ∧

n 1

X ∣Z ) →

1 n 1 n

l1 H(F ) →

n 1

r0 (discussion rate) H(F ) →

n 1 1

r1

Characterize given source . R := closure{(u , ℓ , r ) achievable by some F}

V V V

PX ,Y ,Z

V V V

Related problems Related problems

Private information extraction problem [Asoodeh et al 19] and , , are null V = {1, 2} Z =

1

(X , Y )

2 2

X1 Y1 Z2 Information bottleneck [Tishby et al 99] , , and are null V = {1, 2} X1 Y1 X2 Z2 By restricting the source model, the problem reduces to:

Secure omniscience Secure omniscience

A special scenario of secure information exchange A special scenario of secure information exchange

Z1 ZU

I(F ∧

n 1

X ∣Z ) →

w n w n

ℓ (leakage)

w

X =

w

Z (censored info.)

U

U 1 w

wiretapper users

F1 F

unlimited r1 u =

1

H(Z ∣Z ) (omniscience)

U 1

Zw

Characterize . R :

L = inf{ℓ ∣(u , ℓ , r ) ∈ w V V V

R, u =

i

H(Z ∣Z ) ∀i ∈

V i

A}

helpers in U\A no target info. active users in A

Zh h Fh

unlimited rh

Example Example

With uniformly random and independent bits: , X , X , X

a b c

R =

L

ZU

X , X

a n b n

X , X

b n c n

X +

a n

X +

b n

Xc

n

A = U 1 w 2 3

X +

a n

X , X +

b n b n

Xc

n

X +

a n

Xb

n

F

X +

b n

Xc

n

ℓ = I( ∧ Z ∣Z ) = 0

w n→∞

lim sup n 1

=Zw

n

F

U n w n

X +

a n

X , X +

b n b n

Xc

n

Zw Z1 Z2 Z3 Xa Xb Xc F1 F2

Communication for Omniscience Communication for Omniscience

[Csiszar and Narayan 04]

Z0 ZU

I(F ∧

n 1

X ∣Z ) →

w n w n

ℓ (leakage)

w

X =

w

Z (censored info.)

U

U 1 w

wiretapper users

F0 F

unlimited r1 u =

1

H(Z ∣Z ) (omniscience)

U 1

Zw

Characterize . R :

CO = inf{

r ∣(u , ℓ , r ) ∈ ∑i∈U

i V V V

R, u =

i

H(Z ∣Z ) ∀i ∈

V i

A}

helpers in U\A no target info. active users in A

Zh h Fh

unlimited rh

Minimum leakage vs minimum discussion rate Minimum leakage vs minimum discussion rate

From [Csiszar and Narayan 04], R =

CO

min r r + r ≥ 0, r + r ≥ 1, r + r ≥ 1 {∑

i=1 3 i∣

∣ ∣ ∣ ∣

1 2 1 3 2 3

} ℓ =

w

H(Z ∣Z ) =

3 w

1 ≥ 0 = RL Claim: Any scheme with cannot have (r , r , r ) =

1 2 3

(0, 0, 1) R =

L

• achieving scheme:

RCO F = F =

3

Z3

n

A Zw Z1 Z2 Z3 := {1, 2} ⊆ U := {1, 2, 3} := (X + X , X + X )

a b b c

:= (X , X )

a b

:= (X , X )

b c

:= (X + X + X )

a b c

Proposition if is null. R =

L

RCO Zw Proposition and are not simultaneously achievable in general. RL RCO

solved uniquely by = 1 (r , r , r ) =

1 2 3

(0, 0, 1)

ℓ =

w n→∞

lim sup n 1

=I(F∧Z )=H(F)

U n

I(F ∧ Z ∣Z )

U n w n

Main Results Main Results

Lower bound on minimum leakage Lower bound on minimum leakage

Theorem 1 For the secure omniscience scenario with , for any random variable satisfying the Markov condition . ∣A∣ ≥ 2 RL ≥ H(Z ∣Z ) − C

U w S

≥ R (Z ∣W) − I(Z ∧ Z ∣W)

CO U U w

W I(W ∧ Z ∣Z ) =

U w

I(K ∧

n→∞

lim sup n 1 F, Z ) =

w n

(secrecy) H(K∣Z , F) =

n→∞

lim sup n 1

i n

∀i ∈ A (recoverability) Definition (Multiterminal secret key agreement [Csiszar and Narayan 04])

C :

S =

H(K) s.t.

K,F

sup n

1

Proof Idea Proof Idea

For the first lower bound, similar to an argument in [Csiszar and Narayan 04], by privacy amplication after secure omniscience. C ≥

S

H(Z ∣Z ) −

U w

RL The second lower bound follows from an upper bound on in [Csiszar and Narayan 04]. C ≤

S

H(Z ∣W) −

U

R (Z ∣W),

CO U

CS Theorem 1 For the secure omniscience scenario with , for any random variable satisfying the Markov condition . ∣A∣ ≥ 2 RL ≥ H(Z ∣Z ) − C

U w S

≥ R (Z ∣W) − I(Z ∧ Z ∣W)

CO U U w

W I(W ∧ Z ∣Z ) =

U w

Example Example

A Zw Z1 Z2 Z3 Z4 = U := {1, 2, 3, 4} := X + X + X

a b c

:= Xa := (X , X )

a b

:= (X , X )

b c

:= Xc Is the lower bound achievable?

user 1 is active ∵

RL ≥ − = H(X , X , X ∣X + X + X )

a b c a b c

= 2 H(Z ∣Z )

U w

CS

1≥

= 2 − 1 = 1 C ≤

S

H(Z ) =

1

1

by Theorem 1

Theorem 1 For the secure omniscience scenario with , for any random variable satisfying the Markov condition . ∣A∣ ≥ 2 RL ≥ H(Z ∣Z ) − C

U w S

≥ R (Z ∣W) − I(Z ∧ Z ∣W)

CO U U w

W I(W ∧ Z ∣Z ) =

U w

Main Results Main Results

Upper bound on minimum leakage Upper bound on minimum leakage

Z1 ZU Z1

m

U 1 w F1

′

F′ Zw Zw

m

Theorem 2 For the secure omniscience scenario, where is a public discussion for block length . R ≤

L

[R (Z ∣F ) + m 1

CO U m ′

I(Z ∧

U m

F ∣Z )] ≤

′ w m

RCO F′ m ≥ 1

Proof idea Proof idea

Theorem 2 For the secure omniscience scenario, where is a public discussion for block length . R ≤

L

[R (Z ∣F ) + m 1

CO U m ′

I(Z ∧

U m

F ∣Z )] ≤

′ w m

RCO F′ m ≥ 1

Z , F

1 m ′

ZU Z , F

1 n ′ m

n

U 1 w Zw Z , F

w n ′ m

n

concat. blocks

m n

ℓw

′′ ≤ mℓ

+ r

w ′

∑

i∈U i ′′

Z1 ZU Z1

m

U 1 w F1

′

F′ Zw Zw

m

How to attain omniscience?

F1

′′

F′′ r1

′′

Omniscience possible with . r = ∑i∈U

i ′′

R (Z ∣F )

CO U m ′

ℓ =

w ′

I(Z ∧

m→∞

lim sup m

1 U m

F ∣Z )

′ w m

Example Example

A Zw Z1 Z2 Z3 Z4 = U := {1, 2, 3, 4} := X + X + X

a b c

:= Xa := (X , X )

a b

:= (X , X )

b c

:= Xc Do the upper and lower bounds match in general? RL ≥ 1

by Theorem 1 N.b., already achieves omniscience, so . F ′ R (Z ∣F ) =

CO U m ′

RL ≤ 1

by Theorem 2

does not work but works. m = 1 m = 2

F2

′

F3

′

= + [Xa

(1)

Xa

(2)] Xa

m

[1 1 1 0]

M:=

[Xb

(1)

Xb

(2)] Xb

m

= X + (M + I)X

c m b m

Try Theorem 2 For the secure omniscience scenario, where is a public discussion for block length . R ≤

L

[R (Z ∣F ) + m 1

CO U m ′

I(Z ∧

U m

F ∣Z )] ≤

′ w m

RCO F′ m ≥ 1

Main Results Main Results

Tightness of the upper and lower bounds Tightness of the upper and lower bounds

Theorem 3 For any finite linear source with two users, i.e., , where is the maximum common function of and , i.e., the unique solution to A = U = {1, 2} R =

L

H(Z , Z ∣Z ) −

1 2 w

I(Z ∧

1

Z ∣G)

2

G Zw Z1 J (Z ∧

GK w

Z ) :

1

= H(G).

G:H(G∣Z )=H(G∣Z )=0

w 1

max Theorem 1 With , for any with . ∣A∣ ≥ 2 R ≥

L

H(Z ∣Z ) −

U w

CS W I(W ∧ Z ∣Z ) =

U w

Proof of ≥ Choose , and W = G substitute . C =

S

I(Z ∧

1

Z ∣G)

2

Proof of ≤ Choose , and m = 1 to align with . F′ Zw Theorem 2 with discussion for block length . R ≤

L

[R (Z ∣F ) + m 1

CO U m ′

I(Z ∧

U m

F ∣Z )]

′ w m

F′ m ≥ 1

Bounds do not match in general Bounds do not match in general

C ≤

S

H(Z ) =

1

1 R ≥

L

H(Z ∣Z ) −

U w

C =

S

1 − 1 = 0

A Zw Z1 Z3 := {1, 2} ⊆ U := {1, 2, 3} := X + X

a b

= Z := X

2 a

:= Xb

Claim Minimum leakage is R ≥

L

1. F = F =

3

Xb

n

ℓ =

w

1 Optimal scheme:

Proposition The lower bound on is loose. RL Theorem 1 With , for any with . ∣A∣ ≥ 2 R ≥

L

H(Z ∣Z ) −

U w

CS W I(W ∧ Z ∣Z ) =

U w

Extensions and challenges Extensions and challenges

Tightness for hypergraphical sources can be proved. More explicit characterization using graph entropy is possible. RL Lower bound can be improved for the counter-example. remains unknown for finite linear sources. RL for secure linear function computation, extending [Tyagi et. al 11]. RL

References References

• I. Csiszár and P. Narayan, “Secrecy capacities for multiple terminals,” IEEE Transactions on

Information Theory, vol. 50, no. 12, pp. 3047–3061, Dec. 2004.

• A. Gohari and V. Anantharam, “Information-theoretic key agreement of multiple

terminals—Part I,” IEEE Transactions on Information Theory, vol. 56, no. 8, pp. 3973 – 3996, Aug. 2010.

• S. Asoodeh, M. Diaz, F. Alajaji, and T. Linder, “Estimation efficiency under privacy

constraints,” IEEE Transactions on Information Theory, vol. 65, no. 3, pp. 1512–1534, March 2019.

• N. Tishby, F. C. Pereira, and W. Bialek, “The information bottleneck method,” in Thirty-

Seventh Annual Allerton Conference on Communication, Control, and Computing, Sep. 1999.

• H. Tyagi, P. Narayan, and P. Gupta, “When is a function securely computable?” IEEE

Transactions on Information Theory, vol. 57, no. 10, pp. 6337–6350, 2011.