[PPT] - On Secure Asymmetric Multilevel Diversity Coding Systems Congduan PowerPoint Presentation

SLIDE 1

On Secure Asymmetric Multilevel Diversity Coding Systems

Congduan Li

Sun Yat-sen University

Jun 21-26, 2020 ISIT 2020 (LA, USA, online)

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 1 / 27

SLIDE 2

Outline

1 Motivation 2 System Model 3 An example 4 Extension to general case

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 2 / 27

SLIDE 3

Outline

1 Motivation 2 System Model 3 An example 4 Extension to general case

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 3 / 27

SLIDE 4

Related work

Related work in TransIT [Jiang et al. 2014] Wiretap channel on symmetric multilevel diversity coding systems (SMDCS) Secure rate region obtained Superposition (source separation) coding optimal Natural question: what about asymmetric case? This work: secure rate region for asymmetric MDCS (AMDCS) and

ptimality of achieving coding schemes

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 4 / 27

SLIDE 5

Outline

1 Motivation 2 System Model 3 An example 4 Extension to general case

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 5 / 27

SLIDE 6

Secure AMDCS

Build on the system model of AMDCS by [Mohajer et al. 2010] Impose the secrecy constraints as in [Jiang et al. 2014]

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 6 / 27

SLIDE 7

Secure AMDCS

Independent sources are prioritized: X1 is more important than X2, X2 is more important than X3, and so on. More important sources must be decoded before decoding less important sources. Secrecy key independent to sources

Paths

X1 U1

Secrecy Key and Prioritized Sources Encoders Decoders

X1:S

Fan(D1)

E1 D1 Di EL UL

X1:i Fan(DS)

DS

K X1 X2 . . . XS              K, X1:S

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 7 / 27

SLIDE 8

Secure AMDCS

Sources, together with secrecy key, are coded by multiple encoders Eavesdropper may has access to one of the size m (security level) subset of encoders (wiretap channel II), but can know nothing about the sources

Paths

X1 U1

Secrecy Key and Prioritized Sources Encoders Decoders

X1:S

Fan(D1)

E1 D1 Di EL UL

X1:i Fan(DS)

DS

K X1 X2 . . . XS              K, X1:S

Size m subset

Eavesdropper

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 8 / 27

SLIDE 9

Secure AMDCS

Input to decoder (Fan(Di)) should include at least m + 1 encoders Symmetric case: all decoders with input of m + i encoders can decode X1, . . . , Xi General case: each secure transmission has an associated decoder, and decoder Di can decode X1, . . . , Xi Order: Naturally, a decoder whose input is a subset of another decoder’s input, its decoding level should be smaller

Paths

X1 U1

Secrecy Key and Prioritized Sources Encoders Decoders

X1:S

Fan(D1)

E1 D1 Di EL UL

X1:i Fan(DS)

DS

K X1 X2 . . . XS              K, X1:S

Size m subset

Eavesdropper

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 9 / 27

SLIDE 10

Secure Rate region

Secure rate region: all possible rate and source entropy vectors satisfying all network constraints and secrecy constraints, with existence of proper secrecy key, coding and decoding functions.

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 10 / 27

SLIDE 11

Outline

1 Motivation 2 System Model 3 An example 4 Extension to general case

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 11 / 27

SLIDE 12

AMDCS with 4 encoders and security level 2

4 encoders, 5 sources, 5 decoders Each decoder has a distinct decoding level Order: L1(E1E2E3) = 1 L1(E1E2E4) = 2 L1(E1E3E4) = 3 L1(E2E3E4) = 4 L1(E1E2E3E4) = 5 First obtain the inner bound, superposition secure rate region

X1, X2

K X1 X2 X3 X4 X5

X1 E1 E2 E3 D2 D1 D3 D4

X1, X2, X3 X1, X2, X3, X4 E4

D5

X1, X2, X3, X4, X5 I(X1:5; Ei,j) = 0, i, j ∈ {1, 2, 3, 4}

security :

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 12 / 27

SLIDE 13

Superposition: sources coded separately

Each source, with key, is coded separately Coding rate is sum of sub-rates on each source: Ri = 5

j=1 r j i , i = 1, . . . , 5

For each sub-encoder, it is a solved single-source secrecy problem [Shamir 1979, Cai-Yeung 2011]

}

X1, K X2, K X3, K X4, K X5, K Ei Ri r1

i

r2

i

r3

i

r4

i

r5

i Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 13 / 27

SLIDE 14

Superposition secure rate region

From decoder Di, i = 1, . . . , 4, we have r j

k ≥ H(Xj),

k ∈ Fan(Di), j = 1, . . . , i From decoder D5, we have r j

i + r j k ≥ H(Xj),

i, k ∈ Fan(D5), i = k, j = 1, . . . , 5

X1, X2

K X1 X2 X3 X4 X5

X1 E1 E2 E3 D2 D1 D3 D4

X1, X2, X3 X1, X2, X3, X4 E4

D5

X1, X2, X3, X4, X5 I(X1:5; Ei,j) = 0, i, j ∈ {1, 2, 3, 4}

security :

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 14 / 27

SLIDE 15

Superposition secure rate region

Together with the rate sum equations and eliminate sub-rates, we get the inner bound The achievability can be verified by constructing the codes for each extreme ray of the polyhedra formed by the inequalities

R1 ≥

3

i=1

H(Xi), Rj ≥

4

i=1

H(Xi), j = 2, 3, 4 R1 + Ri ≥ 2

3

j=1

H(Xj) +

5

l=4

H(Xl), i = 2, 3, 4 R2 + Ri ≥ 2

4

j=1

H(Xj) + H(X5), i = 3, 4 R3 + R4 ≥ 2

4

i=1

H(Xi) + H(X5)

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 15 / 27

SLIDE 16

Exact secure rate region

If coding across sources are allowed, the secure rate region is enlarged Some coefficients are changed

R1 ≥

3

i=1

H(Xi), Rj ≥

4

i=1

H(Xi), j = 2, 3, 4 R1 + Ri ≥ 2

3

j=1

H(Xj) +

5

l=4

H(Xl), i = 2, 3, 4 R2 + Ri ≥ 2

4

j=1

H(Xj) + H(X5), i = 3, 4 R3 + R4 ≥ 2

4

i=1

H(Xi) + H(X5) R1 ≥

3

i=1

H(Xi) Rj ≥

4

i=1

H(Xi), j = 2, 3, 4 R1 + Rj ≥ 2

3

i=1

H(Xi) +

5

k=4

H(Xk), j = 2, 3, 4 R2 + Rj ≥ 2

3

i=1

H(Xi) +

5

k=4

H(Xk), j = 3, 4 R3 + R4 ≥ 2

2

i=1

H(Xi) + H(X3) + H(X4) + H(X5) Superposition Exact

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 16 / 27

SLIDE 17

Converse: exact secure rate region

Different inequalities proved by decoding and secure constraints from different subsets of decoders

R1 ≥

3

i=1

H(Xi) Rj ≥

4

i=1

H(Xi), j = 2, 3, 4 R1 + Rj ≥ 2

3

i=1

H(Xi) +

5

k=4

H(Xk), j = 2, 3, 4 R2 + Rj ≥ 2

3

i=1

H(Xi) +

5

k=4

H(Xk), j = 3, 4 R3 + R4 ≥ 2

2

i=1

H(Xi) + H(X3) + H(X4) + H(X5)

X1, X2

K X1 X2 X3 X4 X5

X1 E1 E2 E3 D2 D1 D3 D4

X1, X2, X3 X1, X2, X3, X4 E4

D5

X1, X2, X3, X4, X5

D3 D4 D3, D4, D5

}

D2, D4, D5

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 17 / 27

SLIDE 18

Achievability: exact secure rate region

Need to figure out the new extreme rays when enlarge the rate region New extreme rays are outside the superposition coding region Coding across sources are necessary Construct codes to achieve them It turns out that linear codes are optimal

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 18 / 27

SLIDE 19

Key constraints

Necessary constraints on the size of the key One can define the secure rate region to include the key size, then, the extreme rays will have the necessary key size for the codes Shearer’s Lemma (or Han’s inequality) is used in the proofs

H(K) ≥ 2

3

i=1

H(Xi) + H(X4) + H(X5) H(K) ≥ 2

4

i=1

H(Xi), 2Ri + H(K) ≥ 4

3

j=1

H(Xj) + 2

5

k=4

H(Xk), i = 1, 2 2Ri + H(K) ≥ 4

2

j=1

H(Xj) + 3H(X3) + 2

5

k=4

H(Xk), i = 3, 4

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 19 / 27

SLIDE 20

Outline

1 Motivation 2 System Model 3 An example 4 Extension to general case

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 20 / 27

SLIDE 21

General L encoders with security level L − 2

Secure transmissions include accesses to all size L − 1 subsets of encoders and the whole set of the encoders L + 1 sources and decoders, each decoder has a distinct decoding level from 1 to L + 1 For Di, i = 1, . . . , L, we have Fan(Di) = {E1, . . . , EL} \ {EL−i+1} For DL+1, we have Fan(DL+1) = {E1, . . . , EL}

X1

Decoders

X1:L+1

{E1,...,EL} \ EL

D1 Di

X1:i

DL+1 {E1, . . . , EL} \ EL−i+1

{E1, . . . , EL}

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 21 / 27

SLIDE 22

Superposition secure rate region

R1 ≥

L−1

i=1

H(Xi), Ri ≥

L

j=1

H(Xj), i = 2, . . . , L R1 + Ri ≥ 2

L−1

j=1

H(Xj) +

L+1

l=L

H(Xl), i = 2, . . . , L Ri + Rk ≥

i=k

2

L

j=1

H(Xj) + H(XL+1), i, k ∈ Fan(DL). R1 ≥

3

i=1

H(Xi), Rj ≥

4

i=1

H(Xi), j = 2, 3, 4 R1 + Ri ≥ 2

3

j=1

H(Xj) +

5

l=4

H(Xl), i = 2, 3, 4 R2 + Ri ≥ 2

4

j=1

H(Xj) + H(X5), i = 3, 4 R3 + R4 ≥ 2

4

i=1

H(Xi) + H(X5)

}

4 encoder security level 2 L encoder security level L − 2 Associated constraints on the secrecy key

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 22 / 27

SLIDE 23

Exact secure rate region

R1 ≥

L−1

i=1

H(Xi) Ri ≥

L

j=1

H(Xj), i = 2, . . . , L R1 + Ri ≥ 2

L−1

j=1

H(Xj) +

L+1

k=L

H(Xk), i = 2, . . . , L, R2 + Ri ≥ 2

L−1

j=1

H(Xj) +

L+1

k=L

H(Xk), j = 3, . . . , L, Ri + Rk ≥

i=k

2

L−2

j=1

H(Xj) +

L+1

l=L−1

H(Xl), i, k = 1, 2. R1 ≥

3

i=1

H(Xi) Rj ≥

4

i=1

H(Xi), j = 2, 3, 4 R1 + Rj ≥ 2

3

i=1

H(Xi) +

5

k=4

H(Xk), j = 2, 3, 4 R2 + Rj ≥ 2

3

i=1

H(Xi) +

5

k=4

H(Xk), j = 3, 4 R3 + R4 ≥ 2

2

i=1

H(Xi) + H(X3) + H(X4) + H(X5)

L encoder security level L − 2 4 encoder security level 2

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 23 / 27

SLIDE 24

Key constraints

Does NOT hold in general! A Loose Bound! A special inequality to include the special cases! 4 encoder security level 2 L encoders security level L-2

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 24 / 27

SLIDE 25

Summary

Superposition is NOT optimal for AMDCS in general Tight constraints on the key size for superposition coding obtained Tight secure rate region obtained for a class of AMDCS problems Linear codes are optimal for the class of problems considered An counter-example shows the loose bound on the key constraints Future work: tight key constraints and secure rate region for lower security level

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 25 / 27

SLIDE 26

References

1 J. Jiang, N. Marukala, and T. Liu, “Symmetrical multilevel diversity

coding and subset entropy inequalities,” IEEE Transactions on Information Theory, vol. 60, no. 1, pp. 84-103, Jan 2014.

2 S. Mohajer, C. Tian, and S. Diggavi, “Asymmetric multilevel diversity

coding and asymmetric gaussian multiple descriptions,” IEEE Transactions on Information Theory, vol. 56, no. 9, pp. 4367-4387, 2010.

3 A. Shamir, “How to share a secret,” Commu. Assoc. Comput., vol.

22, pp. 612-613, 1979.

4 N. Cai and R.W. Yeung, “Secure network coding on a wiretap

network,” IEEE Transactions on Information Theory, vol. 57, no. 1,

pp. 424-435, Jan 2011.

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 26 / 27

SLIDE 27

Q & A

Thank you!

Li, et al. (Sun Yat-sen University) Secure AMDCS Jun 21-26, 2020 27 / 27