Multilevel Diversity Coding with Secrecy Constraints - ITA @ UCSD

Multilevel Diversity Coding with Secrecy Constraints A.Balasubramanian, Tie Liu and Scott L.Miller Wireless Communication Laboratory, Electrical and Computer Engineering Department Texas A&M University, Texas, USA, 77840-3128 Email: {anantha,tieliu,smiller}@ece.tamu.edu

Abstract—We consider the problem of communicating multiple sources to a receiver amidst an eavesdropper. The source encoder encodes the sources and sends it through the available L channels to the receiver. The eavesdropper has access to a subset of m channels (m < L). There are totally L − m sources to be communicated to the receiver with the stipulation that the receiver needs to be able to reconstruct sources {1, 2 . . . k}, by accessing any subset of (m + k) channels (1 ≤ k ≤ L − m). We obtain the coding rate region for the case when L = 3, m = 1 and show that separate and secure encoding of sources is optimal for this problem. For the case of arbitrary m, L we derive a lower bound on the sum rate and the result implies a separation scheme to achieve this lower bound.

Index Terms: Secrecy Capacity, Rate Region, Linear Network Codes, Diversity Coding. 1. I NTRODUCTION In a multilevel diversity coding system, an information source is encoded by many encoders and there are multiple decoders with each decoder having access to a subset of encoders [2]. The goal of the multilevel diversity coding system is that the encoding needs to be done in such a way that, each decoder be able to reconstruct the source either perfectly or with some distortion. In the symmetrical multilevel diversity coding problem treated by Roche and Yeung [1], there are multiple information sources and decoders. The decoders are partitioned into multiple levels and the decoder belonging to a particular level should be able to reconstruct a predetermined number of sources. We consider a more generalized version of the problem treated in [1] in this paper: There are multiple information sources that have to be communicated to a legitimate receiver through a total of L separate communication channels. An eavesdropper has access to any subset of m channels (m < L), while the legitimate receiver has access to all L channels. There are a total of L − m sources, that have to be communicated to the legitimate receiver. The source encoder encodes the sources into L packets, possibly of different size and sends them each, through a channel. The encoding is done in such a way that any subset of m packets does not convey any information about any of the sources, while the sources {1, 2, . . . k}(1 ≤ k ≤ L − m) can be reconstructed perfectly by accessing any subset of (m + k) channels. Our main result is an exact characterization of the rate region for the case when L = 3 and m = 1, with two sources. Furthermore, we show that a ‘separate’ secure coding of

W1

R1 R2

XN Encoder

RL

Fig. 1.

. . .

W2

Legitimate Receiver

WL

Secure diversity coding for one source

information sources is optimal for this problem. For the more general case when L and m are arbitrary, we characterize a lower bound on the sum-rate and the result implies a separation strategy to achieve this lower bound. This paper is organized as follows. In Section 2, we treat this problem for the case of one source and provide some preliminary results. Section 3 presents the results for multiple sources wherein the the rate region is derived for the case when L = 3 and m = 1 and a sum-rate bound is derived for the more general case. Finally, we present the conclusions in Section 4. 2. S OME R ESULTS FOR ONE SOURCE The problem considered in this section is shown in Fig. 1. A source encoder is presented with a sequence of source letters from an i.i.d source X, drawn from the alphabet X. For each block of N letters (N arbitrary), there are L outputs, fl (X N ) = Wl ∈ {1, 2 . . . 2N Rl } for l = 1, 2 . . . L. The codeword Wl is sent through the lth communication channel. Let A denote the collection of all subsets of channels of size m (m < L), which the eavesdropper has access to. We assume that the eavesdropper can access any member of A, but no more than one member of A. For A ∈ A, let YA be the codeword transmitted on the channel corresponding to the elements in A. Let B denote the collection of all subsets of channels of size n (m < n ≤ L), from which the legitimate receiver must be able to reconstruct the source, X, with distortion, D0 . Denote R(D0 ) to be minimum number of bits required to reconstruct the source with distortion, D0 [4]. For any B ∈ B, consider all collection of subsets of size ¡ ¢ m and denote it by Υ B . Clearly, [ ¡ ¢ A= Υ B ∀ B∈B

Let RA denote the sum rate of channels corresponding to

elements in A, and HA denote the sum of the entropies of the S1N codewords corresponding to the elements in A. For example, N if A = (1, 2), RA = R1 +R2 , while HA = H(W1 )+H(W2 ), S2 N where R1 , R2 are the rates of channels 1, 2 respectively, and SL−m W1 , W2 are the messages sent along channels 1, 2 respectively. Denote hA to be the equivocation rate when the eavesdropper has access to any A ∈ A and it is desired that: 1 H(X N |YA ) ≥ hA ∀A ∈ A (1) N The aforementioned problem for one source can be succinctly represented by (L, n, m), where L represents the number of packets, n represents the diversity level (i.e., minimum number of packets required to reconstruct the source with distortion D0 ) and m represents the secrecy level of the source. The case of L = n (i.e., no diversity) has been solved in [3]. In what follows, we present results for the general case when m < n ≤ L. ¡ ¢ Theorem 2.1. For a fixed B ∈ B and any A ∈ Υ B , the following holds: ¡ ¢ RAc ≥ hA − H(X) − R(D0 ) where, Ac = B \ A. Proof: N RAc

≥ H Ac ≥ H(YAc ) ≥ H(YAc |YA ) ≥ H(YAc |YA ) − H(YAc |YA , X N ) = = = ≥

i=1

¢¤ ¡ L £ h − H(X) − R(D0 ) n−m

R2 Encoder

RL

Fig. 2.

. . .

W2

Legitimate Receiver

WL

Secure diversity coding of multiple sources

Summing the above over all B ∈ B, ¶ L µ ¶ µ £ ¤¢ L−1 X L n ¡ Ri ≥ h − H(X) − R(D0 ) n − 1 i=1 n n−m which yields (2). Corollary 2.1. The rate region for the case when A = {1}, {2}, {3}, B = {(1, 2), (2, 3), (1, 3)} (i.e., L = 3, n = 2, m = 1) with non-symmetrical secrecy constraints is given by the set of rate pairs (R1 , R2 , R3 ) which satisfies the following: Ri + Rj R1 R2 R3

≥ ≥ ≥ ≥

R(D0 ) for 1 ≤ i < j ≤ 3 max{h2 , h3 } − [H(X) − R(D0 )] max{h1 , h3 } − [H(X) − R(D0 )] max{h1 , h2 } − [H(X) − R(D0 )]

3. S ECURE C ODING FOR MULTIPLE SOURCES

I(YAc ; X N |YA ) I(YB ; X N ) − I(YA ; X N ) I(YB ; X N ) − H(X N ) + H(X N |YA ) £ ¤ N R(D0 ) − H(X) + hA

Theorem 2.2. The sum rate of L channels when hA = h (i.e., symmetrical secrecy constraints), ∀A ∈ A satisfies: Ri ≥

.. .

The proof of Corollary 2.1 is similar to the proof of Theorem 2.1. The achievability can be shown and will not be proved here.

where the last inequality follows from the fact that R(D0 ) is the minimum number of bits required to describe a source at distortion D0 , {X} being i.i.d and from the secrecy requirement in (1).

L X

W1

R1

(2)

In this section, we look at a similar setting of the problem addressed in Section 2, with multiple sources. With the same notations as was used in Section 2, we can represent the problem of K sources by (L, nk , mk ), where nk , mk represents the diversity and secrecy levels respectively for source k (k = 1, 2 . . . K). We provide results for the case when mk = m (i.e., same secrecy levels for all sources), nk = m+k with K = L − m. This is depicted in Fig. 2. This problem for the case of m = 0 (i.e., no secrecy), has been solved in [1], wherein it is shown that separation coding is optimal. Now, we define the problem formally which is solved in this section. Denote the collection of all subsets of sources of size {1, 2 . . . L − m}, by S. For any S ∈ S, the equivocation rate needs to satisfy the following:

¡ ¢ H(S|YA ) = H(S) ∀S ∈ S and ∀A ∈ A. Proof: From Theorem 2.1, we have for any A ∈ Υ B , and for a fixed B ∈ B: Denote by Zm+k the collection of all subsets of channels of ¡ ¢ size (m + k) (1 ≤ k ≤ L − m). Since the legitimate receiver RAc ≥ hA − H(X) − R(D0 ) requires that sources {1, 2 . . . k} need to be reconstructed ¡ ¢ Summing the above over all Υ B , for a fixed B ∈ B, together perfectly (i.e., D0 = 0) when it accesses any Z ∈ Zm+k , with the condition hA = h, ∀A ∈ A yields: it follows that µ ¶ µ ¶ ¡ ¢¤ n−1 n £ H(S1N , S2N . . . SkN |YZ ) = 0 ∀Z ∈ Zm+k and 1 ≤ k ≤ L − m. RB ≥ h − H(X) − R(D0 ) n−m−1 m where, YZ is the codeword transmitted on the channels cor¡ ¢¤ n £ ⇒ RB ≥ h − H(X) − R(D0 ) responding to elements in Z. For simplicity, we henceforth n−m

denote SkN (1 ≤ k ≤ L − m) by Sk . We also define the following: ½ x+y if x + y ≤ 3 x⊕y = x+y−3 if x + y > 3

where, the second inequality follows from Lemma 3.1. Now, we prove (4).

A. Rate region for the case of L = 3 and m = 1

N (Ri + Rj ) ≥ ≥ ≥

Theorem 3.1. The rate region for the case when L = 3, m = 1, with two sources S1 , S2 is given by the rate tuple (R1 , R2 , R3 ) which satisfy the following:

where the second inequality follows from Lemma 3.1, and the last inequality follows from Lemma 3.2. Finally, (5) follows from:

Ri

≥ H(S1 )

for i = 1, 2, 3,

H(Wi ) + H(Wj ) 2H(S1 ) + H(Wi |S1 , Wi⊕1 ) + H(Wj |S1 , Wj⊕1 ) N (2H(S1 ) + H(S2 ))

(3)

N (R1 + R2 + R3 ) ≥

≥ 2H(S1 ) + H(S2 ) 1 ≤ i < j ≤ 3, (4) 3 R1 + R2 + R3 ≥ 3H(S1 ) + H(S2 ). (5) 2 We first prove some lemmas which will be useful in proving Theorem 3.1.

≥

Ri + Rj

Lemma 3.1. For 1 ≤ i ≤ 3 , H(Wi ) ≥ H(S1 ) + H(Wi |S1 , Wi⊕1 )

3H(S1 ) + H(W1 |S1 , W2 ) + H(W2 |S1 , W3 ) + H(W3 |S1 , W1 ) 1 h¡ = 3H(S1 ) + H(W1 |S1 , W2 ) + 2 ¢ ¡ H(W2 |S1 , W3 ) + H(W2 |S1 , W3 ) + ¢ ¡ H(W3 |S1 , W1 ) + H(W3 |S1 , W1 ) + ¢i H(W1 |S1 , W2 ) 3 N (3H(S1 ) + H(S2 )) 2 where the second inequality follows from Lemma 3.1 and the last inequality follows from Lemma 3.2. ≥

Proof: H(Wi )

≥ = = = =

H(Wi |Wi⊕1 ) H(Wi |Wi⊕1 ) + H(S1 |Wi , Wi⊕1 ) H(S1 , Wi |Wi⊕1 ) H(S1 |Wi⊕1 ) + H(Wi |S1 , Wi⊕1 ) H(S1 ) + H(Wi |S1 , Wi⊕1 )

where the first equality follows from the fact that the source S1 must be decodable by any of the two distinct channels and the last equality follows due to the secrecy constraints. Lemma 3.2. For 1 ≤ i ≤ 3 , H(Wi |S1 , Wi⊕1 ) + H(Wi⊕1 |S1 , Wi⊕2 ) ≥ H(S2 ) Proof: ≥ =

H(W1 ) + H(W2 ) + H(W3 )

H(Wi |S1 , Wi⊕1 ) + H(Wi⊕1 |S1 , Wi⊕2 ) H(Wi |S1 , Wi⊕1 , Wi⊕2 ) + H(Wi⊕1 |S1 , Wi⊕2 ) H(Wi , Wi⊕1 |S1 , Wi⊕2 )

1) Separate Secure Coding of Sources is Optimal: We will show that, we can achieve the lower bounds in Theorem 3.1, by designing secure codes for sources S1 and S2 separately. We will use the technique in [1] to prove this claim. We write the rate constraints Ri (1 ≤ i ≤ 3) for sources (S1 , S2 ) as follows: 1≤i≤3 (6) Ri = ri1 + ri2 where, ri1 and ri2 are the rate constraints for sources S1 and S2 respectively. The rate region for source S1 can be obtained from Corollary 2.1, by incorporating the perfect secrecy constraint and D0 = 0 which yields the following: ri1 ≥ H(S1 )

for 1 ≤ i ≤ 3

(7)

= = ≥

Note that source S2 does not have to be endowed with diversity coding and the rate region can be obtained from Theorem 2.1 H(Wi , Wi⊕1 |S1 , Wi⊕2 ) + H(S2 |Wi , Wi⊕1 , Wi⊕2 , S1 ) by incorporating the perfect secrecy constraint and D0 = 0. By doing so, the rate region becomes: H(S2 , Wi , Wi⊕1 |S1 , Wi⊕2 ) H(S2 |S1 , Wi⊕2 ) r12 + r22 ≥ H(S2 )

=

H(S1 , S2 |Wi⊕2 ) − H(S1 |Wi⊕2 )

r22 + r32

≥ H(S2 )

H(S1 , S2 ) − H(S1 )

r12

≥ H(S2 )

(a)

(b)

=

(c)

=

H(S2 )

where, (a) follows due to the fact that the source S2 needs to be decodable from all three channels, (b) due to secrecy constraints and (c) due to the sources being i.i.d. Now, we begin proving Theorem 3.1. (3) can be proved as follows: N Ri ≥ H(Wi ) ≥ H(S1 ) = N H(S1 )

+

r32

(8)

By taking into account (6) through (8), and also by the fact that: 3 r12 + r22 + r32 ≥ H(S2 ) (9) 2 (which follows by summing all equations in (8)), we get the rate tuples (R1 , R2 , R3 ) that satisfy Theorem 3.1.

B. Sum-Rate Lower Bound for arbitrary L and m

=

The sum-rate lower bound for arbitrary L and m with L − m sources is derived. The derivation closely follows the approach presented in [1], with additional secrecy constraints enforced.

L X H(Si−m ) . i−m i=m+1

H(S1 , S2 . . . Sp−m |YA , A ∈ AT ) + p+1−m H(Wj , j ∈ T |S1 , S2 . . . Sp+1−m , YA , A ∈ AT ) i p+1−m (b)

=

Proof: Let T denote a subset of {1, 2 . . . L}. We will prove that for m < r ≤ L, the following holds: PL i=1

H(Wi ) ≥ N L

P T :|T |=r+1−m

Pr

H(Si−m ) i=m+1 i−m

+

L

L (r+1−m )

H(Wi )

≥

NL

i=1

X

(10)

p X H(Si−m ) L +¡ L ¢× i−m p+1−m i=m+1

H(Wj , j ∈ T |S1 , S2 . . . Sp−m , YA , A ∈ AT ) p+1−m

T :|T |=p+1−m

(a)

=

X

NL

p X H(Si−m ) L +¡ L ¢× i − m p+1−m i=m+1

h H(W , j ∈ T |S , S . . . S T j 1 2 p−m , YA , A ∈ A ) + p+1−m

T :|T |=p+1−m

H(Sp+1−m |S1 , S2 . . . Sp−m , Wj , YA , j ∈ T, A ∈ AT ) i p+1−m = X

NL

p X H(Si−m ) L +¡ L ¢× i − m p+1−m i=m+1

H(Sp+1−m , Wj , j ∈ T |S1 . . . Sp−m , YA , A ∈ AT ) p+1−m

T :|T |=p+1−m

= X

h H(S

T :|T |=p+1−m

p X L H(Si−m ) NL +¡ L ¢× i − m p+1−m i=m+1

. . . Sp−m , YA , A ∈ AT ) + p+1−m

p+1−m |S1 , S2

H(Wj , j ∈ T |S1 , S2 . . . Sp+1−m , YA , A ∈ AT ) i p+1−m

p X H(Si−m ) L +¡ L ¢× i−m p+1−m i=m+1 h H(S X p+1−m ) + p+1−m

H(Wj , j ∈ T |S1 , S2 . . . Sp+1−m , YA , A ∈ AT ) i p+1−m

where, AT denotes the collection of all subsets of channels that the eavesdropper has access to, not contain j ¡ ¢ which does | T for j ∈ T . Clearly, |AT | = L−|T , where |A | represents the m cardinality of the set AT . Assume that (10) is true for r = p. We will now show that (10) holds for r = p + 1. L X

NL

T :|T |=p+1−m

×

H(Wj ,j∈T |S1 ,S2 ...Sr−m ,YA ,A∈AT ) r+1−m

p X L H(Si−m ) +¡ L ¢× i − m p+1−m i=m+1 h H(S , S . . . S T X 1 2 p+1−m |YA , A ∈ A ) − p+1−m

T :|T |=p+1−m

Theorem 3.2. For the case of arbitrary L and m, with L − m sources, the sum-rate satisfies the following: R1 + . . . RL ≥ L

NL

=

NL

p+1 X L H(Si−m ) +¡ L ¢× i−m p+1−m i=m+1

X

H(Wj , j ∈ T |S1 , S2 . . . Sp+1−m , YA , A ∈ AT ) p+1−m

T :|T |=p+1−m (c)

≥

NL

p+1 X H(Si−m ) L +¡ L ¢× i − m p+2−m i=m+1

X

H(Wj , j ∈ T |S1 , S2 . . . Sp+1−m , YA , A ∈ AT ) p+2−m

T :|T |=p+2−m

where, (a) follows from the decodability of the source Sp+1−m given a subset of message {W1 , . . . WL } of size (p + 1), (b) follows due to secrecy constraints and the fact that all sources are i.i.d and (c) from the application of Han’s inequality [4]. 4. CONCLUSION In this paper, we have studied the problem of secure diversity coding, with multiple sources. The rate region was derived for a simple case of the problem, while a sum-rate lower bound was provided for the general case. R EFERENCES [1] J.R.Roche, Raymond W.Yeung and Ka Pun Hau, “Symmetrical Multilevel Diversity Coding,” IEEE Trans. Information Theory, vol. 43, no. 3, pp. 1059-1064, May 1997. [2] R.W.Yeung, “Multilevel diversity coding with distortion,”, IEEE Trans. Information Theory, vol.41, pp.412422, Mar. 1995. [3] Ning Cai and Raymond W. Yeung, “Secure Network Coding,” submitted to the IEEE Trans. Inf. Theory, Feb 2008. [4] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991