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Generalized Secure Distributed Source Coding with Side Information

In this paper, new inner and outer bounds on the achievable compression-equivocation rate region for generalized secure data compression with side information are given that do not match in general. In this setup, two senders, Alice and Charlie intend to transmit information to Bob via channels with limited capacity so that he can reliably reconstruct their observations. The eavesdropper, Eve, has access to one of the channels at each instant and is interested in the source of the same channel at the time. Bob and Eve also have their own observations which are correlated with Alice's and Charlie's observations. In this model, two equivocation and compression rates are defined with respect to the sources of Alice and Charlie. Furthermore, different special cases are discussed where the inner and outer bounds match. Our model covers the previously obtained results as well.


Introduction
Recently, much theoretical research has been dedicated to the problems of nodes with dense distribution and resource constraints. In such networks, nodes compress correlated data separately, without collaboration, which is known as distributed source coding. The nature of sensitive data in these networks necessitates secure compression while meeting quality of service requirement.
In the classical wiretap channel model, considered by Wyner in [1], nonzero secrecy rate can be achieved without using a secure key if the intended receiver has a better quality communication channel than the eavesdropper. The idea of generating a shared secret key from the correlated observations has been explored vastly so far. Maurer [2] and Ahlswede and Csiszar [3] were researchers in this subject. On the other hand, in some applications like sensor networks, it is needed that the correlated information from different sources can be reconstructed by a node which has some other source of information. Simultaneously, the information sources should be kept secret from an eavesdropper as much as possible.
In this paper, we explore the above mentioned problem, where two nodes must separately compress their sources in such a way that the eavesdropper with side information learns as little as possible about them. In our model, a general secure distributed compression problem is considered in which two transmitters Alice and Charlie, with correlated observations, intend to send information to a receiver, Bob, over noiseless channels with limited capacity in such a way that he can reconstruct both sources reliably. Also, there is an eavesdropper, Eve, who listens to either Alice or Charlie's channel, one at a time, and when she listens to each of the sender's channel, she is only interested in learning information about that sender's source. Bob and Eve have their own side information correlated to Alice's and Charlie's observations.
In the sequel, we aim at information source when we use the term source.

Related Works
In [4], Yamamoto addresses lossy compression with security constraints over a noisy broadcast channel in which the users share a secure key. It was shown that if the source is compressed at the first step and then encrypted using the secure key and finally transmitted over the noisy channel using wiretap channel code, the optimal strategy is selected. In some other works, the communication channels are considered noiseless. For example in [5] and [6], Yamamoto investigated the scheme where a sender observes the outcomes of two correlated sources and wishes to send information in such a way that one of the sources can be reconstructed at the receiver but the other is kept as secret as possible. A simplified but significant problem has been addressed by Prabhakaran and Ramchandran in [7] where Alice intends to send information to Bob to enable him to reconstruct her source and keep eavesdropper, Eve, as ignorant as possible about her source. In this problem, Bob and Eve have access to side information arbitrarily correlated to Alice's source, and the minimum leakage rate in secure lossless compression is explored. The significance of [7] is explaining the point that in the case of arbitrarily correlated side information at the eavesdropper, the usual Slepian-Wolf compression is not always sufficient.
Secure lossless compression of two correlated sources is investigated in [8], where the related information of each source is sent over one channel and the eavesdropper has access to only one of the channels stream at any instant and wishes to get information about the sender's source of that channel at that time. The problem explored in [8], by Luh and Kundur, is a simplified case in which eavesdropper has no side information and so Slepian-Wolf coding suffices to setup minimum leakage. In this situation, the compression-equivocation capacity region is given. In another scenario in [ Eve has access to only one of the channels stream at any instant and wishes to get information about the sender's source of that channel at that time. So, in this situation, two equivocation rates are defined and the inner and outer bounds on achievable compression-equivocation rates ( , , , ) R are the capacity of Alice's and Charlie's channels and A ∆ and C ∆ are the equivocations of Eve with respect to Alice's and Charlie's sources, respectively. In this generalized scenario, the Slepian-Wolf coding alone is not optimal and must be combined with random coding. Also, different cases are discussed and it can be seen that our results contain the results of the above mentioned references as the special cases.
The paper is organized as follows. The generalized model is introduced in section 2. In section 3, inner and outer bounds of compression-equivocation rate region are given in theorems 1 and 2, respectively, which generalize the well known Slepian-Wolf region to include secrecy constraints. Different scenarios based on the availability of the side information at the nodes are considered in section 4. The conclusion and the proofs of the theorems are included in section 5 and appendix, respectively.

Generalized Model
In the model shown in Fig.1 , respectively, and the error probability is defined as Pr{ ( ( ), ( )) ( , The closure of all achievable rate quadruples ( , , , ) A C A C R R ∆ ∆ is compression-equivocation capacity region for which the inner and outer bounds are represented in section 3.

Generalized Secure Compression of Sources
If security is ignored in the problem described in the previous section, then it is reduced to a Slepian-Wolf

Theorem 1 (inner bound):
In the described setup ( , , , ) ( ; ) ( , ; ) The detailed proofs of both theorems are given in the appendix but some discussion is followed.
In the mentioned problem, Alice and Charlie attempt to increase equivocation of Eve with respect to their own sources. In [10] Charlie's channel was secure and he used his channel capacity as much as possible so that Alice could keep her source as secret as possible from Eve. A significant difference of our problem with the problem in [10] is that Charlie is also attentive of his source security and so, most usage of his channel capacity is not a good strategy, necessarily. This makes the problem more complex and according to equations (6) and (16), there is a trade-off between the equivocation rates. In fact, in the achievability scheme, it can be assumed that first, Alice and Charlie transmit the auxiliary random variables U and V , respectively, which have distributions ( ) p u a and ( ) p v c . After that, they launch to transmit remainder of information which is required for Bob to reconstruct both sources. For this purpose, according to Slepian-Wolf theorem, Alice and Charlie should transmit some information with the overall rate ( , , , ) H A C B U V at least, which is sent via random binning. This rate is divided between Alice and Charlie and this division determines the trade-off according to equation (6).
Using the auxiliary random variables can potentially result in higher equivocation rates. This fact is shown in  We prove this corollary for auxiliary random variable U . For the random variable V the proof is similar.
From equation (12) of theorem 2, the equivocation rate A ∆ will be upper bounded as:

Special Cases
Now, we investigate some special cases. The first and also the simplest case is when eavesdropper has no side information or in other words, E is constant.

Case 1.Eavesdropper with no side information:
In this situation, the following corollary can be deduced.

Corollary 4.1:
When there is no side information at Eve, the inner and outer bounds match each other. The compression-equivocation capacity region is characterized by: The achievability and converse of the corollary can be considered as a special case of theorems 1and 2 by setting the auxiliary random variable U and V constant and using Slepian-Wolf binning. It can be seen that this special case coincides in [8, theorem 1].
Some other special cases can be assumed when side information of Bob or Eve or both is accessible at Alice or Charlie. Hence, three cases can be considered when Alice has access to Bob's or Eve's side information or both. For simplicity, in these three cases, it is assumed that the channels from Alice and Charlie to Bob have infinite capacity and only the equivocation rates are considered.

Case 2.Eavesdropper's side information at Alice:
In this case, it is assumed that Eavesdropper's side information (source E ) is available at Alice and the following corollary can be obtained.

Corollary 4.2:
If Eve's side information is available at Alice, ( , , , ) It can be shown that in this case, the best strategy is to establish U E = and V as constant value and transmit remainder information via Slepian-Wolf random binning. The rate of this remainder information is ( , , ) H A C B E . This rate must be sent by Alice and Charlie and depending on the portion of transmitted information by each of them, there is a tradeoff between the equivocation rates.
We note that when Alice has access to Eve's side information, the source A will be replaced with , A E . Similarly equations (28) and (29) can be deduced.
It should be noted that the availability of either Eve's or Bob's side information at Alice enlarges the space of auxiliary random variable U and can potentially result in higher equivocation rates at the eavesdropper. From the Slepian-Wolf source coding, it is known that the availability of receiver's side information at the senders does not result in compression rates improvement, but this is not true in the case of equivocation rates. We show via an example that availability of Bob's side information at Alice increases equivocation rate of Eve with respect to Alice's source. This example is an extension of the example in lemma 4.2 in [10].
It is evident that when A is independent of ( , ) B C and Alice does not have access to B , then =0 A ∆ . However when she has access to B , she can use it as a key to encrypt her message.

Corollary 4.3:
When A is independent of ( , ) B C , B is independent of E and Alice has access to B , then  Another special case occurs when Eve's side information is available at Bob in addition to his source B .

Case 5.Eve's side information at
The achievability can be obtained from theorem 1 by replacing B with ( , ) B E and considering the auxiliary random variables U and V constant. The converse is followed by theorem 2.

Conclusion
In this paper, secure distributed compression of two sources was considered. In the studied model, the eavesdropper had access to side information and intercepted one of the channels at a time. In this model, two equivocation and compression rates were defined. The inner and outer bounds of rate quadruples ( , , , ) Availability of Bob's or Eve's side information or both at Alice and availability of Eve's side information at Bob provide other special cases where compression-equivocation capacity region is given for each case. These cases were in agreement with the results obtained in this subject previously.
As the future work, the same problem can be regarded when the communication is two-way via public channel, i.e. Bob can also transmit data to Alice and Charlie. This leads in less leakage rate. In addition, the problem can be more generalized in a way that either there are multiple receiver with different side information that should decode the Alice's and Charlie's sources or there are multiple eavesdroppers with different side information that listen to one of the channels at a time.

Appendix: Proofs
Proof of Theorem 1 1 In proof of achievability, we consider two rate quadruples ( , , , ) in which U and V are random variables with the distribution of theorem 1.
For a typical observation of N A , Alice finds a jointly typical sequence ( )   Now, the equivocation rates can be lower bounded as:     It can be seen that similar to the previous decoding scheme, Bob can reliably decode N A and N C with high probability. Now, the same parameters can be calculated (the procedure is the same as before and so details are

R I A U B H A U V B H A V B I U V B R I C V U B H C V A B H C U B I A C U V B R R H
It can be seen that with this coding scheme, equations (2) In both above coding schemes, the sum rates of compression rates and equivocation rates are the same. Now, achievability of the two rate quadruples is proved. We avoid detailed proof of achievability of the total region of theorem 1and content ourselves with a proof scheme.
In both of the above coding scheme, at the first step, Alice and Charlie use auxiliary random variables U and

R H A B I A V U B H A V B I U V B
3-Alice sends information with the rate of ( ; ) ( ) -(

, ) I A U B H A B H A U B
= at the first step and then information with the rate of ( , , ) H A U B C at the second step. Charlie sends information with the rate of It can be seen that in the first and second cases the individual equivocation rates are the same but the transmitted rates are different. Using time sharing according to theorem g. of [11], the other rate quadruples between these points are achievable. The same is true for third and forth cases and hence, the achievability of the theorem 1 region is satisfied.

Proof of theorem 2
The stochastic functions A F and C F are defined as ( ) Fano's inequality leads to α is a non-negative function with lim ( ) x 0 x 0 α → = . Now, we define: It can be seen that ( , , ) are Markov chains. We have: Similarly, it can be deduced that:   In a similar way it can be shown that: