Enhancing Secrecy Rate Region for Recent Messages for a Slotted Multiple Access Wiretap Channel to Shannon Capacity Region

Enhancing Secrecy Rate Region for Recent Messages for a Slotted Multiple Access Wiretap Channel to Shannon Capacity Region

Shahid M Shah, Student Member, IEEE, and Vinod Sharma, Senior Member, IEEE Part of the paper was presented in IEEE Wireless Communication and Networking Conference (WCNC), March 2015, New Orleans, LA, USA.Shahid M Shah and Vinod Sharma are with Electrical communication Department, Indian Institute of Science, Bangalore, India.
Abstract

Security constraint results in rate-loss in wiretap channels. In this paper we propose a coding scheme for two user Multiple Access Channel with Wiretap (MAC-WT), where previous messages are used as a key to enhance the secrecy rates of both the users until we achieve the usual capacity region of a Multiple Access Channel (MAC) without the wiretapper (Shannon capacity region). With this scheme all the messages transmitted in the recent past are secure with respect to all the information of the eavesdropper till now. To achieve this goal we introduce secret key buffers at both the users, as well as at the legitimate receiver (Bob). Finally we consider a fading MAC-WT and show that with this coding/decoding scheme we can achieve the capacity region of a fading MAC (in ergodic sense).

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Enhancing Secrecy Rate Region for Recent Messages for a Slotted Multiple Access Wiretap Channel to Shannon Capacity Region


Shahid M Shah, Student Member, IEEE, and Vinod Sharma, Senior Member, IEEE


00footnotetext: Part of the paper was presented in IEEE Wireless Communication and Networking Conference (WCNC), March 2015, New Orleans, LA, USA.00footnotetext: Shahid M Shah and Vinod Sharma are with Electrical communication Department, Indian Institute of Science, Bangalore, India.

Index Terms

Physical layer security, Multiple Access Channel, Wiretap Channel, Strong Secrecy, Resolvability, Rate loss

Wyner in his seminal paper [1] on a degraded wiretap channel proved that by assigning multiple codewords to a single message, we can achieve reliability as well as security in a point to point channel. He characterized secrecy capacity for this channel. After a couple of decades of this work when wireless revolution began, researchers started extending Wyner’s coding scheme (wiretap coding) in different directions. A single user fading wiretap channel was studied in [2], [3]. A secret key buffer was used in [4] to mitigate the fluctuations in the secrecy capacity due to variations in the channel gain with time.

A multiple access channel with security constraints was studied in [5] and [6]. In [5] the transmitting users treat each other as eavesdroppers and an achievable secrecy rate region is characterized. In some special cases the secrecy capacity region is also found. In [6] the authors consider the eavesdropper to be listening at the receiving end. The authors provide an achievable secrecy-rate region. The secrecy-capacity region is not known for such a MAC. The same authors also studied a fading MAC with full channel state information (CSI) of Eve known at the transmitters. In [7] this work is extended to the case when the CSI of Eve is not known at the transmitters. For a detailed review on information theoretic security, see [8], [9], and [10].

In all these works a notion of weak secrecy was used, i.e., if is the message transmitted and Eve receives for a codeword of length channel uses, then , as . This notion of secrecy is not stringent enough in various cases [9]. Maurer in [11] proposed a notion of strong secrecy: as . For a point to point channel, he showed that it can be achieved without any change in secrecy capacity. Since then other methods have been proposed for achieving strong secrecy [12], [13] and [14]. The methods of [12] and [14] have been used to obtain strong secrecy for a MAC-WT in [15] and [16] respectively.

In all these works we observe that security is achieved at the cost of transmission rate. For a single user AWGN wiretap channel if is the capacity of the legitimate receiver (Bob) and is the capacity of Eve’s channel, then the secrecy capacity of this channel is , where ([17]). In recent years some work has been done to mitigate the secrecy-rate loss. Feedback channel is used in [18] and [19] to enhance the secrecy rate, and under certain conditions the authors prove that the secrecy capacity can approach the main channel capacity. In [20] the authors assume that the transmitter (Alice) and Bob have access to a secret key, and then they propose a coding scheme which utilizes that key to enhance the secrecy rate. Secure Multiplex scheme has been proposed in [21] which achieves Shannon channel capacity for a point to point wiretap channel. In this model multiple messages are transmitted. The authors show that the mutual information of the currently transmitted message with respect to (w.r.t.) all the information received by Eve goes to zero as the codeword length .

Shah et al. in [22] propose a simple coding scheme, without any feedback channel or access to some key, and enhance the secrecy capacity of a wiretap channel to the Shannon capacity of the main channel. In this work also, only the message currently being transmitted is secure w.r.t. all the information possessed by Eve. In [23] we extended the coding scheme of [22] to a multiple access wiretap channel and showed that we can achieve Shannon capacity region of the MAC as the secrecy rate region, while keeping currently transmitted message secure w.r.t. all the information of Eve. In this paper we extend the coding/decoding schemes of [22] and [23] to a multiple access wiretap channel and prove that we can achieve Shannon capacity region of the MAC as the secrecy-rate region while keeping all recent messages secure w.r.t. the information possessed by Eve till present. Finally we achieve the same for a fading MAC-WT.

Rest of the paper is organised as follows. In Section 1 we define the channel model and recall some previous results which will be used in this paper. We extend our coding/decoding scheme of [22] to two user discrete memoryless MAC-WT (DM-MAC-WT) in Section 2 and prove the achievability of Shannon capacity region, under the security constraint that only the currently transmitted message is secured w.r.t. all the data received by Eve. In Section 3 we consider a two user DM-MAC-WT where each user, receiver, as well as Eve have infinite length buffers to store previous messages. We propose a coding scheme to enhance the secrecy-rate region to Shannon capacity region of the usual MAC, this time with security constraint that all recent messages are secure w.r.t. all the information possessed by Eve. In Section 4 we consider a two user fading MAC-WT and extend the coding scheme of previous sections to enhance the secrecy-rate region of the fading MAC-WT to the Shannon Capacity region of the MAC in the ergodic sense. Section 5 concludes the paper. The Appendix at the end contains several lemmas used in the proofs of the main theorems.

In this paper random variables will be denoted by capital letters etc., vectors will be denoted with upperbar letters, e.g., , scalar constants will be denoted by lower case letters etc.

1 Multiple Access Wiretap Channel

Encoder 1Encoder 2BobEve
Fig. 1: Discrete Memoryless Multiple Access Wiretap Channel

A discrete memoryless multiple access channel with a wiretapper and two users is considered (Fig. 1). The channel is represented by transition probability matrix where , is the channel input from user , , is the channel output to Bob and is the channel output to Eve. The sets are finite. The two users want to send messages and to Bob reliably, while keeping Eve ignorant about the messages.

Definition 1.1.

For a MAC-WT, a codebook consists of (1) message sets and of cardinality and , (2) messages and , which are uniformly distributed over the corresponding message sets and and are independent of each other, (3) two stochastic encoders,

(1)

and (4) a decoder at Bob,

(2)

The decoded messages are denoted by .

The average probability of error at Bob is

(3)

and leakage rate at Eve is

(4)

Leakage Rate: In [6] the authors have defined two types of security requirements depending upon the trust of the transmitting users on each other. If each user is conservative such that when the other user is transmitting then it may compromise with Eve and provide Eve with its codeword, then individual leakage constraints

(5)
(6)

are relevant, where denotes the codeword for user .

In a scenario where users trust each other, collective leakage

(7)

is relevant. Since, and hence also where denotes that random variable is independent of ,

(8)

and hence, if individual leakage rates are small then so is the collective leakage rate. In this paper we consider the secrecy notion (1).

Definition 1.2.

The secrecy-rates are achievable if there exists a sequence of codes with as and

(9)

The secrecy-capacity region is the closure of the convex hull of achievable secrecy-rate pairs .

In [6], a coding scheme to obtain the following rate region was proposed.

Theorem 1.1.

Rates are achievable with , if there exist independent random variables as channel inputs satisfying

(10)

where and are the corresponding symbols received by Bob and Eve.

The secrecy capacity region for a MAC-WT is not known. If the secrecy constraint is not there then the capacity region for a MAC is obtained from the convex closure of the regions in Theorem 1 without the terms on the right side of (10) (Fig.2) [24]. In the next section we show that we can attain the capacity region of a MAC even when some secrecy constraints are satisfied.

Fig. 2: Capacity region and Secrecy Rate region of MAC

2 Enhancing the Secrecy-Rate Region of MAC-WT

In this section we extend the coding-decoding scheme of [22] for a point-to-point channel to enhance the achievable secrecy rates for a MAC-WT. We recall that in [22] the system is slotted with a slot consisting of channel uses. The first message is transmitted by using the wiretap code of [1] in slot 1. In the next slot we use the message transmitted in slot 1 as a key along with wiretap code and transmit two messages in that slot (keeping the number of channel uses same). Hence the secrecy-rate gets doubled. We continue to use the message transmitted in the previous slot as a key and wiretap coding, increasing the transmission rate till we achieve a secrecy rate equal to the main channel capacity. From then onwards we use only the previous message as key and no wiretap coding. This scheme guarantees that the message which is currently being transmitted is secure w.r.t. all the Eve’s outputs, i.e., if message is transmitted in slot then

(11)

as the codeword length , where is the data received by Eve in slot .

In the following, not only we extend this coding scheme to a MAC-WT but also modify it so that it can be used to improve its secrecy criterion (11) and for fading channels as well. The secrecy criterion used is the following: If user transmits message in slot , we need

(12)

for any given . This will be strengthened to strong secrecy, at the end of the section (See the next section for further strengthening of their criteria). We modify message sets and encoders and decoders with respect to Section 1 as follows.

Each slot has channel uses and is divided into two parts. The first part has channel uses and the second , . The message sets are for users , where satisfy (10) for some . The encoders have two parts for both users,

(13)
(14)

where , and are the sets of secret keys generated for the respective user, are the wiretap encoders corresponding to each user as in [6] and are the usual deterministic encoders corresponding to each user in the usual MAC. User may transmit multiple messages from in a slot. In the first part of each slot of length, one message from may be transmitted using wiretap coding via (denoted by in slot ) and in the second part multiple messages from may be transmitted (denoted by ) using messages transmitted in previous slots as keys. The overall message transmitted in slot by user is .

The following is our main result.

Theorem 2.1.

The secrecy-rate region satisfying (12) is the usual MAC region without Eve, i.e., it is the closure of convex hull of all rate pairs satisfying

(15)

for some independent random variables .

Proof.

We fix distributions . Initially we take . In slot 1, user selects message to be transmitted confidentially in the first part of the slot, while the second part is not used. Both the users use the wiretap coding scheme of [6]. Hence the rate pair satisfies (10) and . In slot 2, the two users select two messages each, and to be transmitted. Both users use the wiretap coding scheme (as in [6]) for the first part of the message, i.e., , and for the second part user first takes of with the previous message, i.e., . This ed message is transmitted over the MAC-WT using a usual MAC coding scheme ([24], [25]). Hence the secure rate achievable in both parts of slot 2 satisfies (10) for both the users. This is also the overall rate of slot 2.

In slot 3, in the first part the rate satisfies (10) via wiretap coding. But in the second part we with and are able to send two messages and hence double the rate of (10) (assuming via (10) is within the range of (15)). We continue like this (Fig 3).

Define

(16)

where is the smallest integer . In slot the rate of user 1 in the second part of the slot satisfies,

(17)

Similarly we define as

(18)

In slot , the rate satisfies

(19)

In slot , the sum-rate will satisfy

(20)

After some slot, say, , the sum-rate will get saturated by sum-capacity term, i.e., , and hence thereafter the rate pair in the second part of the slot will be at a boundary point of (15) and the overall rate for the slot is the average in the first part and the second part of the slot.

In slot , (where ) to transmit a message pair , where , we use wiretap coding for and for the second part, we it with the previous message i.e., , and transmit the overall codeword over the MAC-WT. (Fig. 3)

To get the overall rate of a slot close to that in (15), we make . By taking large enough, we can come arbitrarily close to the boundary of (15).

For this coding scheme, . A convex combination of the rates in (15) can be obtained by time sharing. Now we show that our coding/decoding scheme also satisfies (12).

Wiretap Coding only

Wiretap Coding

Secret Key

0

1

2

3

Fig. 3: Coding Scheme to achieve Shannon Capacity region in MAC

Leakage Rate Analysis: Before we proceed, we define the notation to be used here. For user , the codeword sent in slot will be represented by . Correspondingly, and will represent -length and -length codewords of user in slot . When we consider to be 1 or 2, will be taken as 2 or 1 respectively. In slot , the noisy codeword received by Eve is , where is the sequence corresponding to the wiretap coding part and is corresponding to the part (in which the previous message is used as a key).

In slot 1, since wiretap coding of [6] is employed, the leakage rate satisfies,

(21)

For slot 2 we show, for user 1,

(22)

Similarly one can show for user 2.

We first note that

(23)

where follows from wiretap coding and follows by the fact that , and .

Next consider

(24)

We get upper bounds on and . The first term,

(25)

where follows because . Furthermore,

(26)

where follows since and follows since the first part of the message is encoded via the usual coding scheme for MAC-WT.
Also,

where follows since .

From (24), (25) and (26) we have .

Next consider,

(27)

We have,

and follows since ; holds because