# On The Achievable Rate Region of a New Wiretap Channel With Side Information

###### Abstract

A new applicable wiretap channel with separated side information is considered here which consist of a sender, a legitimate receiver and
a wiretapper. In the considered scenario, the links from the transmitter to the legitimate receiver and the eavesdropper experience different conditions or channel states. So, the legitimate receiver and the wiretapper listen to the transmitted signal through the channels with different channel states which may have some correlation to each other. It is assumed that the transmitter knows the state of the main channel non-causally and uses this knowledge to encode its message. The state of the wiretap channel is not known anywhere. An achievable equivocation rate region is derived for this model and is compared to the existing works. In some special cases, the results are extended to the Gaussian wiretap channel.

## I Introduction

Secure communication from an information theoretic perspective was first studied by Shannon in his famous paper [1], where a noiseless channel model was assumed with an eavesdropper which has an identical copy of the encrypted message as a legitimate receiver, and the sufficient and necessary condition for perfect secrecy using information theoretic concepts were established. In the Shannon’s model, a source message is encrypted to a ciphertext by a key shared by the transmitter and the receiver. An eavesdropper, which knows the family of encryption functions, i.e., keys and the probability of choosing the keys, may intercept the ciphertext . The system is considered to be perfectly secure if the a posteriori probabilities of for all would be equal to the a priori probabilities, i.e., . Alternatively, Shannon proved that the perfect secrecy can be achieved only when the secret key is at least as long as the plaintext message or more precisely, when .

The wiretap channel was first introduced and studied by Wyner in his fundamental paper [2] which is the most basic physical layer model explains the communication security’s problems. In his model, the transmitter wishes to transmit a source signal, i.e., a confidential message, to a legitimate receiver in a way that this message be kept secret from an eavesdropper. In this model illustrated in Fig. 1, despite of the Shannon’s model, it is assumed that the channel to the eavesdropper is a physically degraded version of the channel to the legitimate receiver. In other words, the channel’s output at the eavesdropper may be a noisy version of the channel output at the legitimate receiver. On the other hand, the transmitter communicates to the intended receiver through the main channel which may be noisy or noiseless, but the wiretapper receives a noisy copy of the message through a wiretap channel which is a cascade of the main channel. In addition, Wyner [2] assumed that the eavesdropper knows the transmitter’s encoding-decoding scheme. So, the objective is maximizing the rate of reliable communication such that the wiretapper realizes as little as possible about the source output. The information leakage was measured by equivocation rate as , where and are represented the message set and the channel output at the wiretapper, respectively. Eavesdropper is assumed to be a passive receiver which does not transmit any signal over the channel. Furthermore, Wyner [2] proposed a basic principle coding strategy to achieve secure communication for wiretap channels which is based on the fact that the eavesdropper is not able to decode any information more than it’s channel capacity.

Csiszár and Körner generalized the Wyner’s wiretap channel [3]. In their model, it is assumed that the wiretap channel’s output is not necessarily a degraded version of the legitimate receiver’s one. They showed that the secrecy capacity can be expressed as , where , and are the channel input, the channel output in the legitimate receiver and the channel output at the wiretapper, respectively. Moreover, the maximization is over all random variables in joint distribution with , and such that forms a Markov Chain.

Using the channel state information in communication channel models was introduced by Shannon in his landmark paper [4], where he assumed the availability of Channel Side Information at the Transmitter (CSIT). Gel’fand and Pinsker in their essential work [5] proved that the capacity of the state-dependent discrete memoryless channel with non-causally CSIT is given by , where the maximum is taken over all input distribution with a finite alphabet auxiliary random variable .

Costa in his well known paper named Writing on Dirty Paper, extended this result to the Gaussian channel and showed that for this channel, interference did not affect the capacity [6]. He chose and maximized the Gel’fand and Pinsker’s capacity over all quantity of and proved that for this value of , the capacity of the channel reduces to the channel without states. The dirty paper channel was extended to the basic Gaussian wiretap channel with side information by Mitrpant and et al. [7], in which an achievable and upper bound for this channel has been introduced.

Chen and Vinck investigated Wyner’s wiretap channel with side information [8] (Fig. 2). Their results are based on the previous wiretap channel’s results in [2], [3], [7] and the discrete memoryless channel with state information [5]. They gave an achievable rate region which is established using a combination of the Gel’fand-Pinsker coding and the Wyner’s wiretap coding. They extended their results to the Gaussian wiretap channel with side information using the same technique like dirty paper channel [8].

Furthermore, there were some different works on the wiretap channel with and without side information. The work [10] studied the two way wiretap channel. The Gaussian wiretap channel with m-pam inputs was considered in [11] and the secrecy capacity of the Gaussian MIMO multi-receiver wiretap channel was investigated by [12]. Liu et al. in [9], studied the two-sided channel state problem in the discrete memoryless wiretap channel, where as shown in Fig. 3, the information of the two-sided channel states are available at the transmitter and the main receiver, respectively. In addition, in their scenario the wiretap channel is not necessarily a degraded version of the main channel. An achievable rate equivocation region for this general case is given in [9]. Khisti et. al., considered the secret-key agreement problem in the wiretap channel [13], [14]. In their model, the transmitter communicates to the legitimate receiver and the eavesdropper over a discrete memoryless wiretap channel with a memoryless state sequence. The transmitter and the legitimate receiver generate a shared secret key that remains secret from the eavesdropper. The results are comparable to the wiretap channel introduced by [8]. Recently, an improved lower bound for the wiretap channel with causal state information at the transmitter and receiver has been reported in [15], where the achievability of the rate region is proved using block Markov coding, Shannon strategy, and key generation from the common state information [4]. The state sequence available at the end of each block, is used to generate a key which is used to enhance the transmission rate of the confidential message in the following block.

In this paper, we introduce a new wiretap channel model with side information, in which the wiretapper’s messages is not a degraded version of the legitimate receiver’s one. On the other hand, the transmitter sends its message through the main and the wiretap channels. So, the receiver and the wiretapper listen to the sent message from the separated channels with different characteristics, i.e., different channel states. This model is a general case of Chen–Vinck [8] and Wyner wiretap channel [2] and reduces to these channels in special cases. We extend our model to the Gaussian wiretap channel where the states of the main and wiretapper channels are different with some correlation coefficients. In the Gaussian case, if the correlation coefficients are equal to one, our channel reduces to Chen–Vinck’s channel. The proposed channel is illustrated in Fig. 4.

The rest of the paper is organized as follows. In Section II, the channel model is introduced. The main results are presented in Section III. In Section IV, the proof of the main results are given. In Section V, the results are extended to the Gaussian case and the paper is concluded in the last section.

## Ii Channel Model and Preliminaries

First, we clear our notation in this paper. Let be a finite set. Denote its cardinality by . If we consider , the members of will be written as , where subscripted letters denote the components and superscripted letters denote the vector. A similar convention applies to random vectors and random variables, which are denoted by uppercase letters.

Consider the situation shown in Fig. 4. Assume that the state information of the main channel, i.e., the channel from the transmitter to the legitimate receiver, is known at the encoder non-causally but the state of the wiretapper’s channel is unknown and the channels’ states, i.e. , , , are independent and identically distributed (i.i.d), but and are correlated. The transmitter sends the message to the legitimate receiver in channel uses. Based on the and , the encoder generates the codeword and transmits it on the main and the wiretap channels. The decoder at the legitimate receiver makes an estimation of the transmitted message based on the received message . The corresponding output at the wiretapper is . The channels are memoryless, i.e.,

(1) | |||

(2) |

Assume that is uniformly distributed on , so . The average probability of error is given by

(3) |

We define the rate of the transmission to the intended receiver to be

(4) |

and the fractional equivocation wiretapper to be

(5) |

Obviously, we have .

##
Iii Main Results:

outer and inner bounds

Like [8], we say that the pair is achievable, if for all , there exists an encoder-decoder pair such that

(6) |

Definition 1: The secrecy capacity is the maximum such that is achievable.

Definition 2: We denote

(7) | |||||

(8) | |||||

(9) |

where is an auxiliary random variable such that forms a Markov chain. Now, consider the following result:

Theorem 1: For the discrete memoryless channel with side information shown in Fig. 4, we denote as the set of points with , and . Let

(10) |

Then the set , defined as following, is achievable:

(11) |

The region is achievable if we limit the cardinality of by the constraint .

###### Proof.

The proof of the theorem is relegated to the next Section. The constraint is implied by lemma 3 of [16]. ∎

Remark 1: The point in with is of considerable interest. These situations correspond to the perfect secrecy situation, defined as

(12) |

The following theorem bounds the secrecy capacity of the proposed wiretap channel with the side information.

Theorem 2: For the discrete memoryless wiretap channel with side information, shown in Fig. 4, we have

(13) |

where is the capacity of the main channel.

###### Proof.

From Theorem 1, we have and from the result by Csiszár and Körner [3] we have . This completes the proof. ∎

## Iv The Proof of Theorem 1

In this Section, we prove the achievability of the region . We prove that the rate equivocation pairs and are achievable and then by implying time–sharing, achievability of the region is proved.

### Iv-a , 1) is Achievable

First we construct random codebooks by the following generation steps:

#### Iv-A1 Codebook Generation

.

a. Generate i.i.d sequences , according to the distribution .

b. Partition these sequences into bins where . Index each bin by . Thus each bin contains sequences.

c. Distribute sequences randomly into subbin such that every subbin contains sequences. Then index each subbin which contains by

#### Iv-A2 Encoding

To transmit message thorough the main channel with interference , the transmitter finds -th bin of the sequence such that . We use to denote the strong typical set based on the distribution , otherwise choose . The transmitter sends the associated jointly typical generated according to

#### Iv-A3 Decoding

The intended receiver receives according to the distribution . Then it looks for the unique sequence such that and the index of the bin containing is declared as the transmitted message.

#### Iv-A4 Wiretapper

The wiretapper receives a sequence according to .

Now, we prove that is achievable. As the first step we should prove that , as . Our encoding-decoding strategy is similar to the one used in [8] and it is easy to show that the information rate in the main channel is achievable. For more detail see Appendix A in [8]. As the second step, we should prove that , as . In this step, we consider the uncertainty of the message to the wiretapper. So we have

where

follows from the fact that ;

is because of the fact that and

follows from the fact that , and

To compute the second term in (IV-A4), we should bound the entropy of the codeword conditioned on the bin , subbin and the wiretapper’s received signal . We consider the subbin in bin as a codebook, in the codebook as the input message and as the result of passing through the wiretap channel. From , the decoder estimates the sent message . Let be the decoder and the estimate be . Define the probability of error

(15) |

By Fano’s inequality [17], we have

(16) |

Hence

(17) |

Now, we should prove that for arbitrary , . The proof is similar to the one in [8]. Thus, we have bounded for given arbitrary small and .

Thus we derive that , as .

### Iv-B is Achievable

From the (7)- (9), it is derived that if , then the equivocation rate pair is equal with . So, we should prove that if , then is achievable. In this case, when , we have

(19) | |||||

(20) |

Now we introduce the encoding and decoding strategy.

#### Iv-B1 Codebook Generation

.

a. Generate i.i.d sequences , according to the distribution .

b. Partition these sequences into bins where . Index each bin by . Thus each bin contains sequences.

c. Distribute sequences randomly into subbins such that every subbin contains sequences. Then index each subbin containing by .

#### Iv-B2 Encoding

To transmit message thorough the main channel with interference , transmitter finds bin for a sequence such that , otherwise choose .

#### Iv-B3 Decoding

The intended receiver receives according to the distribution . Then the receiver looks for the unique sequence such that and the index bin of the bin containing declares as the message index.

#### Iv-B4 Wiretapper

The wiretapper receives a sequence according to .

To prove that is achievable, first we should prove that , as . The proof is similar to the one in Section IV-A. Then we should prove that , as . For this purpose we can follow the strategy in Section IV-A. So we have

(21) | |||||

and for the second term in (21) like (15) – (17) we have

(22) |

So, combining the above results, we have

(23) |

Thus we have , as .

## V A New Gaussian Wiretap Channel

In this Section we extend Theorem 1 to the Gaussian case like the approach taken in [8], using the same auxiliary random variable . For the new Gaussian wiretap channel shown in Fig. 5, we have the following results based on Theorem 1.

Theorem 3: (Theorem 1 in Gaussian case For the Gaussian wiretap channel shown in Fig. 5) Using the auxiliary random variable , where is a real number and is the correlation coefficient of and , we denote as the set of points with , , , where and are defined in (7) and (8). By defining

(24) |

the set , defined as follows, is achievable:

(25) |

###### Proof.

The proof is similar to the proof of Theorem 1. We only need to show that is achievable for the specified and . Assuming transmitter has the power constraint , the side information in the main channel satisfies , the wiretap channel has the side information, satisfying , , and represent the correlation coefficient between , and and (see Appendix A), we use some modification in the proof of as follows.

In the codebook generation, sequence are generated according to , where for all . In the encoding process, . The intended receiver observes and the wiretapper observes . As a source constraint, we should introduce potential error , which represents in the encoding process and does not satisfy the power constraint.

Then, provided that there is at least one sequence jointly typical with , the probability of error tends to zero. Therefore, the modifications do not influence the achievability proof of . Assuming is arbitrarily small, since . ∎

Now, we calculate , and , with respect to . We have

(26) | |||||

Then, we introduce Leakage Function which is defined as . Thus, we have

(29) | |||||

Hence

(30) |

and we can find two points and in which

(31) |

Furthermore, there is a point in which is maximized, i.e.,

(32) |

where

(33) |

Now, we want to study the leakage function. So, denote and . Because of the complexity of the results, we consider two special cases.

### V-a Case I

As the first condition, we assume that , and . In this case our model reduces to the channel introduced [8] and we have

(34) |

which is maximized by as described in [7] and achieves , in which is the maximum rate of the main channel. It can be found easily that is an increasing function with respect to as , a decreasing function with respect to as .

Similarly, the rate has two extremum points in and and it can be shown that is a decreasing function with respect to as