Wiretap Channel with Rate-Limited Channel State Information

. We revisit a channel coding problem where the channel state information (CSI) is rate-limited (or coded) and available to the channel encoder. A wiretapper is added into this model, and the confidential message is intended only for the legal receiver and should be kept from being eavesdropped by the wiretapper. Equivocation analysis is provided to evaluate the level of information leakage to the wiretapper. We characterize an achievable rate-equivocation region as well as an outer bound for this security model. To achieve the rate-equivocation triples, we propose an efficient coding scheme, in which the coded CSI serves as the CSI for the channel encoder, based on Gel’fand and Pinsker’s coding and Wyner’s random coding . Furthermore, an example of Gaussian wiretap channel with rate-limited CSI is presented, of which a lower bound on the secrecy capacity is obtained. By simulation, we find there exists an optimal rate of the coded CSI at which the biggest secrecy transmission rate of the Gaussian case is achieved.


Introduction
The problem of coding for channels with CSI has been studied actively.In these models, CSI is generated by nature and provided to the transmitter and/or to the receiver in a causal or noncausal manner.Shannon first studied the discrete memoryless channel (DMC) with causal CSI only at the encoder and got its capacity [1].It was found that the capacity was the same as the capacity of the DMC without CSI.The problem where CSI was only noncausally known to the channel encoder was solved by Gel'fand and Pinsker [2].They showed a different capacity expression from [1] and provided a different coding strategy which was now known as Gel'fand and Pinsker's coding.The key idea of Gel'fand and Pinsker's coding is that the codeword chosen for the transmitted message is jointly typical with the state sequence.Actually, the capacity for the causal case in [1] can be got from [2] by letting the auxiliary random variable be independent of CSI.Some other extended channel models with CSI at both the encoder and the decoder were studied in [3][4][5][6][7].Among them, Heegard and El Gamal first investigated a more practical channel with the unique feature that CSI at encoder and decoder was subject to a rate constraint [3].The motivation of this unique feature, that is, rate-limited CSI, was that CSI was transmitted over way-side channels for which only limited resources (bandwidth, operation time, memory, etc.) of the system were allocated.They only gave inner bounds on the capacity region for this model.Rosenzweig et al. then revisited this model and gave the capacity region of the model with rate-limited CSI at encoder and full CSI at the decoder [4].For the model where CSI was fully known at the encoder and rate-limited at the decoder, capacity region was obtained in [5].
It is known that secure information transmission is an essential communication requirement.The above statedependent channel models [1][2][3][4][5][6][7] considered no secrecy constraint.Recently, the works [8][9][10][11][12][13][14][15] have introduced the wiretapper in channels with CSI to model a safe communication model.Chen and Vinck explored the discrete memoryless wiretap channel with noncausal CSI and presented an achievable rate-equivocation region [8].The region was established using a combination of Gel'fand and Pinsker's coding and Wyner's random coding.They showed that CSI helped to get a larger secrecy capacity for the Gaussian wiretap channel.In [9], an achievable rate-equivocation region for the Gaussian wiretap with side information was given.The authors proposed a perfect-secrecy-achieving coding strategy for the model based on code-partition technique.The code-partition technique divided a random code into bins so that high rates could be achieved with asymptotic perfect secrecy.Dai and Luo improved the results of [8] by providing upper bounds on the secrecy capacity [10].Liu and Chen got a lower bound on the secrecy capacity of the model where CSI was available noncausally at both the encoder and the decoder [11], and an upper bound was established by [12].Action-dependent channel models involving secrecy were studied in [13][14][15].Dai et al. provided the inner and outer bounds on the capacity-equivocation region for the wiretap channel with action-dependent states [13].Then, they extended the model by adding a noiseless feedback link between the transmitter and receiver [14], in which the feedback served as a secret key.We restricted the rate of the feedback by introducing a rate-limited feedback link and obtained the corresponding capacity-equivocation region.Then, we explored information embedding on the actions in wiretap channel where part of the message was embedded on the actions [15] and characterized the tradeoff between the sum secrecy rate and the information embedding rate.
We are motivated to revisit a state-dependent channel model by adding a wiretapper (shown in Figure 1), that is, discrete memoryless wiretap channel with rate-limited channel state information.The CSI is known to the encoder and not known to the decoder.In our setup, the CSI at the encoder is subject to a rate constraint   , as in the work [3].Note that the CSI in previous works considered that secrecy constraint [8,10] was rate-unlimited, which is different from our rate-limited setup.However, we will later see that the model in [8,10] is actually a special case of our model.In addition, the confidential message is intended only for the receiver and should be kept secret from the wiretapper as much as possible.
To the best of our knowledge, the model in Figure 1 involving secure information transmission has not been explored.Our goal is to characterize the inner and outer bounds on the capacity-equivocation region of the model.Particularly, we are interested in getting the corresponding bounds on the secrecy capacity for which perfect secrecy is realized.The perfect secrecy means that no information is leaked to the wiretapper.We also provide efficient coding schemes to achieve the rate-equivocation triple by means of Gel'fand and Pinsker's coding and Wyner's random coding.Moreover, an example of Gaussian wiretap channel with ratelimited CSI is given and its lower bound on the secrecy capacity is also calculated.The simulation results show that there exists an optimal rate of the coded CSI at which the biggest secrecy transmission rate of the Gaussian example is achieved.
The remainder of the paper is organized as follows.Section 2 describes the wiretap channel with rate-limited CSI and outlines the inner and outer bounds on the capacityequivocation region.Section 3 calculates the corresponding bounds on the secrecy capacity and presents an example.Section 4 proposes a coding scheme to achieve the rateequivocation triples and gives the outer bound proof.We conclude in Section 5 with a summary of the whole work and some future directions.

Channel Model and Main Results
In this section, the model of wiretap channel with rate-limited CSI is described.Then, we present the inner and outer bounds on the capacity-equivocation region.

Channel Model.
In this paper, calligraphic letters, for example, X, are used to denote the finite sets.We use ‖X‖ to denote the cardinality of the set X. We use    to denote the ( −  + 1) vectors (  ,  +1 , . . .,   ) of random variables for 1 ≤  ≤  and will always drop the subscript when  = 1.Besides, for  ∼ () and 0 ≤  ≤ 1, the set of the typical -sequences   is defined as T   () = {  : |( |   ) − ()| ≤ () for all  ∈ X}, where ( |   ) denotes the frequency of occurrences of letter  in the sequence   .The set of the jointly typical sequences, for example, T   (), follows similarly.The base of the log function in this paper is 2.
The channel model is described as follows.We consider the rate-limited CSI setup where a rate-limited description of the channel states is provided to the channel encoder.This setup is motivated by the limited capacity of the channel over which the channel states are transmitted.The input of the state encoder is the channel state   which is independently and identically distributed (i.i.d) ∼ (), and the output is   ∈ {1, 2, . . ., 2 is the probability that the messages  and   are encoded as the channel input   .The decoder  3 : Y  → M maps the output sequence   into decoded message M. Before stating the main results, we give the definitions of "achievable" and "secrecy capacity" as follows.
Definition 1.A rate-equivocation triple (,   ,   ) is said to be achievable for the model in Figure 1 if there exists a lim where  is an arbitrary small positive real number,  is the rate of the message ,   is the rate of the coded message   , and   is the rate of equivocation.The capacity-equivocation region is defined as the convex closure of all achievable rateequivocation triples (,   ,   ).
Definition 2. The secrecy capacity is the maximum rate at which the confidential message can be sent to the receiver in perfect secrecy with the constraint on   .The secrecy capacity is where R is the capacity-equivocation region and Γ is the constraint condition of rate   .

Main Results.
We first give an achievable rateequivocation region for the wiretap channel with rate-limited CSI and then present an outer bound.Some comments on the theorems are given subsequently.Further discussion about the results and comparison with other existing models are shown in Section 3.

Theorem 3. An inner bound on the capacity-equivocation region of the wiretap channel with rate-limited CSI is the set
where  → (, ) →  →  form a Markov chain.
We have the following comments: (1) The sets R  and R  are convex; the proof is similar to the proof of Proposition 1 in [5].
(2) Theorem 3 tells that any rate-equivocation triple belonging to R  is achievable.Theorem 4 tells that all achievable rate-equivocation triples are contained in R  .The capacity-equivocation region of the model in Figure 1 is between R  and R  . (

Discussion and Gaussian Example
In this section, we first calculate the lower and upper bounds on the secrecy capacity of the model in Figure 1.Subsequently, we compare our results with other existing state-dependent channel models.Then, we provide a (physically) degraded example of Gaussian wiretap channel with rate-limited CSI and calculate a lower bound on the secrecy capacity subject to a rate constraint on   .By simulation, we find that there exists an optimal rate of the coded CSI at which the biggest secrecy transmission rate of the Gaussian example is achieved.
Proof.We first prove (8).According to the definition of formula ( 4 Based on the definition of secrecy capacity, (9), we prove (8).
The comparison of the secrecy capacity between [10] and our results is given as follows.We first present Dai's main results in [10].
Dai and Luo [10] characterized the lower and upper bounds on the secrecy capacity of wiretap channel with full and noncausal CSI at the encoder, depicted in Figure 2, as  The comparison is listed below.
(1) The lower bounds (7) and ( 12) share the same expression.We note that, in the rate-limited CSI setup, CSI is not directly known to the channel encoder.The CSI is encoded by the state encoder, and it is the coded version that serves as the channel states to the channel encoder.In the coding scheme (presented in Section 4), the coded CSI  serves as the CSI  in the jointly typical encoding.Based on this observation, it is easy to see that ( 7) and ( 12) share the same expression (without considering the constraint on rate of the coded CSI).
(2) For deriving the upper bounds ( 8) and ( 13), the auxiliary random variables are different.In [10], there are three auxiliary random variables  2 ,  1 , and , while there are four auxiliary random variables  2 ,  1 , , and  in our derivations.The detailed expression of these random variables is given in Section 4. ( where ( 14) is from the Markov chain  →  → .If we let  be independent of , the conditional mutual information (;  | ) = (; ).In this case, it is easy to see   ≤  dai .

Gaussian Example.
In this subsection, we look at the (physically) degraded Gaussian wiretap channel with ratelimited CSI shown in Figure 3.We treat  1 ∼ N(0,  1 ) and  2 ∼ N(0,  2 ) in Figure 3 as noise.Let  ∼ N(0, ) and  = .Similarly to [8,9], consider  =  + , where  and  are independent from each other and  is distributed as  ∼ N(0, ).The parameter  is to be determined.We assume that the correlation coefficient of  and  is   .Then, using similar derivations in [8,9], we have where Note that the expressions of (; ), (; ), (; ) are the same as those in [8,9], and the difference is that here we have the rate constraint on   ; that is,   ≥ Λ 2 .
Applying Corollary 5, we get the lower bound on the secrecy capacity of the Gaussian case as which is subject to the constraint   ≥ Λ 2 .The graph of  lG () as a function of  for fixed , ,  1 ,  2 , and   is presented in Figure 4.It can be seen that, for the six cases in Figure 4, the lower bound on the secrecy capacity achieves the maximum value at different values of .By setting a proper value to the parameter , we can achieve the maximum value of  lG ().Besides, we see that when the noise power of the wiretap channel decreases (or the noise power of the main channel increases),  lG () is reduced.This is straightforward.
It can be seen that, in general, when   is small,  lG (  ) increases with   .However, there exists an optimal value of   at which  lG (  ) is the biggest.When   is bigger than the optimal value,  lG (  ) decreases.On the one hand, we see that, for case 1 where variable  is independent of , secrecy rate  lG (  ) decreases sharply with the CSI rate   (when   passes the optimal value).On the other hand, when variable  is not independent of  (case 2-5),  lG (  ) decreases slowly with   .
Besides, comparing cases 2 and 3, we find that the smaller the noise power is, the higher the secrecy rate  lG (  ) is.This means if information transmission happens in good channels, we can achieve higher secrecy transmission rate.
Furthermore, comparing cases 4 and 5, it can be seen that when the main channel is less noisy than the wiretap channel, higher secrecy rate  lG (  ) is achieved.This result is straightforward since the main channel is better than the wiretap channel.

Proof of Theorems 3 and 4
In this section, two theorems in Section 2 are proved.To show Theorem 3, we construct a coding scheme to achieve the rate-equivocation triple in R  based on Gel'fand and Pinsker's coding and Wyner's random coding and give the equivocation analysis in Section 4.1.Then, we prove Theorem 4 and give the identification of the auxiliary random variables in Section 4.2.
To send , the transmitter tries to find a V  (  ,  sb ,  V ) such that (V  (  ,  sb ,  V ),   (  )) ∈ T   in the bin indexed by   = .Then, the input sequence of the main channel is generated by To decode the message, the decoder finds a unique codeword V  ( t , tsb , tV ) such that (V  ( t , tsb , tV ),   ) ∈ T   and outputs m = t .Since the number of the codewords V  is 2 ((;)−) , this decoding step succeeds with high probability.Moreover, the number of   (  ) is 2   ≥ 2 (;) , so we can find a   (  ) such that (  (  ),   ) ∈ T   with high probability.Similarly, since each bin contains 2 ((;)−−) = 2 ((;)+−) codewords V  , the error probability of finding a V  (  ,  sb ,  V ) such that (V  (  ,  sb ,  V ),   (  )) ∈ T   in a given bin indexed by   approaches zero.
We focus on analyzing the uncertainty of  given the wiretapper's observation   .Consider where ( 19) is from the Markov chain  →   →   and (20) is from the fact that the codewords V  are i.i.d and the channels are discrete memoryless.
Condition ( 24) is proved as follows.Consider In the above derivation, formula (30) is from the fact that the message is independent of the state sequence.Therefore, we have (;   ) = 0. Formula (31) is from the fact that (;   |   ) = 0 which is proved in (35).Consider

Condition (25) is proved as follows. Consider
where ( 37) is from the fact that   is independent of  −1 .Condition ( 26) is proved as follows.Consider The first two terms in (41) are calculated as follows.From (32), one has Similarly, we can get Then, we substitute (42) and ( 43

Conclusion
This paper has investigated the problem of rate-limited CSI in state-dependent channels under secrecy constraint.It is an extension of the rate-limited CSI setup without secrecy.This model is more general than the existing wiretap channel with full CSI at the transmitter.We proposed an achievable rate-equivocation region and an outer bound on the capacityequivocation region.To achieve the rates, a proper coding scheme is constructed based on Gel'fand and Pinsker's coding and Wyner's random coding.
Our setup can be probably extended in the following directions.
(i) In this current model as well as [1][2][3][4][5][6][7], the CSI is generated by nature.However, in some practical communication systems, the transmitter can take actions to affect the formation of the CSI.This idea called action-dependent states is from [18].We can further explore the impact of the action-dependent states in the wiretap channel with rate-limited CSI.
(ii) Only inner and outer bounds on the capacityequivocation region are obtained at present.We can try to find some special cases where the two bounds match.
(iii) In our model, the CSI is rate-limited and known to the encoder.We can study the model where the CSI is rate-limited and known to the decoder (or both the decoder and the encoder) under the secrecy constraint.

2 MathematicalFigure 1 :
Figure 1: Wiretap channel with rate-limited channel state information at the encoder.

Figure 2 :
Figure 2: Wiretap channel with full CSI at the encoder.

Figure 3 :
Figure 3: Gaussian wiretap channel with rate-limited channel state information at the encoder.
}.The channel encoder encodes the messages  and   into   .The main channel is a DMC with discrete input alphabet X × S and output alphabet Y.The channel is memoryless in the sense that (  |   ,   ) = ∏  =1 (  |   ,   ), where   ∈ Y  ,   ∈ X  , and   ∈ S  .The receiver decodes the message with the observation   .The output of the decoder is M. The probability of error for the decoder is defined as   = Pr{ M ̸ = }.The wiretap channel is also a DMC with input   and output   .The wiretap channel is memoryless in the sense that (  |   ) = ∏  =1 (  |   ), where   ∈ Z  .The uncertainty of the message for the wiretapper is measured by lim  → ∞ Δ = lim  → ∞ (( |   )/)., 2   , ) code for the model in Figure1includes a state encoder, channel encoder, and decoder.The state encoder  1 : S  → M  maps the state sequence   into   , where   ∈ M  = {1, 2, . . ., 2   }.The stochastic channel encoder  2 is specified by a matrix of conditional probability distributions  2 (  | ,   ), where This reminds us that it is sufficient to show that (,   ,   = (; )−max{(; ), (; )}) is achievable.
3.1.Discussion and Comparison.The lower and upper bounds on the secrecy capacity are presented in Corollary 5.