Power Allocation Strategy for Secret Key Generation Method in Wireless Communications

: Secret key generation (SKG) is an emerging technology to secure wireless communication from attackers. Therefore, the SKG at the physical layer is an alternate solution over traditional cryptographic methods due to wireless channels’ uncertainty. However, the physical layer secret key generation (PHY-SKG) depends on two fundamental parameters, i.e., coherence time and power allocation. The coherence time for PHY-SKG is not applicable to secure wireless channels. This is because coherence time is for a certain period of time. Thus, legitimateusers generate the secret keys (SKs) with a shorter key length in size. Hence, an attacker can quickly get information about the SKs. Consequently, the attacker can easily get valuable information from authentic users. Therefore, we considered the scheme of power allocation to enhance the secret key generation rate (SKGR) between legitimate users. Hence, we pro-pose an alternativemethod, i.e., a power allocation,to improve the SKGR. Our results show 72% higher SKGR in bits/sec by increasing power transmission. addition, the power transmission based on important parameters, and power loss factor impacts power transmission and, consequently, impacts the SKGR.


Introduction
Security is profoundly important due to the rapid increase in wireless communication. In 2018, Ericsson announced that 5G subscribers would hit 1.9 billion by the end of 2024. It is also predicted that the networks would hold 35% of data and serve 65% of the global population [1]. Researchers are exploring different ways to meet new technological requirements, such as increasing bandwidth performance, coverage areas, and latency. However, security problems have not yet matured in wireless communications. Multiple attacks, such as impersonation, eavesdropping, and information modification, may endanger wireless communications. Such attacks target the authentic users to extract secret information between authentic users. Traditional cryptographic methods are usually used to secure data based on secret keys (SKs) between authentic users [2]. However, this technique is less attractive for distributed systems since mobile devices have minimal computational resources, unlike centralized networks [3,4]. Furthermore, the SKs are dependent on every user to keep the public key certificate in traditional cryptographic [5]. Hence, mobile devices can't carry a public key certificate in a distributed network due to limited resources [6][7][8][9].
Alternatively, Shannon's well-known work showed that channel reciprocity among authentic users at the physical layer (PHY) had achieved special consideration [10,11]. The channel reciprocity involves generating the SKs using the channel randomness between communicating parties [12,13]. However, SKG at the physical layer is essential in identifying the information based on channel state information (CSI). It offers the opportunity to imitate or require the characteristics of channels [14]. For example, the channel randomness of authentic users is unknown to unauthorized users [15]. In addition, PHY-SKG may not require any computational complexity due to channel randomness. Also, no key management scheme is needed for PHY-SKG [16]. Furthermore, the PHY-SKG leverages the dynamic channel variations to alleviate the complication by enabling the one-time pad scheme [17]. PHY-SKG overcomes the key distribution problem, and hence, the keys are distributed dynamically based on wireless channel reciprocity.
Keeping the above discussion, the researchers in [18] proposed the received signal strength (RSS) technique for SKG. The researchers implemented the RSS technique to improve the SKG rate [19]. Nonetheless, RSS-based SKG is not feasible for a distributed network. It is because RSS requires advanced algorithms to deliver a satisfactory SKGR. The authors in [20] suggested a relay-based SKGR scheme and addressed an idle intruder's optimal power distribution. The work indicated that SKs are generated with the help of the relay node. However, relay-based SKG is not feasible for the generation of secret keys. This is due to the reason that authentic users must also secure SKs against relay nodes. In another article, the authors have proposed the PHY-SKG scheme in [21] that takes advantage of power and error correction by exploiting RSS. Again, leveraging the RSS for SKG is not feasible because the RSS technique generates a low rate of SKs that restricts their use. To solve this problem, we consider CSI an alternative method to generate SKs in wireless communications.
Moreover, in [22], the authors considered the PHY characteristic of wireless channels, i.e., time allocation for maximizing group SKs. In [23], the authors proposed a reinforcement learning technique to generate SKs in vehicular communications. The proposed method is applicable in a dynamic environment. However, the authors did not find the channel's variations due to vehicles' high speed. Alternatively, we believe that the power allocation strategy enhances SKGR instead of coherence time. Essentially, the duration of coherency is for a particular time duration. Since users utilize shorter SKs in a practical scenario, the attacker can quickly collect SKs among legitimate users.
Nonetheless, the research indicates that low SKGR is the main limitation for PHY-SKG [24]. Furthermore, due to the limited period, i.e., coherence time, legitimate users produce shorter SKs. Hence, we investigate the impact of power allocations on generating and improving SKGR. Our significant achievements are summarized as follows.
• Due to the limited time duration, the authentic users generate shorter length of SKs. Therefore, an attacker can get information about SKs among legitimate users. Conversely, we examine the power allocation strategy to generate SKs. We illustrate a power allocation scheme to investigate SKGR. • Our results show 72% higher SKGR (bits/sec) at higher power allocation than low power allocation. To prove our result, we also analyze other factors, such as epsilon ( ), and the power factor loss (α).
The rest of our paper is organized as follows. We illustrate the system model, formulation, and proposed solution in Section 2. The simulation results are discussed in Section 3. Section 4 concludes the paper.

System Model, Formulation, and Proposed Solution
In the system model, two authentic users, i.e., u 1 and u 2 are considered. First, u 1 transmits the signal S u 1 . The receiver u 2 , receives the signal R u 2 = G 1 S u 1 + n u 2 . Here, G 1 represents the channel gain while n u 2 represents the noise factor at u 2 . Likewise, u 2 transmits the signals S u 2 and u 1 receives the signal, i.e., R u 1 = G 2 S u 2 + n u 1 . Here, G 2 represents the gain of channel while n u 1 denotes the noise factor at u 1 . The authentic users, i.e., u 1 and u 2 assume the channel gain G 1 and G 2 , respectively. Furthermore, we assume S u 1 be the transmitted signal by u 1 . Hence, the channel gain at u 2 is where S * u 1 is the conjugate of S u 1 . Similarly, the E at u 1 is The fundamental description of SKGR between u 1 and u 2 is described as the mutual information MI E u 1 , E u 2 and coherence time, T [10,14], i.e., Since and The correlation coefficient between E u 1 and E u 2 is Therefore, the covariance matrix of and The entropy can be calculated by Substituting (4), (5), and (9) into (10) leads to Let p be the transmitted power, and T denotes the channel's coherence time. Due to an optimal coherence time length T 2 , the signal is S u 1 , S u 2 2 = p T 2 . Thus, (12) indicates that γ u 1 , u 2 is equal to the coherence time and power allocation. The SKGR is low if the coherence time increases and vice versa. On the other side, if we increase the power transmission, the SKGR increases proportionally. In this work, we suppose that the power is distributed and allocated equally to both u 1 and u 2 . In a realistic scenario, users send data over several links concurrently by allocating power. From (12), it is also indicated that SKGR can be calculated by taking power allocation into account, i.e., max p u 1 , p u 2 γ u 1 , u 2 (13) where p u 1T , and p u 2T , are the total powers transmitted by u 1 and u 2 , respectively. Nonetheless, to get the optimal solution, we need the validation of convexity and concavity of our objective functions, as mentioned in (14). From (14), the objective function's problem for power maximization is non-concave. However, (13) indicates the objective function is concave with respect to p, and finding the optimal solution is difficult. Therefore, we optimize the γ u 1 , u 2 as a function of p u 1 , and p u 2 , respectively. Furthermore, we also considered Lagrangian form on γ u 1 , u 2 as a function of p according to [20], i.e., and assume the conditions of Karush Kuhn Tucker (KKT) as It is observed from (16) that 1 > 2 ≥ 0. Therefore, p u 1T − p u 1 = 0. It is noted that the transmitter uses power higher than zero, i.e., 2 ≥ 0, while comparing KKT parameters as indicated in [24][25][26]. It is assumed that the power distribution at the transmitter side (p u 2 ) is initially distributed equally. Therefore, as outlined in Algorithm 1, we can resolve the proposed power allocation strategy in Algorithm 2. The symmetric method of SKG helps us to rewrite the Lagrangian for all steps of the power allocation process in a similar way. From the power allocation of u 1 , the following optimization problem at u 2 is given by Now, we apply the same approach as discuss for the power allocation of u 1 , and is given by Algorithm 1 can be updated for the transmitter u 1 based on (18). Consequently, a locally optimal solution can be achieved on both sides [22]. Thus, the power allocation of u 1 is discussed in Algorithm 2.

Simulation Results
We figure out the SKGR by considering power allocation and to exploit different parameters in our simulation results. The coherence time is set to T = 20. The variances, i.e., v 1 & v n are set to 1 [27]. The α is initialized to 4. In our simulation results, Fig. 1 shows that if we increase the power transmission, the SKGR increases accordingly. For example, at 50 mW, the SKGR is 17 bits/s when the distance between u 1 and u 2 is 70 m. Furthermore, the SKGR is 26 bits/s at 20 m between legitimate users by considering 50 mW power. This indicates that more power is needed to produce higher SKs. Nonetheless, when we increase the distance between u 1 and u 2 , the SKGR decreases because of the large distances between legitimate users. This is because when the distance increases, the SNR decreases between u 1 and u 2 , and hence, SKGR decreases. Nonetheless, the result also reveals that the SKGR rises with increasing power, regardless of the distances between u 1 and u 2 . It proves that even though the distance can impact the SKGR because of increased power, the SKGR increases. For illustration, the SKGR is approximately equal to 25-26 bit/s at 250 mW. The results also indicate that the SKGR depends not only on the coherence time but also on the transmission power. We also analyze the power transmission versus by varying the power loss factor α (ranges from 2.5 to 4). The result is significant to our work due to the power transmission's symmetry as given in (16). The change in causes an increase in transmission power, as depicted in Fig. 2. Nevertheless, a higher power factor loss (α) would result in low transmission power than a low power factor loss (α). The transmission power at = 0.5 is 100 mW at α is 4. In addition, the transmission power is approximately 148, 152, and 154mW at α = 3.5, α = 3 and α = 2.5, respectively, at = 0.5. Hence, the value of will consequently impact the power transmission at a different value of the power loss factor (α). Ultimately, it influences SKGR. This is because if power increases or decreases, the SKGR will increase or decrease accordingly.  Finally, by varying the value of , we also investigate SKGR with respect to transmission power. It is noticed from Fig. 3 that preferably, we can get the highest SKGR when the value of = 0. With the rise in transmitting power at different values of , the SKGR improves.
For example, the SKGR is about 27, 23, and 20 bits/s at 100 mW, for = 1, 0.5, and 0, respectively. The reason is that a higher value of impacts power transmission and subsequently impacts the SKGR. Nevertheless, by increasing the transmitting power, the SKGR is also improved. For instance, when the value of = 1, the SKGR is approximately 40.7% greater at 250 from 50 mW. The SKGR is also 18.5% higher at 250 from 50 mW at = 0.5.

Conclusion
We introduced a mechanism to generate SKs and enhance the SKGR with power allocation. It guarantees the reliability of decentralized wireless networks. From the existing works, it is noticed that the coherence time for SKGR may not always be possible because coherence time produces a small length of SKs. Consequently, the intruders can easily obtain the SKs between authentic users. Therefore, we consider the power allocation scheme to generate SKs and enhance SKGR. Our research has shown that we can get a higher SKGR by increasing the transmitting power. The simulation results showed that SKGR is approximately 72% higher at higher transmission power. We also considered the value of , distance and power factor loss, α to verify the power allocation concept on SKGR. The SKGR is approximately 40.7% greater at 250 from 50 mW at epsilon = 1. The value of SKGR is reduced to 18.5% at the same power transmission when epsilonis 0.5. Furthermore, the transmission power is also measured against the different power loss factor values, i.e., 3.5, 3, and 2.5, respectively. Hence, it is concluded that the value of epsilon and power loss factor impacts power transmission and, consequently, impacts the SKGR.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.