Physical Layer Security in IRS-Assisted Cache-Enabled Satellite Communication Networks

This paper presents a comprehensive analysis of the physical layer security performance of a cache-enabled satellite communication network that incorporates intelligent reflecting surfaces (IRS) in the presence of a passive eavesdropper. In the proposed system, content caches are deployed at both the ground station and the satellite, which can improve system performance by reducing latency and transmission overhead. Moreover, the use of IRS provides an additional layer of security by enabling the manipulation of the reflected signals to impede eavesdropping. Practical channel models are used to derive connection probability and secrecy probability for both the ground station-IRS-user and the satellite-IRS-user links. The obtained results are then used to evaluate the system’s secure transmission probability, which is maximized subject to the caching probabilities and transmission rate constraints. The paper presents numerical results to demonstrate the accuracy of the analysis and the effectiveness of deploying IRS and caching to support secure content delivery. The findings provide valuable insights into the potential benefits of utilizing IRS and caching technologies in satellite communication networks for improved physical layer security.


I. INTRODUCTION
I NTEGRATING satellite communications (satcom) into ter- restrial networks is a vital solution to deliver the Internet of Everything.To overcome high service latency and pricey bandwidth issues in satcom, edge caching, which allows popular contents to be pre-stored at edge devices during off-peak hours, has emerged as a promising solution [1].Edge caching in satellite-terrestrial network (STN) has been proved to effectively offload the backhaul of terrestrial networks [2], reduce the content delivery time [3], [4], enhance the outage probability [5], and the successful delivery probability [6], etc. Nonetheless, satcom is more exposed to security vulnerabilities due to the openness of transmission medium as well as the large beam coverage area of satellites.To provide wireless transmission confidentiality, two types of approaches are typically employed: the upper-layer cryptographic encryption and the physical layer security (PLS) techniques.The latter approach has attracted much research attention for two reasons.Firstly, unlike cryptography, it does not rely on computational complexity, ensuring security is not compromised by powerful unauthorized devices.Secondly, PLS is highly scalable, making it suitable for a network with decentralized devices of varying computational capabilities and power.This makes it useful for both direct secure data communication and for distributing cryptographic keys in the network [7].Recently, there have been numerous works investigating PLS in cache-enabled wireless networks, for example see [8], [9], [10], [11], [12], [13], [14] and references therein.Enabling the caching capability at base stations (BS) and users in a wireless heterogeneous network, [8] designed two transmission schemes at symbol and bit level to improve the transmission security.In these transmission schemes, the BS transmits a combination at symbol and bit level of a requested file with a pre-cached file at user to degrade the eavesdropper's channel.Studying the PLS in cache-enable cellular networks where the caching ability is enabled at micro BSs, [9] and [10] designed the cache placement and file delivery to achieve secure transmissions against randomly distributed eavesdroppers.The secrecy throughput and secrecy energy efficiency have been investigated in [9], while [10] analyzed the secure content delivery probability.The multi-input multi-output transmission in a multi-cell network has been secured by deploying caching to enhance the secrecy rate in [11].The PLS in a large-scale edge caching network was studied in [12] with a focus on the secure content delivery probability problem.Reference [13] proposed a two-hop edge caching scheme in 6G networks to protect data against being eavesdropped, and the PLS in cacheenabled mmwave heterogeneous networks was studied in [14].While these studies have investigated PLS in cache-enabled wireless networks, none of them have focused specifically on integrating satcom into terrestrial networks.The integration of satcom into terrestrial networks presents inherent challenges such as channel modeling for satellite and terrestrial communications links, transmission schemes, and caching designs, among others [1].These challenges make it necessary to explore the potential of PLS in cache-enabled STN to ensure secure and efficient data transmission.Thus, further research is needed to investigate the feasibility and effectiveness of PLS in STN, which would help address the security concerns associated with the openness of the transmission medium and large beam coverage area of satellites.
The recent deployment of intelligent reflecting surfaces (IRS) in wireless communication systems has led to more efficient PLS techniques [15].An IRS is a metasurface composed of low-power passive reflecting elements that can be reconfigured to modify the amplitude and/or phase of incident signals, reflecting them to a desired location.In PLS systems, the IRS is utilized to enhance the signal-to-noise ratio (SNR) at authorized users while simultaneously decreasing the SNR at eavesdroppers.IRS is particularly compelling for wireless systems as it can reshape the wireless environment without incurring huge complexity and cost to enhance PLS [15].However, research on IRS-assisted PLS systems has not yet matured enough.There are only a few works on IRSassisted terrestrial PLS systems [16], [17], [18], [19], [20], [21].It is important to note that the IRS-related channels in all of these works are investigated with terrestrial channel characteristics.PLS with IRS-space related channels has not yet been addressed.
Motivated by the above discussions, this paper investigates the potential of PLS in a two-tier cache-enabled IRS-assisted STN, where content caches are deployed at both the satellite and ground station.Such systems can find applications in a variety of scenarios where reliable, high-speed connectivity is necessary.By enabling cache at both the BS and satellite, and allowing the IRS to assist both the BS-user link and the satellite-user link, they can provide efficient use of radio resources, reduced communication latency, and increased system capacity.The two-cache tier configuration has been proposed in previous works to improve satellite bandwidth consumption [3] and service delivery time [6].To improve the cache hit ratio, probabilistic caching is employed at both cache tiers in this study.While this work does not explicitly consider the green communication potential of the system, it should be noted that a cache-enabled IRS-assisted STN can significantly benefit green communication by reducing energy consumption through content caching at the satellite and ground station and enabling passive beamforming with IRS, thus eliminating the need for active transmitting and receiving elements.However, this work focuses on the overall system secure transmission analysis of novel cooperative two-hop content delivery schemes and the optimization of caching probabilities and transmission rates.The main contributions of this paper are summarized as follows.
1) We propose an IRS-assisted cache-enabled STN where a probabilistic caching policy is employed at the two cache tiers.Additionally, a novel two-hop content delivery scheme is introduced, with the IRS providing assistance in securing the transmission from the ground station and satellite to the user.
2) We evaluate the connection probability and secrecy probability of content delivery over two cascaded fading channels: the Rayleigh-Rayleigh fading channel for the ground station-IRS-user link, and the Rayleigh-Shadowed-Rician fading channel for the satellite-IRS-user link, with provided asymptotic and closed-form expressions.Based on these results, we assess the system's secure transmission probability in combination with the probabilistic caching policy.
3) We jointly design the transmission rates and caching probability to maximize the system's secure transmission probability.The non-convex optimization problem is decomposed into two steps.In the first step, we find the transmission rates that maximize the secure transmission probability in the ground station-IRS-user and satellite-IRS-user schemes.In the second step, we derive the caching probability that maximizes the secure transmission probability.
4) We evaluate the PLS performance for the two-tier cacheenabled STN with and without the IRS, as well as the IRS-assisted single-tier versus two-tier cache-enabled STN.The numerical results confirm the validity of the theoretical analysis and verify the independence in optimizing the redundant rate and caching probability.The importance of optimizing the redundant rate to achieve the highest secure transmission probability while saving resources is also highlighted.The results demonstrate that having IRS and two cache tiers enabled can improve the secrecy probability by almost 50% compared to scenarios without IRS and single cache tier.
The remainder of this paper is organized as follows.Section II describes the proposed system model and the two-hop content delivery scheme.Section III analyzes the connection probability and secrecy probability of the transmission for ground station-IRS-user and satellite-IRS-user cases.Section IV jointly designs the transmission rates and the caching probability to maximize the overall system secure transmission probability.The theoretical analysis is validated by numerical results in Section V, and conclusions are made in Section VI.

A. Network Model
We consider a two-tier cache-enabled satellite-terrestrial system in Fig. 1, assisted by an IRS.The system comprises a gateway, a GEO satellite (S), a ground station (G), an IRS, a user (U), and an eavesdropper (E), all equipped with single antennas.The IRS is composed of M passive reflecting elements, each capable of independently adjusting the phase of incoming signals.We assume that there is no direct link between the gateway and the ground station due to their remote locations.Additionally, we assume that there is no direct link between the satellite and the users.This can occur in various scenarios where the signal from the satellite is obstructed or weakened, making it difficult or impossible for users to establish a direct link with the satellite.In such cases, an IRS can be used to reflect the satellite signal and establish a link with the users.
There are two caching tiers in the proposed system, one at the ground station G and the other at the satellite S, each with a storage capacity of C 1 and C 2 bits, respectively.To efficiently serve the user's requests, both the satellite and the ground station can pre-cache the content during off-peak hours from the gateway.The file library is hosted by the gateway and consists of N files denoted as W 1 , . . ., W N , where each file has an equal size of F bits. 1 The user requests a file W n with probability q n that follows the Zipf distribution [22], i.e., q n = n −α /( N m=1 m −α ), where α ∈ (0, 1).If the requested file is available in one of the cache tiers, a cache hit event occurs, and the file is served directly from that cache.Otherwise, a cache miss event occurs, and the requested content is served from the gateway since it is not possible to pre-cache all contents due to limited caching capacity.Let A i = {a i,1 , . . ., a i,n , . . ., a i,N }, i = 1, 2 denote the caching probabilities with a i,n being the probability that file W n is pre-fetched at cache C i .The caching probabilities are subject to the condition that the sum of the probabilities of all files stored in a cache tier should not exceed its storage capacity, i.e., N n=1 a i,n ≤ C i /F , (i = 1, 2) and 0 ≤ a i,n ≤ 1, (i = 1, 2; n = 1, . . ., N ).The whole-file caching strategy is assumed in this work.

B. Channel Model
We adopt block-based communications, where a transmission section is accomplished within a coherence time T (seconds).The channel fading consists of large-scale and small-scale fading.The large-scale fading is modeled by the distance-dependent power-law path-loss attenuation d −α i , where d denotes the distance, and α i represents the path-loss exponent.Since the satellite employed is geostationary, its distance to the devices on the ground can be considered timeinvariant.For small-scale fading, shadowed-Rician fading [23] and Rayleigh fading models have been widely adopted for satcom and terrestrial channels.Let hmn denote the small-scale fading coefficient of the link between nodes m and n.
The satellite link composes of multipath fading, which consists of one line-of-sight (LOS) and multiple weak scatter component, and shadow fading, which has LOS and multiplicative shadow fading [23].The satellite channel fading coefficient is governed by the following distribution with coefficients , where b 1 represents the average power of the scatter components; Ω 1 represents the average power of the LOS component; m 1 is the Nakagami parameter; 1 F 1 (•; •; •) is the confluent hypergeometric function of the first kind [24, eq. ( 9.210.1)].
The terrestrial link is modeled as the independent and identically distributed Rayleigh fading channel with the probability density function of where hmn is the average channel power gain taking into the effects of small-scale fading.
We assume that the satellite and ground station have access to only the statistical channel state information (CSI) instead of instantaneous CSI for both the main and wiretap channels,2 hence fixed rates are employed as discussed in the following.

C. Security Performance Metrics
To secure the content delivery, the well-known Wyner's wiretap encoding scheme [25] is employed, where the confidential information is encoded and redundant information is embedded to confuse the eavesdroppers.Let R s and R e respectively denote the secrecy data rate and the redundant information rate.The transmission rate of the codewords is defined as If the achievable rate of the legitimate link is more than R t , the user can recover the secret message and a connection is established.The probability of this event is called connection probability (CP) p c .If the achievable rate of the wiretap link does not exceed R e , the eavesdropper is not able to decode the secret message, then the transmission is secured.The probability that this event is referred as secrecy probability (SP) p s [10], [26].Mathematically, we have: where γ u and γ e represent the SNRs of the user and the eavesdropper links, respectively.The content delivery is secured only if both the connectivity and secrecy are guaranteed simultaneously.

D. Content Delivery Scheme
We consider the following transmission scheme.When the ground station receives a request for file W n , it acts based on the file caching status.
1) Case 1: If W n is cached locally, the ground station will directly transmit the file to the user (one-hop transmission).2) Case 2: If file W n is not cached at ground station, it will forward the request to satellite.There are two cases: • If the file is cached at the satellite, which will transmit the file to the user via IRS (one-hop transmission).• Otherwise, it will be retrieved from gateway, then relayed through satellite to the user (two-hop transmission).We can see that a secure transmission happens only when the user establishes a reliable connection with ground station or satellite, and the confidentiality of the requested file is guaranteed during the transmission.Hence, the probability of secure one-hop transmission event for file W n is p secure,1 (n) = a 1,n p c,1 p s,1 + a 2,n p c,2 p s,2 .The event of secure two-hop transmission for file W n is equivalent to that the gateway-satellite link is able to support the transmission of the whole file W n within a half transmission block, and the transmission from satellite to user is secured, and that file W n is not cached at both satellite and ground station.Let γ Gw denotes the SNR of the gateway-satellite link, the probability that the link can support whole file transmission within a half block is p Gw = P( 12 TB Gw log 2 (1+γ Gw ) ≥ F ).The probability of secure transmission from satellite to user is p c,2 p s,2 .The probability that file W n is not cached at both tiers is 1 − a 1,n − a 2,n , note that for effective cache utilization we do not cache the same file at both tiers.Hence, the probability of secure two-hop transmission event can be expressed as The CP and SP, p c,1 and p s,1 , are calculated as in (3) The secure transmission probability (STP) of the system can be expressed as: where q n represents the probability that file W n is requested.

III. SECURE TRANSMISSION PROBABILITY ANALYSIS
With the assistance of IRS, the signal propagates to the user by the adjusted reflections of the IRS.The IRS is assumed to have perfect knowledge on the CSI of the user.

1) Ground Station-IRS-User.
When file W n is served directly from ground station, the received signal at user consists of the direct signal transmitted by ground station and the reflected signal from IRS, which can be expressed as where h GU is the channel coefficient of the direct G-U link; h GI i and h I i U respectively represent the channel coefficients of the indirect link from G to the i-th element of IRS to U; φ 1,i is the adjustable phase induced by the i-th reflecting element of IRS; x 1 represents the transmit signal by ground station with transmit power P 1 ; and w n ∼ CN(0, σ 2 n,U ) is the additive white Gaussian noise (AWGN) at U with zero mean and variance σ 2 n,U .The channel coefficients comprises of large-scale fading and small-scale fading, i.e., h GU = hGU d −αg GU (6) where hGU is modeled as in (2), d GU denotes the distance between G and U, and α g is the terrestrial path-loss exponent.h GI i and h I i U are modeled similarly.The reflected signal from IRS, , is modeled as when each of the IRS elements receives the incoming signal from the ground station independently and reflects the signal with adjusted phase to the user.Then, the received signal from all IRS elements are modeled as a superposition of all the corresponding reflected signals [27].
The result CP and SP of this case can be expressed as where R t,1 , R e,1 represent the codeword transmitting rate and redundant rate in Case 1, respectively; h GE , h I i E are respectively the channel coefficients of G-E and the i-th element of IRS-E links; σ 2 n,E is the variance of the AWGN at E. Note that φ 1,i is adjusted to maximize the SNR at the user, i.e., φ 1,i can be equal to negative of phase h GI i h I i U , hence, p c,1 (and following p c,2 ) is independent of φ 1,i .
The user receives only the reflected signal from IRS in this case.Let x 2 denote the transmit signal by satellite with transmit power P 2 , the CP and SP of this case are where φ 2,i represents the phase shift of the i-th element of IRS in Case 2;

SI
is the channel coefficients of S-IRS link in which hSI i follows shadowed-Rician fading in (1), d SI is the distance between satellite and IRS, α s is the satellite path-loss exponent; R t,2 and R e,2 are respectively the threshold transmitting rate and redundant rate for secure transmission of Case 2.

A. Connection Probability Analysis
Case 1: Ground Station-IRS-User.The connection probability is found in a closed-form using the moment-matching method. Let Theorem 1: For a given z 1 , the connection probability is obtained as follows, with the shape parameter Var (Xc) and the scale parameter w c = Var (Xc ) E(Xc ) where Δi . Here, dt denotes the lower incomplete Gamma function, and Γ(a) denotes the Gamma function.
Proof: Please see Appendix A.

Since the term |
is the squared sum of product of exponential and shadowed-Rician random variables, it is challenging to derive a closed-form expression for p c,2 .However, we can derive the asymptotic expression of p c,2 as follows.
Corollary 1: In high SNR regime, the connection probability p c,2 is obtained as follows, where , and β(1, n) denotes the Beta function. Proof: The connection probability of the Satellite-IRS-user case is expressed using integral form as where (.) n denotes the Pochhammer symbol, K n (.) is the modified Bessel function of the second kind with order n [24, eq.( 8.432)], and ui = .
It is worth noting that the infinite summation term in (10) converges, and its numerical value can be found through partial sum method.
Remark 1: The results from Theorem 1 and Corollary 1 demonstrate the significant impact of transmit power and transmission rate on the connection probability.Increasing transmit power leads to an increase in connection probability, while increasing transmission rate results in a decrease in connection probability for both cases.

B. Secrecy Probability Analysis
The SP derivations are similar to that of CP above.However, it is worth noting that the IRS does not have the eavesdropper's CSI.There is no reflection adjustment at IRS in order to maximize the SNR at eavesdropper; Hence, the value of φ j ,i , j = 1, 2 (the reflection adjustment at IRS to maximize the SNR at user) remain the same as in the connection probability analysis.
Case 1: Ground Station-IRS-User. Let Based on the analysis for (9), the secrecy probability for a given s 1 is given by with the shape parameter Var(Xs ) and the scale parameter w s = Var(Xs ) E(Xs ) , where n,E P 2 (2 2R e,2 − 1), from (8) the secrecy probability is p s,2 = P{| M i=1 h SI i h I i E e j φ 2,i | 2 ≤ s 2 }.Proposition 3: Similar to the analysis of (10), in high SNR regime, p s,2 can be given as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Remark 2:
The results indicate that the secrecy probability decreases with an increase in the transmit power, while the connection probability and secrecy probability are both affected by the redundant information rate.Specifically, increasing the redundant information rate leads to a rise in the secrecy probability and a reduction in the connection probability.

IV. SECURE CONTENT DELIVERY PROBABILITY MAXIMIZATION
The redundant rate and caching probability play crucial roles in enhancing the system STP.R e prompts the trade-off between transmission reliability and secrecy since it affects both the CP and SP.The caching probability directly affects the STP since it triggers the probability of adopting one-hop or two-hop transmission schemes.Hence, in this section the optimal redundant rate R e,i , i = 1, 2 and caching probability A i , i = 1, 2 are jointly designed to maximize the system STP.The optimization problem can be formed as max This is a non-convex optimization problem.Observing p secure , (p c,i p s,i ) and q n are all independent of a i,n ; Hence, the optimization of the redundant rate R e,i and the caching probability A i is independent of each other.The maximizing p secure problem can be decomposed into two steps: (i) determining the optimal R e,i to maximize (p c,i p s,i ) for each case in the transmission scheme; (ii) designing the optimal A i for the two cache tiers to maximize p secure .

A. Optimization of Redundant Rate R e,i
The first-step problem of ( 15) can be formulated as max R e,i ≥0 Let Case 1: Ground Station-IRS-User.Problem ( 16) can be recast as Γ(ks ) (17) where n,E P 1 ws , and B 1 = 1+β s,1 .Solution to problem (17) is presented in the following theorem.

B. Optimization of Caching Probability A i
The second-step problem of ( 15) can be formulated as Remark 3: Function p secure in ( 23) is a monotonically increasing multivariable function of a i,n .Hence, the optimal solutions for (23) can be found using exhaustive search.
Proof: We have where ϕ 1 and ϕ 2 are found in Section IV-A; p Gw is obtained as follows, Here, h GwS denotes the channel coefficient of the gatewaysatellite link; B Gw denotes the bandwidth of the gatewaysatellite link; P 0 is the gateway transmit power; σ 2 n,S represents the variance of the AWGN at satellite; (m 2 ; b 2 ; Ω 2 ) is the satellite uplink channel parameter as in (1); 2 F 2 is the confluent hypergeometric function of the second kind.F γ Gw (γ th ) is found following [30, eq. ( 3)].

V. NUMERICAL RESULTS
In this section, the numerical results are presented to validate the theoretical analysis.The simulation is done using Monté Carlo method.Throughout the experiments in this paper, the value of key parameters are always set as in Table I, unless otherwise specified.The user and eavesdropper are randomly placed inside a circle with radius 10 km and center at ground station G.
Fig. 2 and Fig. 3 depict the CP and SP when the satellite and ground station transmit powers vary.The results from Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Monté Carlo simulation match well with the theoretical values.The asymptotic p c,2 in (10) and p s,2 in ( 14) achieve high accuracy comparing to their exact values in the integral forms.As expected in the theoretical analysis, when the transmit power increases, the CP increases and the SP decreases.It is also observed from Fig. 2 that the satellite-IRSuser scheme provide better reliability performance than that of the ground station-IRS-user scheme.For transmission secrecy in Fig. 3, the ground station-IRS-user scheme performs better; and both schemes yield higher SP in the low transmit power regime but get small value of SP in the high transmit power regime.
In Figs. 4 and 5, the CP and SP are plotted as functions of the redundant rate R e,i .The high accuracies of the theoretical results to the simulation and the asymptotic values to the exact values are confirmed.The behavior of CP and SP of the two schemes is expected as the CP decreases and the SP increases when increasing the redundant rate.The CP in ground station-IRS-user case reaches saturation at R e,1 = 6.5 bits/s/Hz , and the SP does at R e,1 = 8 bits/s/Hz .In satellite-IRSuser case, the CP starts to decrease very slowly when R e,2 passes 12.5bits/s/Hz , and the SP reaches saturation at R e,2 = 11.5 bits/s/Hz .System secure transmission probability vs. number of reflective elements on IRS.
The system STP in (4) versus the redundant rate with different caching probability is depicted in Fig. 6.It is observed that p secure increases significantly when the redundant rates pass 5 bits/s/Hz, and p secure reaches is maximal value at R e,i = 11.25 bits/s/Hz .As indicated by the dash line in Fig. 6, at the maximum p secure , the value of R e,i is not affected by different caching probability of [A 1 , A 2 ] i , which verifies the independence in optimizing the redundant rate and the caching probability.Fig. 7 shows the relationship between the number of IRS reflective elements and the STP.Larger number of reflective elements improves the system transmission security.When R e,1 and R e,2 are set to 5 bits/s/Hz, double the reflective elements enhances the p secure by up to 25% in the less-than-50-element regime.When R e,1 and R e,2 are set to 11.25 bits/s/Hz, which is the value where p secure is highest in Fig. 6, double the reflective elements can improve the p secure by up to 18% in the more-than-50-element regime.Note that even the p secure is improved when using larger IRS, the p secure is in a low range of probability when R e,1 and R e,2 are set to 5 bits/s/Hz and in a higher range for the other case.This implies the importance of optimizing the redundant rates R e,i to not only achieve the maximum p secure but also save the resources.The secure transmission probability of ground station-IRS-user scheme with and without IRS.To confirm the effect IRS in system on the transmission secrecy, the simulation is set up the ground station-IRS-user scheme with and without IRS.The enabled system without IRS is adopted from [9], [10], [13] with single base station.The results are presented in Fig. 8.The secrecy probability in case of having only the direct link from ground station to user is significantly less than the one having IRS.Having IRS can improve the secrecy probability by at most 50% when R e,1 passes 8 bits/s/Hz.
In Fig. 9, the secrecy performance between two-tier versus single-tier cache-enabled systems is compared.The singletier cache-enabled system is adopted from [2], [31] where the caching capability is only enabled at ground station with the caching probability A 1 .It is observed that our proposed system outperforms the single-tier cache-enabled system by almost 50% at the maximum p secure .And the result once again confirms the independence in optimizing the redundant rate and the caching probability of single-tier and/or two-tier cache.
Fig. 10 shows how the optimal redundant rates solution R * e,i is influenced by the secrecy rates.These are the solutions for the first-step optimization problem in (16).Corresponding to the optimal R * e,i , the optimal system STP is depicted in bits/s/Hz , the optimal p * secure is at 80% with R * e,1 = 5.5 bits/s/Hz and R * e,2 = 11 bits/s/Hz , while the non-optimized p secure (in Fig. 6) only achieves its highest value at 72% with R e,1 = R e,2 = 11.25 bits/s/Hz .To further compare the optimal probabilistic caching policy, the system STP under ground station and satellite most popular caching policies [6] are also provided in Fig. 11.As expected, the optimal probabilistic caching policy gives the best performance.The ground station most popular caching policy performance is not as good as that of the ground station most popular caching policy and surpasses its opponent when the secrecy rates are more than 3.5 bits/s/Hz.

VI. CONCLUSION
In this paper, we proposed a novel two-hop content delivery scheme in an IRS-assisted cache-enabled STN with probabilistic caching policies at both the satellite and ground station.We evaluated the system's connection probability and secrecy probability using asymptotic and closed-form expressions.By Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
jointly designing the transmission rates and caching probability, we were able to maximize the system's secure transmission probability.Our numerical results confirmed the validity of the theoretical analysis and demonstrated that having an IRS and two cache tiers can improve the secrecy probability by almost 50%.Moreover, our results emphasized the importance of optimizing the redundant rate to achieve the highest secure transmission probability while saving resources.

APPENDIX A PROOF
Let denote hmn the average channel power gain of the link between node m and n, then where Δ i = hI i U hGI i e j φ 1,i and 2 r 1 ,...,r M = 2! r 1 !•••rM ! .The second moment of X c , denoted as E(X 2 c ), is computed as follows, The density function f V i (v ) is found using inverse Mellin transform as follows.
Upon substituting ( 29) and ( 30) into the Mellin transform of , the density function f V i (v i ) can be expressed as where K n (.) is the modified Bessel function of the second kind.
Since {V i } M i=1 are independent and following the factorization theorem, the join density function can be given by Substituting ( 31) and ( 32) into (28) then changing IU e j φ 2,i b 1 i arrives at (11).

Fig. 8 .
Fig.8.The secure transmission probability of ground station-IRS-user scheme with and without IRS.

Fig.Fig. 11 .
Fig. The optimal redundant rate, R e,i with respect to secrecy rate.

Fig. 11 .
Fig.11.These optimal p * secure values are obtained as the solutions for the optimization problem in(15) after achieving the optimal caching probability [A * 1 , A * 2 ].At R s,1 = R s,2 = 1 bits/s/Hz , the optimal p * secure is at 80% with R * e,1 = 5.5 bits/s/Hz and R * e,2 = 11 bits/s/Hz , while the non-optimized p secure (in Fig.6) only achieves its highest value at 72% with R e,1 = R e,2 = 11.25 bits/s/Hz .To further compare the optimal probabilistic caching policy, the system STP under ground station and satellite most popular caching policies[6] are also provided in Fig.11.As expected, the optimal probabilistic caching policy gives the best performance.The ground station most popular caching policy performance is not as good as that of the ground station most popular caching policy and surpasses its opponent when the secrecy rates are more than 3.5 bits/s/Hz.

TABLE I PARAMETERS
USED FOR NUMERICAL RESULTS