Asymptotic Bound of Secrecy Capacity for MIMOME-Based Transceivers : A Suboptimally Tractable Solution for Imperfect CSI

The paper principally proposes a suboptimally closed-form solution in terms of a general asymptotic bound of the secrecy capacity in relation to MIMOME-based transceivers. Such pivotal solution is essentially tight as well, fundamentally originating from the principle convexity.The resultant novelty, per se, is strictly necessary since the absolutely central criterion imperfect knowledge of the wiretap channel at the transmitter should also be highly regarded. Meanwhile, ellipsoidal channel uncertainty set-driven strategies are physically taken into consideration. Our proposed solution is capable of perfectly being applied for other general equilibria such as multiuser ones. In fact, this in principle addresses an entirely appropriate alternative for worst-case method-driven algorithms utilising some provable inequality-based mathematical expressions. Our framework is adequately guaranteed regarding a totally acceptable outage probability (as 1 − preciseness coefficient).The relative value is almost 10% for the estimation error values (EEVs) ⩽ 0.5 for 2 × 2-based transceivers, which is noticeably reinforced at nearly 5% for EEVs ⩽ 0.9 for the case 4 × 4. Furthermore, our proposed scheme basically guarantees the secrecy outage probability (SOP) less than 0.05% for the case of having EEVs ⩽ 0.3, for the higher power regime.


Introduction
In order to highlight a significantly more favourable system performance in terms of higher security in wireless networks, a programmable beamformer is set up at the transmitter [1][2][3][4].Such technologically advanced equipment (precoder) is undoubtedly able to manage the users' links in totally different paradigms, originated from the fact of resource allocation.In the context of the forenamed widely supported principle, it is also highly motivated to provide the fashions convex, without loss of optimality.
The convexity cited above has a vast range of beneficial knock-on effects such as markedly more adequate convergence in design problem.Thus, the enlightening concept convexity is highly emphasised in order to guarantee an absolute robustness in terms of tractability in association with the optimisation problem.
The optimisation problem (where the precoder is computed) optimally/suboptimally features a guarantee of the quality of service (QoS) in multiuser games, for example, multi-input multioutput multiantenna eavesdropper-(MIMOME-) based ones.QoS should be fairly taken into account among all players, since eavesdroppers (Eaves) may be licensed or unauthorised ones in the mentioned futuristic transceivers.
The suboptimality expressed above advances a trade-off between two criteria such as convexity which suffers from, for example, channel uncertainties (originated from imperfect channel state information, which necessarily causes information deficiency) and a tractable response.In this paper, such trade-off is addressed as well.
1.1.Related Work.In this part, we generally provide a comprehensive overview over the existing work in relation to secrecy enhancement.
1.1.2.Algorithms: An Overview.In [14], a secure game theorydriven transmission framework was conducted for MIMO-MEs proposing three algorithms aimed at maximising sum secrecy rate.A trade-off was theoretically defined in [15] for the principles sum secrecy rate reinforcement and energy efficiency which could provide a balance in physical-layersecurity-based transceiver designs.In [16], three general approaches were proposed in order to choose a paradigm for two-objective design problems, namely, Nash bargaining (the most efficient one), Jain's index, and the Kalai-Smorodinsky bargaining.In [17], utilising a dynamic programming algorithm, a multilayer wiretap coding framework was fundamentally proposed.
1.1.3.Closed-Form Solutions: An Overview.Regarding mainto-eavesdropper ratio, SOP's diversity order remaining fixed was proven in [18].In [19], taking into account various receiver arrangements for MIMOME transceivers over Nakagami-m fading channels, SOP was precisely computed.In [20], it was proven that the intercept probability does not rely on Eave' numbers in a cooperative-based network; instead, it relies specifically upon the number of relays.Asymptotic bounds for the outage probability defined for the secrecy rate and the achievable diversity were explored in [21].Investigating the information's exponential decay rate intercepted by Eave when using an average wiretap channel code, a tight and achievable secrecy rate was mathematically derived in [22].Taking into account the generalised degrees of freedom, an investigation was basically provided in relation to the optimality of treating interference as noise over ×2channels in [23].Integrating a chaos stream cipher with the secure polar coding strategy, a general secure framework was technically fulfilled in [24].In other words, we mathematically provide a general solution as a newly discovered trade-off irrespective of the kind of fading channels, the type of strategies, and so on.Such novel trade-off is able to be deployed to other schemes and has not been examined so far, which simultaneously is satisfied with a tight (correct) response and a deniable SOP (computation error).The mentioned trade-off is interpreted with the aim of near-optimally finding the upper-bound (worst-case) of Eave's received power.
1.4.Notation and Organisation.(⋅) ↘ means that a term falls, whereas (⋅) ↗ means that a term rises.Additionally, A, a, and  stand for, respectively, matrices, vectors, and scalars.⊗ stands for the Kronecker product.The other symbols and notations utilised throughout the paper are listed in Notations.Finally, all matrices and vectors are, respectively,  ×  and  × 1, without loss of generality.
The rest of the paper is organised as follows.Firstly, the main problem and the proposed solution are addressed in Section 2. Additionally, the simulation-based analytical results are discussed in more detail in Section 3. Finally, conclusion and proofs are given in Section 4 and the appendix, respectively.

Problem Description
A MIMOME-based transceiver is literally given in Figure 1.The MIMO channels between the legitimate transmitter and the legitimate user as well as Eave are principally denoted by H and G (please note that our scheme undoubtedly satisfies H and G of sizes  ×  and  ̸ = ; however, we considered them as  ×  (square) for more convenience), in a Down-link paradigm.For G, we experience imperfect knowledge (which degrades QoS): that is, G ≜ G 1 ± Γ (our scheme is not applicable for some other frameworks [25,26] in which, for example, the channel matrix is an arbitrary known matrix physically multiplied by a clear matrix [27, p. 4]).Now, regarding the threshold , for G we merely define the ellipsoidal uncertainty set Γ ≜ {Γ ∈ C × : ‖Γ‖ 2 ⩽ }.Instantaneous Ergodic capacity and also the undesired capacity (in connection with Eave) are, respectively, determined as with respect to the noise variance terms  2  and  2  at the receiver sides and the transmit covariance matrix Q ≜ {xx  }.
The mapping decomposition {xx  }  → Q cited above fundamentally requires the constraint rank(Q) = 1 [28][29][30][31].Inversely, Q being rank-1 unfavourably causes nonconvexity [29][30][31].Of course under some circumstances, the aforementioned constraint holds and it can be alternatively dropped (e.g., in [32] in which nearly 99.79% rank-1 results in the simulations were witnessed).However, in order to guarantee the considerably more acceptable system with the higher reliability, we should ensure the design problem further.
As newly recently proposed in [33,34], the rank-1 constraint can be basically relaxed as the dual constraints where A 1 and A 2 are slack variable matrices, correspondingly.Fully take into account the optimisation problem below as Problem P 1 where  Transmit is a threshold for the transmit power.
Lemma 1 (see [35]).As described in [35], multiplying (pre or post one) by any matrix (of any rank) does not cause any change on positive semidefinite matrices' rank (see, e.g., [36, 9.6.10]for the proof).1).See, for example, [36, 9.6.10]as the positive definite property.Corollary 3.After carefully writing the dual Lagrange problem, then principally taking the derivative with respect to the defined Lagrange variables, then unifying the resultant term by zero, and finally right multiplying both sides of the resultant mathematical expression by Q, a positive semidefinite matrix would theoretically hold.Regarding Lemma 1, the exact value in relation to the rank can be conveniently calculated.
According to the penalty method (see, e.g., [17]), Problem P 3 is alternatively expressed as maximise for which we have P 3 is nonconvex, per se, owing to the lack of control on the error in the estimation of G, originated from the channel uncertainty parameter Γ.Since P 3 has no closed-form solution with respect to its existing format, consequently, it needs an entirely appropriate alternative.See the following theorem.

Theorem 6. P 3 unhesitatingly has a solvable and relaxed alternative as Problem
in which while the upper-bound of ( 10) can be conveniently expressed as See the Appendix for the proof.Therefore, the upper-bound of ( 10) is recasted to B as well.See the following lemma.Lemma 7. Defining the deterministic value  New (Q, G 1 ; ) instead of  Old (Q, G 1 ; Γ) as its worst-case (upper-bound) and simultaneously maximising − New (Q, G 1 ; ) provide the asymptotic bound of the optimisation problem in terms of , which efficiently guarantees convexity and then tractability.Indeed, we merely provided a technical segregation between the terms   and   (in the Appendix), which chiefly provided an opportunity where the inaccessible and unrecognised (in relation to the ellipsoidal channel uncertainty sets) term   is generally formulated from another viewpoint.
Lemma 8.According to Schur product theorem [38,Theorem 7.5.3],Q⊙Q would be inevitably a positive semidefinite matrix; therefore, the key features (intrinsic physical and statistical properties) sufficiently remain stable.

Results
In this section, we analyse the system performance as well.
Some parameters and assumptions in the simulations are as follows. Transmit is equal to SNR.Correlations at the MIMO antenna array are defined as 0%.The number of randomly generated Rayleigh fading channel realisations is 1000., , and  are separately defined for each figure.
Figure 2 strongly proves the tremendously sensible reliability in terms of the noticeably acceptable outage probability error (which is mathematically defined in ( 12)), which is plotted versus the  regime while changing .The relative content is repeated in the subfigure (b) for 3 dimensions against  and .Meanwhile, we should inform that  = 0 technically means the perfect CSI case.
Analytically, the outage probability error is written as according to which preciseness coefficient is calculated as 1 − O Prob .
It should be noted that, for  = 4 or higher, our proposed solution is completely trustable or even for  = 2 while having the lower error rates in the estimation, that is, ∀ ⩽ 0.5.Inversely, for  = 1 (Single-Input Single-Output schemes) such closed-from solution has a remarkably less reliable performance.The case 1 × 1 is discovered in detail in the second subsection.
Figure 3 repeats the previous one for 1 − O Prob in the context of preciseness coefficient.As can be seen, for  = 2 and ∀ ⩽ 0.5, preciseness coefficient is higher than 90%, which is enhanced at ⩾95% for higher s.Again, the relative content is repeated in the subfigure (b) for 3 dimensions against  and .Meanwhile, we should inform that  = 0 technically means the perfect CSI case.
Figure 4 shows the observably more efficient secrecy capacity for our approach compared to the worst-case method prevalently utilised in the literature [39][40][41], while changing  and  and also having  = 50mWatt.Such strong reinforcement is effectively justified by the markedly more adequate convergence for P 4 , analytically caused by convexity comprehensively originated from solvability.Worst-case method in our simulation is applied according to S-Lemma (see, e.g., [39, eq. 20-21], [40, eq. 11-15], [41, eq.58-61]) which indicates that (5b) can be relaxed as In the linear matrix inequality-based expression ( 13),  1 is the slack variable.
In Figure 5, the secrecy capacity for our approach compared to the worst-case method is depicted against SNR (dB), while changing  and  and also having  = 100 mWatt.It is revealed that the higher  we adjust, the larger values for the secrecy capacity can be mathematically achieved.This point can be theoretically analysed as follows.For example, assume that the general convex optimisation problem maximise ⩾0 M 1 =  1  −  1 , subjected to  1  ⩽ .In general, M 1 ↗ when  ↗.It is sufficient to write Lagrange function for the problem, after taking the derivative with respect to one Lagrange variable in connection with the unique constraint and also  and then unifying the resultant terms by zero; M 1 = ( 1 / 2 − 1).Indeed, M 1 ∝ .In other words, the higher  physically causes the bigger search region.Finally, how to take the derivative needed for our scheme is essentially obvious according to the fact , G}, which is proven with regard to (/X)Tr{AXB} = A  B  , which is given in [36, eq. 101].
Figure 6 illustrates SOP against SNR (dB) as well for the different values of , , and .Analytically, SOP is expressed as in which   and   conventionally denote the optimised secrecy capacity achieved, respectively, from the case of applying our scheme (SNR − D − 2 ) (see (8a)) and of not applying our scheme (SNR − D − 1 ) (see (6a)).SOP fairly seems acceptable even for the case 1 × 1, which is technically less than 0.0005 for the case where  = 50 mWatt and  = 0.3, ∀SNR ∈ [12 dB, ∞).   Figure 6 is briefly analysed in Table 1.In this table, the values in connection with  are in terms of mWatt which are ignored (not written) for more convenience.Indeed, the asymptotic bound of the secrecy capacity is technically equal to SNR while mathematically subtracting Eave's received power.The penalty in relation to Eave's received power is basically equal to min{,  New ()}, which would be evidently fixed at  in our simulations in the high SNR regime; that is, −10 dB for the case where  = 100 mWatt and ∀SNR ∈ [17 dB, ∞).
In Figure 7, the penalty function in connection with Eave's received power (undesired power) against SNR (dB) is shown while changing .It is in parallel with Figure 6.

Complexity.
In relation to the complexity, we should compare (11) with (13).To this end, among the methods in the    literature (e.g., [5,32,42]), we use the procedure of counting the flip-flops, as described in [5,Table 3].In this method, for example, the complexity of the matrix product for two matrices of sizes, respectively,  ×  and  ×  is O((2 − 1)).
Regarding an extreme range of operations in (13), our scheme has a more favourable complexity, which is ≈O(4    With respect to the main advantages thoroughly discussed in relation to the proposed scheme, its outperformance, correctness, and effectiveness are obviously highlighted over the existing methods (conventionally utilised in the literature, specifically the worst-case one).

Conclusion
An asymptotic bound for the secrecy capacity in the context of resource allocation was suboptimally defined for MIMOME-based systems for a general channel uncertainty fashion, supporting one theorem and also some lemmas and corollaries.The suboptimality cited above was principally highlighted in terms of a significantly acceptable trade-off between tractability and a less outage probability.Indeed, the deniable values obtained for the relative outage probability guaranteed our framework, which also fundamentally highlighted a tight and acceptable response for the system as well.The relative outage probability was almost 5% for EEVs ⩽ 0.9 for 4 × 4 MIMO antenna arrays.Our proposed framework is (A. 3) The proof can be conveniently completed regarding the detailed Remarks A.1, A.2, and A.3.
Remark A.1.The fourth line in (A.1) is literally approximated while generally ignoring L(⋅).Succinctly speaking, the general perspective and theoretical background behind the forenamed negligence lie mostly in the influence of the norms of the relative terms.The aforementioned logic can be theoretically observed in Figure 2.
Remark A.2.The logic behind the fifth line in (A.1) comes technically from the physical and casual behaviour of both precoder and error matrices.Indeed, since the forenamed matrices are positive semidefinite, hence, they should principally exist.
3 dimensions: against  and

)
Corollary 5. Simultaneously using Corollary 2 and Lemma 4, and since the log(⋅) function has no technical influence on minimisation or maximisation in principle, we can accordingly maximise Tr{HQH  }, as well as maximising −Tr{GQG  }, which is in correspondence with the signal-to-noise ratio (SNR) and Eave's received power, respectively.Now, with the aid of  instead of Tr{HQH  }, and defining  as a threshold for the maximum of Eave's received power, Problem P 2 is
According to O Prob , the reliability for  = 1 is crucially lower than that of the cases with larger values for .However, because of the term SNR which is extremely larger than D − 2 (see (8a)), SOP can be guaranteed instead, which perfectly highlights the importance of the outperformance, correctness, and effectiveness of our work.Feature 2. For the case 1×1, convexity theoretically is satisfied.