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IEEJ Transactions on Electrical and Electronic Engineering

Quantum Computing Resistant Secure Communication Over Modified Nakagami‐q Fading Wireless Channel

Md. Sohidul Islam, Yosuke Sugiura, Tetsuya Shimamura

Published Apr 5, 2026• 0 citations
Publisher

Abstract

Quantum computing threatens traditional encryption methods like RSA and ECC, as quantum algorithms such as Shor's algorithm can break them efficiently. Physical Layer Security (PLS) offers a resilient alternative by leveraging the inherent properties of communication channels, making it immune to quantum advancements. This paper investigates the secrecy capacity of modified Nakagami‐ q fading channels in the presence of a multi‐antenna eavesdropper. The multiple antennas at the receivers are assumed to be independent and identically distributed (i.i.d.). We apply a small limit argument approximation (SLAA) to the zeroth‐order modified Bessel function of the first kind within the conventional Nakagami‐ q probability density function. This modification eliminates the mathematical intractability associated with the Bessel function, enabling the derivation of novel, simple, and closed‐form analytical expressions for key secrecy metrics. Specifically, we derive and analyze the probability of positive secrecy capacity, secure outage probability, and ergodic secrecy capacity. The validity of the proposed approximation is confirmed through high‐fidelity Monte Carlo simulations ( samples), which demonstrate a negligible Root Mean Square Error (RMSE) of from the conventional model. Results show that the system maintains positive secrecy capacity even when the eavesdropper has a higher SNR than the legitimate channel, particularly in low‐SNR regions where eavesdropper antenna count has minimal impact. Special cases, such as Rayleigh fading with single‐antenna setups, further validate these findings. This research highlights the robustness of PLS in protecting communications against quantum threats and provides a mathematically tractable framework for secrecy analysis in generalized fading environments. © 2026 Institute of Electrical Engineers of Japan and Wiley Periodicals LLC.

DOI
10.1002/tee.70308

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In this paper, we consider an independent and identically distributed Nakagami-q fading single-input multiple-output (SIMO) channel, where the transmitter sends some confidential message to the legitimate receiver in the presence of an eavesdropper. Both the receiver and the eavesdropper have multiple antennas. We formulated the probability of non-zero secrecy mutual information, secure outage probability, and ergodic secrecy mutual information for the SIMO wireless communication system. The calculations have been carried out using small limit argument approximation on zeroth-order modified Bessel function of first kind. This approximation provides decent results in the low signal-to-noise-ratio (SNR) regime. In the low-SNR regime, either the transmission power is small or the bandwidth is large. In either case, it is difficult for an eavesdropper to detect the signals in this regime. We present that in the presence of fading, positive secrecy mutual information is achievable even when the eavesdropper has better SNR than the main channel. We also found in our work that at low-SNR regime, the effect of number of eavesdropper's channel on average secrecy mutual information is almost negligible.
In this study, the authors investigate the secrecy performance at the physical layer of wireless communication systems where the channels are modelled as cascaded fading channels, in which the fading effects are worse than those in the cellular communication channels like Rayleigh and Nakagami‐ m . Analytic results are derived for the cascaded Nakagami‐ m fading channel model comprising the cascaded Rayleigh, 2*Nakagami‐ m , the generalised‐ K , and the conventional cellular channel models as well. By evaluating the information‐theoretic physical layer security performance, they obtain the closed‐form expressions of the average secrecy capacity and probability of the positive secrecy capacity in case of that channel‐state‐information of the eavesdropper is known by the source. Then they obtain the exact closed‐form secrecy outage probability (SOP) when the eavesdropper is passive. Analytic results are computed in terms of the bivariate and univariate Meijer's G functions and later on are successfully validated through Monte Carlo simulations. The simulation results show that the secrecy performance worsens with the increasing cascading degree of the fading channels. For example, in 2*Nakagami‐ m fading channels where the cascading degree is 2, extra 4 dB signal‐to‐noise ratio is required to achieve the same SOP value in comparison with the classical Nakagami‐ m fading channels.
Different from conventional key‐based cryptography schemes, physical‐layer security (PLS) techniques have drawn much attention recently to realise unconditional security from the information theory perspective. As an important performance metric in PLS, the ergodic secrecy rate (ESR) for a multi‐input multi‐output wireless communication network over a Nakagami fading channel is analysed. The network is consisted of a multi‐antenna transmitter (Alice), a multi‐antenna legitimate receiver (Bob), and a multi‐antenna eavesdropper (Eve). By using the selective transmission (ST) at Alice and the maximum ratio combining (MRC) at Bob and Eve, an exact expression of the ESR is derived. However, due to the infinite summation, it is very hard to evaluate the ESR performance. To reduce computational complexity and obtain more insights, a lower bound of the ESR is then obtained, which is in a closed form. As special cases, the lower bounds of the ESR for the signal‐antenna scenario and Rayleigh fading channel are also obtained, respectively. Numerical results show that the derived expressions of the ESR and its lower bound are very accurate to evaluate system performance.
Multicasting over wireless channels is a challenging task now-a-days as conventional frameworks can not ensure the secrecy of transmitted information. Hence, in this paper, a secure multicast framework is presented over Generalized Nakagami-m fading channel in the appearance of multiple eavesdroppers and key concentration is given to the transmission of information at a secrecy rate so that no eavesdropper can decode any information from the multicast channels. The impact of the system parameters on the secrecy performance of proposed system is analyzed by obtaining closed-from mathematical expressions of some secrecy measures such as probability of nonzero secrecy multicast capacity and secure outage probability for multicasting. The generalization of the proposed model is achieved by obtaining Rayleigh, Nakagami- m, Gamma, Half-normal, Weibull, Exponential, etc. fading channels as a special case of Generalized Nakagami- m fading channel. Finally, the correctness of the derived expressions is authorized using Monte-Carlo simulations.
Nakagami-m fading channel is chosen to analyze the secrecy capacity for fading channels since the Nakagami-m distribution can model fading conditions, which are more or less severe than that of Rayleigh and has the advantage of including Rayleigh as a special case. At first, secrecy capacity is defined in case of full channel state information (CSI) at the transmitter, where transmitter has access to both the main channel and eavesdropper channel gains. Secondly, secrecy capacity is defined with only main channel CSI at the transmitter. Then, optimal power allocation at the transmitter that achieves the secrecy capacity is derived for both the cases. Moreover, secrecy capacity is defined under open-loop transmission scheme, and the exact closed form analytical expression for the lower bound of ergodic secrecy capacity is derived for Nakagami-m fading single-input multiple-output (SIMO) channel. In addition, secrecy capacity is defined for the AWGN channel in order to realize the information-theoretic security of wireless channels with no fading. Finally, analytical expressions for the probability of non-zero secrecy capacity and secure outage probability are derived in order to investigate the secure outage performance of fading channels.
Physical layer security (PLS) has been shown to hold promise as a new paradigm for securing wireless links. In contrast with the conventional cryptographic techniques, PLS methods exploit the random fading in wireless channels to provide link security. As the channel dynamics prevent a constant rate of secure communications between the legitimate terminals, the outage probability of the achievable secrecy rate is used as a measure of the secrecy performance. This work investigates the secrecy outage probability of a simultaneous wireless information and power transfer (SWIPT) system, which operates in the presence of multiple eavesdroppers that also have the energy harvesting capability. The loss in secrecy performance due to eavesdropper collusion, i.e., information sharing between the eavesdroppers to decode the secret message, is also analyzed. We derive closed‐form expressions for the secrecy outage probability for Nakagami‐m fading on the links and imperfect channel estimation at the receivers. Our analysis considers different combinations of the separated and the integrated SWIPT receiver architectures at the receivers. Numerical results are provided to validate our analysis.