# Gaussian-Distributed Spread-Spectrum for Covert Communications
## 1 Introduction  
Secure communication schemes are essential for transmitting mission-critical information securely and privately, especially in hostile environments. Traditional methods to achievesecure communication can be broadly categorised into three main groups [1]: cryptographic methods, steganographic methods [2; 3; 4], and physical layer security techniques [5]. Cryptographic methods use encryption and decryption algorithms to prevent unauthorised access, disclosure, and alteration of data. Steganographic methods are primarily focused on concealing confidential information within non-sensitive objects, such as images, audio, and video files. On the other hand, physical layer security techniques aim to bolster the security of wireless communication systems by leveraging the unique properties of wireless channels. However, none of these methods can conceal the evidence of communication or protect against detection of the transmission. Therefore, an adversary can potentially intercept the transmission and take advantage of any vulnerabilities in the communication protocol to gain access to the secure data or launch electronic attacks against the user [6]. The operational needs of secure communication in contested electromagnetic environments go beyond just protecting the transmitted content; it also requires the concealment of the transmission behaviour [7]. This paper presents a new signalling scheme for covert communication. Covert communication, also known as communication with a low probability of detection (LPD), is centered around hiding any evidence of communication to avoid detection. This is achieved by reducing the received signal-to-noise ratio (SNR) at the eavesdropper [8; 9; 10]. Research published in the field of LPD communications can be categorised into two main areas: (i) information-theoretic aspects of LPD communications and (ii) designing waveforms for LPD communications. Information-theoretic studies focus on determining the fundamental limits of LPD communication in terms of the amount of information that can be conveyed from a transmitter to a receiver subject to a constraint on adversary's detection error probability [11; 12]. The authors in [13] present a square root law (SRL) that defines the constraints and performance limits of LPD communication for the additive white Gaussian noise (AWGN) channel. The SRL law states that covert and reliable communication can be achieved provided no more than \(\O\sqrt{n}\) bits are transmitted in \(n\) channel uses. This gives an information rate of \(\O(1/\sqrt{n})\), which approaches 0 as \(n\) goes to infinity. Hence, recent studies have focused on developing LPD schemes to obtain a positive information rate [14]. This paper focuses on designing waveforms for LPD communication. For LPD communications, it is desirable to transmit with minimal PSD to hide the transmitted signal under the receiver's noise floor [7] and have random like characteristics such as non-repetitive features [15], making the signal indistinguishable from thermal white noise present at any receiver [16; 17; 18]. These properties are needed, as without them the waveform has the potential to be detected using advanced signal processing techniques such as cyclostationary analysis, higher-order moments analysis, energy detection methods, and time-frequency transforms [19; 20; 21; 22; 23]. Existing covert communication schemes use spread-spectrum, chaotic theory, or a combination of both to achieve covert communication. The proposed schemes vary: utilising machine learning [18], using noise envelopes to mask the signal [24], chaotic spreading and modulation [25; 26], and using the message itself to spread the signal [27]. In addition to different waveform design approaches, there are also other methods proposed in the literature that exploit the uncertainties in the eavesdropping channel [28], noise power [29], transmit time [30], and interference power from friendly jammers [31] to reduce signal detectability and improve information rate [32].