Hopf-Hopf bifurcation analysis and chaotic delayed-DNA audio encryption using cubic nonlinear optoelectronic oscillator

Abstract

The study presents a hybrid model for encryption which is based on the utilization of nonlinear chaotic behaviors in a cubic optoelectronic oscillator (OEO) as well as biomimetic DNA computation to achieve secure and lossless protection of audio data. Using a model of the oscillator that had a delayed feedback loop exhibited a strength of dynamical phenomena including Hopf–Hopf bifurcation, quasi-periodicity, and chaos, which will be applied for the generation of cryptographic keys. Bifurcation analysis was performed using the multiple scales method (MMS) in combination with DDE-BIFTOOL, and the analyses were able to observe the dynamical behaviors of the system through bifurcations while mapping high-entropy key sequences. The chaotic key sequences will drive DNA level permutation, substitution, and complementation processes to perform nonlinear diffusion and confusion of the audio data. Statistical tests demonstrated excellent encryption efficacy, with entropy values approaching , NSCR above , and UACI around confirming adequate randomness and resistance to statistical and differential attacks. In addition, resulting decrypted signals had negligible MSE and high PSNR dB), ensuring complete lossless recovery and accuracy.

Introduction

Optoelectronic oscillators (OEOs) have been a major focus of research in the last 50 years due to their diverse applications in modern industrial and technological systems, such as telecommunications, radar, microwave photonics, and high, precision frequency generation. These systems achieve the generation of highly spectrally stable and pure oscillations by combining optical and electronic components with time, delay elements in a unique way^1,2. OEOs exploit the complementary advantages of photonic and electronic subsystems and, therefore, can obtain outstanding characteristics such as ultra, low phase noise, very high frequency stability, and extremely wide operational bandwidths.

However, the use of time, delay feedback loops results in intrinsic nonlinearities that can have a significant impact on the dynamics of the system. The time delays are known to cause a variety of complex behaviors, such as multistability, bifurcations, quasi, periodicity, and chaos, which may lead to an improved functionality or a performance degradation of the system if they are not properly controlled^3,4. For this reason, it is important to understand and control the nonlinear dynamics of OEOs so that they can be used safely in high, speed signal processing and secure communication systems. OEOs have one of the most remarkable features in that they can skip the normal breather routes to chaos by going through extensive regions of high, frequency limit, cycle oscillations with short intervals of crenelated oscillatory states in between, which makes them very promising for the.

Chaos is a pervasive phenomenon that can be found in natural, industrial, and artificially created systems. Since the discovery of the Lorenz chaotic attractor, the field of nonlinear dynamics and chaos theory has been extensively studied in different scientific disciplines^5–7. Several fundamental chaotic systems have been introduced and experimentally verified, such as the Rssler system⁸, Chuas circuit⁹, and the Chen system¹⁰. These frameworks have been instrumental in illuminating the standard mechanisms that lead to chaos and have opened up a multitude of practical applications including secure communications, random number generation, and signal masking. A fascinating research area that has attracted a lot of attention recently is the cubic nonlinear optoelectronic oscillator (NEO). The buzz around these devices has been primarily brought about by significant advances in materials science, photonics integration, and device engineering. The main focus of several studies since 2020 has been the role of time, delay feedback loops in improving the stability, tunability, and performance of such oscillators¹¹. The insertion of delayed feedback has been evidenced to profoundly influence the rich dynamical behavior of optoelectronic systems. Through this, different steady states, periodic oscillations, and even chaotic regimes can be reached by a controlled manner. More specifically, time delays have been found to extend regions where oscillations are stable and also open a pathway for adaptable frequency tuning. Furthermore, time delays provide an alternative yet effective way in controlling bifurcation scenarios.

Deeper explorations of cubic nonlinear time, delay systems have shown that complex oscillatory patterns result from the interplay of nonlinearity and delayed feedback in these systems. The works recognize the phenomena of frequency pulling, multiple attractors coexistence, and chaos onset among others, pointing to the extreme sensitivity of parameter changes in such systems¹². The achieved dynamical complexity is particularly advantageous for usage in secure communication, analog signal processing, and information encryption. Additionally, recent exhaustive reviews on time, delayed optoelectronic oscillators stress the importance of going beyond existing theoretical models that would better account for the practical implementations of devices and new physical prototypes¹³. The intense interaction between the cubic nonlinearity, the delayed feedback, and the optoelectronic architectures thus paves the way for novel photonic system designs and impels the investigation of new materials and oscillator setups that can harness such complex dynamics.

In parallel with the evolution of nonlinear photonic systems, the digital era has been flourishing at an unprecedented pace. This has, in effect, greatly facilitated the storage, copying, and sharing of multimedia content, with the audio data being the most prominent beneficiary. Though this progress has been a boon for the sharing of information and the cultural exchange, it has also been a cause of concern for data security, privacy protection, and copyright infringement issues. Audio signals, as compared to text or static images, are often larger in terms of data size, have high redundancy, and exhibit strong temporal correlations between adjacent samples, which is why they are more susceptible to unauthorized access and are prone to be manipulated^14,15. On the other hand, conventional cryptographic algorithms such as DES, AES, and RSA, while being extremely secure for general data, are computationally heavy and thus are not suitable for real, time audio encryption as a result of their high power consumption and processing overhead.

To overcome these constraints, researchers have devised an extensive array of encryption algorithms designed explicitly for multimedia and audio data^16–21. A large number of these techniques exploit the nature of chaotic systems, which are extremely sensitive to initial conditions, have ergodic behavior, and produce pseudo, random sequences, in order to formulate encryption schemes that are both efficient and secure. In these kinds of solutions, chaotic sequences are generally utilized to produce keystreams that latterly are employed for encryption as well as for the secure transmission of audio signals^22,23. One example is the work of Wu et al., who presented a dual, channel voice encryption scheme based on two, dimensional chaotic maps, thus, achieving a drastic reduction of the temporal correlations in audio signals²⁴. In a similar manner, Rahul et al. created a chaos, based audio encryption system which uses biometric images and the SHA, 256 hash algorithm for the purpose of key security²⁵. Besides that, employing DNA and RNA encoding methods along with chaos theory has resulted in very effective means of multimedia security that can withstand statistical, differential, and noise attacks^26–28. To emphasize, Wang et al. formulated a hybrid hyper, chaotic system by merging logistic and sine maps, and they established its capability for image and multimedia encryption via dynamic RNA, based diffusion processes²⁹.

Within this context, the present research uses DDE-BIFTOOL for the calculation of two, parameter bifurcation diagrams and the localization of HHBP in a cubic nonlinear OEO with a single delayed feedback loop³⁰. The method of multiple scales (MMS) serves to accurately determine the double Hopf bifurcation points that are then unfolded and categorized by the normal form theory^31,32,33,34. The thorough bifurcation and stability investigation reveals the nonlinear dynamics of cubic OEOs more clearly. The next section presents the mathematical formulation and basic analysis of the proposed delayed cubic nonlinear OEO model³⁵,

1

The cubic nonlinear optoelectronic oscillator is a DDE model. The state variables of the optoelectronic system are and , are described in³⁵.

Current chaotic or DNA-based encryption methods mainly depend on low-dimensional chaotic maps with a limited key space and a lack of complexity needed for strong security. Many encryption works do not even consider the real, physical realizability of chaotic sources. To date, there is a lack of literature that bridges optoelectronic oscillator-based delayed chaos behavior with DNA-level operations (computational) with a bifurcation analysis to establish a link between physical dynamics and cryptographic strength. The rapid growth of multimedia data sharing audio in particular is driving the development of encryption methods that provide both privacy and quality. Traditional cryptosystems such as AES and RSA are strong in a mathematical sense, but they are high overhead and are not accommodating to the latency and temporal redundancy common to audio data. Similarly, optoelectronic oscillators and a range of chaotic dynamical systems have shown substantial success in producing rich and unpredictable high dimensional signals suitable for encryption. With these challenges in mind, this study ties together optical chaos from delayed nonlinear oscillators with biomimetic DNA computing to create a lightweight yet secure audio encryption scheme based on a high entropy, exact, and physically-realizable approach.

This study demonstrates an original combination of nonlinear optoelectronic dynamics and DNA-inspired computing for the encryption of multimedia content. The originality of the study is establishing a direct connection between the Hopf–Hopf bifurcation dynamics of a cubic nonlinear optoelectronic oscillator (OEO) and a source of high-entropy chaotic sequences for cryptographic use. In this research, we do not rely on low-dimensional chaotic maps as is common practice in chaotic systems, but explore the nonlinear OEO as a physical realization of delayed feedback that enhances unpredictability and security. This paper proposes a chaotic delayed hybrid–DNA encryption scheme whereby the time-delayed chaotic trajectories drive the DNA-level processes of permutation, substitution, and complementation to generate non-linear diffusion and confusion. The study presents a comprehensive bifurcation study and MMS modeling that provides a theoretical foundation for the system’s complex dynamics. A simulation study illustrates the efficacy of the cryptographic system and shows a significant advancement from previous work. The proposed method reaches maximum entropy (~ 7.99), low correlation, and an NSCR above 99.9%, demonstrating robustness against statistical, brute-force and differential attacks. Additionally, lossless decryption is achieved with low residual noise/error and very high PSNR (> 58 dB) thus ensures audio fidelity during transmission. Unlike conventional chaotic audio encryption relying solely on mathematical maps, this work leverages a physically realizable cubic OEO with Hopf–Hopf delayed chaos, extracting high–entropy keys generated from real dynamical bifurcation sources rather than synthetic maps, which to the best of our knowledge, has never been reported.

The rest of the document is structured as follows: section “Existence of the Hopf-Hopf bifurcation” examines the existence of HHB in the cubic nonlinear OEO with a single delayed feedback loop and establishes the stability conditions of the equilibrium points using methods of multiple scales (MMS) and DDE-BIFTOOL. In section  “Classification and unfolding of HHBP through MMS”, the Hopf–Hopf bifurcation is classified and unfolded through a normal form analysis and the emergent dynamical behaviors are discussed (periodic, quasi-periodic, chaotic). Section “Numerical simulations” offers detailed numerical simulations that demonstrate continuity with the analytical results and shows the transitions of the system dynamics using bifurcation diagrams and eigenvalue distributions. Section “Methodology” describes the hybrid chaotic delayed–DNA audio encryption framework proposed, specifically the key generation mechanism, audio preprocessing, DNA encoding and encryption and decryption algorithms and key sensitivity. Section “Experimental results and performance analysis” hops between the experimental results and performance evaluation period that includes waveform, spectrogram, entropy, correlation, histogram, differential and statistical analyses. There is also an evaluation of time complexity, key space and reconstruction quality. Section “Conclusion and future recommendation” concludes the study with a summary of the results, emphasis on the importance of combining nonlinear optical chaos with DNA computation, and directions for future research on the broadened scope of this framework into other multimedia security applications.

Existence of the Hopf-Hopf bifurcation

In this part, Eq. (1) is discussed. Equation (1) presents three equilibriums , and . We decided to analyze the Hopf bifurcation behavior (HHB) around the non, trivial equilibrium point since the complex equilibrium points did not show the required conditions for a HHB. The linearized system at this nontrivial equilibrium is given by the following

2

The characteristic equation associated with Eq. (2) is given by:

3

First of all, it is obvious that cannot be considered an eigenvalue of Eq. (3). In order to find the possible periodic solutions which bifurcate from HHBP, we shall take . Replacing in Eq. (3) leads to the result:

4

The real part and imaginary part of Eq. (4) is:

5

Equation (3) has two different pairs of purely imaginary roots . Suppose that the HHB takes place at ,. The value of ∆ for the necessary condition of the HHB at the bifurcation parameter.

where which satisfies After replacing with in Eq. (3), it can be seen that

6

The coordinates are the HHBP for Eq. (1) when based on the above computations. At this stage, the characteristic Eq. (3) produces complex conjugate roots with and no roots with positive real parts.

Classification and unfolding of HHBP through MMS

In this part, we will analyze the Hopf-Hopf bifurcation of the system (1) near the unique equilibrium , using the approach of multiple time scales (MMS). To analyze the equation around the Hopf-Hopf bifurcation point, we use and Eq. (1) reads

7

where are the unfolding parameters.

Equation (7) is governed by the following Eq. 

8

where are two-dimensional vectors,

For MMS, the used new time scales and the time derivatives are initially defined as

9a

9b

Using alternative time scales, the periodic and delayed solutions of Eq. (7) can be asymptotically expanded up to the third order and represented as follows:

10a

10b

where , perturbation and .

The second, order approximation of the slow flow modulation function can finally be written as

11

Here, () indicates the derivatives concerning the initial time t and are complex numbers.

In order to carry out the normal form analysis, the complex variables in Eq. (11) are changed into polar form

12

The real and imaginary parts of and () are denoted by , respectively. The amplitude is represented by , the phase angle by (), and the other values are real numbers. To form the normal form in polar coordinates, and the real and imaginary parts are taken correspondingly.

13a

13b

13c

13d

Equation (13a) and (13b) serve as a direct indicator of the stability and bifurcation of the system, which is original, implied. To be more precise, Eqs. (13a) and (13b) may possess a maximum of four equilibria, i.e.

14

Their corresponding eigenvalues are

15

16

17

18

where.

The local asymptotic behavior around the positive equilibrium point is basically determined by the real parts of the eigenvalues that correspond to . That is, if both eigenvalues are negative then any perturbation in the neighborhood of disappears exponentially with time. Hence, the equilibrium is locally asymptotically stable. On the other hand, if at least one of the eigenvalues has a positive real part, then small perturbations around increase with time and thus, the corresponding trajectories move away from the equilibrium. Thus, the positive equilibrium ceases to be stable and becomes an unstable point. Moreover, the whole t of findings from the current research lead to the conclusion that not only the existence but also the stability and qualitative features of all the equilibrium states depend heavily on the system parameters and for . Changes in these control parameters signal the beginning of different dynamical regimes thus, determining the behavior of the system close to the critical parameter values. Hence, these parameters are the main factors in the identification and description of the associated higher, order Hopf bifurcation points (HHBPs) which direct the changes from stable equilibria to oscillatory solutions and instability regions as well as the intermediates between these.

Numerical simulations

In this section, we utilize WinPP, a numerical simulation tool designed for nonlinear differential equations, to confirm the results of the previous theoretical studies. The parameters are outlined below:

19

In the HHB, and are the bifurcation parameters. The DDE-BIFTOOL tool is employed to generate (, ) plane bifurcation diagrams (BD). Figure 1 shows the (, ) plane with the critical curves for HHBPs.

Fig. 1.

Critical curves.

According to the data in Fig. 1, the system in Eq. (2) has two HHBP: . The real part of the eigenvalues of the linearized System (2) at the two points and is illustrated in Fig. 2. The eigenvalues at are and those at are .

Fig. 2.

The eigenvalue distribution for the HHBP sites for Sys. (2)

Based on previous studies, the 4-D normal form at the first HHB with frequencies and was derived. The bifurcation point can be thought of as the origin of new phenomena, as it is the point at which two oscillatory modes coincide and, thereby, dynamic behaviors with features like quasi, periodicity and amplitude modulation can unfold in the vicinity. Numerically, all of these will be dealt with in detail in the next section.

where and are the unfolding parameters. The BD for is illustrated in Fig. 3.

Fig. 3.

BD near .

To begin, in Region I, the trivial equilibrium remain stable. is come and stable in Region II, while is saddled. In Region III, begins as a saddle, remains an unstable node and remains stable. In Region IV,and are stable, whereas remains unstable node and the come is saddle. This suggests that there exists bistability of periodic solutions in System (1). In Region V, remains stable, whereas remains unstable node, is saddle and is disappears. In Region VI, becomes stable, whereas is saddle. Stable local solutions exist in all Regions I-VI. Figure 4 illustrates the corresponding numerical simulations.

Fig. 4.

Simulation assessments of System (1).

For the HHBP , with and . Similarly, the Fig. 5 depicts the branch and the classification of the HHBP .

Fig. 5.

BD of .

To begin, in Region I, the trivial equilibrium remain stable. is come and stable in Region II, while is saddled. In Region III, begins as a saddle, remains an unstable node and remains stable. In Region IV,and are stable, whereas remains unstable node and the come is saddle. This suggests that there exists bistability of periodic solutions in System (1). In Region V, remains stable, whereas remains unstable node, is saddle and is disappears. In Region VI, becomes stable, whereas is saddle. Stable local solutions exist in all Regions I-VI. Figure 4 illustrates the corresponding numerical simulations. The corresponding numerical simulations capturing these phenomena are presented in Fig. 6, which serves as a visual demonstration that the analytical results agree with these conditions, and also reveals the intricate stability switching features in the system.

Fig. 6.

Numerical solutions of System (1) for .

Example 2. In this example, we use and as the bifurcation parameters. Figure 7 illustrates the critical curves of Hopf bifurcation and their intersection points in the plane, with the following parameters,

Fig. 7.

Critical curves of HHBPs and the distributions of the eigenvalues of of system (2) in the .

Regarding the Hopf-Hopf bifurcation point we can also compute the distributions of the eigenvalues of , the eigenvalues have positive real components. Hence, we employ exclusively scientific computations to investigate the complex dynamics of System (1) in its surrounds.

To illustrate the complex dynamics, we will provide the time history, phase portraits, and the associated Poincaré sections of System (1) in the vicinity of the HHBP . Figure 8 illustrates complex dynamics around the Hopf-Hopf bifurcation point in System (1). As a consequence, System (1) shows a variety of complex dynamical phenomena, like periodic solution, Period-1, quasi-periodic solutions, phase-locked solutions and chaos, for various parameters.

Fig. 8.

Numerical simulations of System (1) for in consequently and their Poincare Sections in (a3)-(h3), with Section for initial functions with where .

The Fig. 9 illustrated the lyapunov exponent plot for system (1).

Fig. 9.

Lyapunov exponent.

Methodology

Core features of the suggested ciphering method

The suggested ciphering method employs a chaotic delayed differential framework, DNA coding, and bit-level processes in a combination to produce secure lossless audio encoding. The essential features are composed of:

1. a chaotic delayed process that creates pseudo-random sequences;

2. a chaos-keys expansion framework that is very sensitive to its initial conditions and;

3. DNA encoding to perform nonlinear substitution, permutation, and complementation on the audio signal.

Chaotic delayed system

The encryption algorithm being put forth utilizes a chaotic delayed differential system previously defined in Eq. (1) as its core pseudo-random generator. This system has a deterministic yet un-predictable nature, demonstrating sensitivity to initial conditions and nonlinear delayed feedback. The delayed term provides the system with memory in its dynamics, resulting in greater complexity and the possibility of hyper-chaotic behavior.

The system has control parameters and that specify the degree of nonlinearity and delay coupling, ensuring that the change of even a small amount in these variables, or in the initial conditions and , result in completely different trajectories providing excellent key diversity which is a good property of a chaotic sequence to be applied in a cryptographic system.

To construct the key stream, System (1) is solved numerically in MATLAB using the dde23 solver over a fixed time horizon. The trajectories produced as functions of time, and , are uniformly sampled and mapped to . The normalized sequence of is then used to derive three independent keys which drive the encryption process:

• Substitution key () – calculated as , where the result produces integers in the set corresponding to DNA arithmetic operations;

• Permutation key () – obtained by sorting the chaotic sequence in ascending order and defining the index vector of sorted as the pattern for the permutation;

• Complement flag () – a binary control sequence defined by if , and 0 otherwise to indicate whether a nucleotide will be complemented.

The chaotic delayed system exhibits characteristics of high unpredictability and extreme sensitivity to initial parameters that, along with the proposed key generation strategy, demonstrate strong diffusion and confusion characteristics, and a solid resistance to brute-force, statistical, and differential attacks. Therefore, it would be a suitable pseudo-random source to generate dynamic keys that can be used to drive the DNA-based encryption process that follows.

Audio pre-processing

Plain audio signals are first read and normalized, to ensure that both amplitude and sampling characteristics are as consistent as possible. If it is stereo, the audio signal is turned into mono, converted to a sample rate.

The normalized waveform is then quantized to an 8-bit unsigned integer, the obtained data is then turned into binary bit-streams. Each set of samples generates an 8-bit vector for DNA mapping.

DNA sequence encoding

DNA encoding is a biomimicry-inspired data representation which maps binary information onto the four nucleotides adenine (A), cytosine (C), guanine (G), and thymine (T). The process of DNA encoding maps every two bits into one nucleotide using the pre-defined mapping.

The binary bitstream is transformed into a nucleotide sequence that will be denoted as:

DNA arithmetic operations are defined as modular base-4 addition and subtraction, while the complement operation follows the mapping provided in the previous section:

These biologically inspired operations insert nonlinearity and diffusion into the encryption process.

DNA arithmetic refers to modular–4 operations performed on nucleotides after mapping 2-bit binary blocks to DNA bases. For any nucleotide encoded as , substitution is performed using modular addition: . This differs from XOR-based binary DNA logic—allowing nonlinear substitution at symbol level, which provides stronger diffusion.

Chaotic-based key generation

Pseudo-random data values are derived from the chaotic sequence and used to control all stages of encryption. Normalized values are generated:

• Permutation sequence () - reordering of the nucleotide;

• Substitution sequence () - used in DNA modular addition/subtraction;

• Complement flag () - whether the nucleotide needs to be complemented.

Any slight perturbation (e.g., ) in any initial condition or parameter results in an entire different key stream being generated, demonstrating key sensitivity that is extreme and a desirable feature in cryptosystems.

Encryption process

The DNA-based audio encrypting passes through three main operations:

1. Permutation: The sequence is permuted using the permutation indices associated with destroying the local correlation amongst neighbors in the audio.

2. DNA Arithmetic: Each permuted nucleotide is modified through modular addition with a substitution key at

   

3. Conditional Complementation: For values where , the nucleotide is complemented at . The resulting sequence De is the encrypted DNA data.

Finally, the De is decoded back to binary format, grouped into bytes, and converted to waveforms back in the original amplitude range. This output is saved into a binary file and the encrypted .wav file.

Decryption method

Decryption involves the reverse order of operations for each of the three operations, utilizing the same system parameters and initial conditions:

1. Inverse complementing and associated with .

2. Mod-4 DNA subtraction and associated with .

3. Inverse permutation in association with .

Once the bytes are reconstructed, they are rescaled to form original audio samples, resulting in a decrypted audio signal that is identical to the plaintext, provided the keys are equal.

Key sensitivity and lossless recovery

The tests performed on the proposed chaotic DNA scheme, indicate that lossless decryption with mean-square error approaching 0 between the original and decrypted signals is achievable. The chaotic generator is highly sensitive in terms of the initial state and all parameters meaning that even the smallest change to a parameter or initial state will make decryption impossible. This guarantees that the system is robust against brute-force attacks, chosen-plaintext, and statistical attacks.

Experimental results and performance analysis

Waveform analysis

Waveform analysis, which refers to the examination of the temporal waveform of an encrypted audio signal, is a very significant technique in the determination of an encryption method’s effectiveness. The reason for carrying out waveform analysis is to check if the encryption scheme hides the original audio structure well enough so that even a visual inspection of the encrypted waveform reveals no recognizable information of the audio signal that existed previously. Figure 10 shows the spectrograms of the original audio, encrypted audio, and decrypted audio for the five different test samples, named as signals 1 to 5. Each row represents one audio sample, while the three columns indicate the original, encrypted, and decrypted audio, respectively. The original audio waveforms (1i–5i) exhibit clear amplitude fluctuations and temporal structures associated with the input audios clearly depict amplitude fluctuations and temporal structures that can be attributed to the input audios. In comparison, the encrypted waveforms (1ii–5ii) appear to be completely random because they do not show any distinguishable temporal or amplitude pattern; thus indicating that the process has achieved its purpose of concealing the original content of the signal. The decrypted waveforms (1iii–5iii) are almost identical to their original counterpart, allowing us to be assured that the proposed encryption–decryption procedure is accurate and reversible. This figure provides visual evidence that the scheme provides an extremely high level of confidentiality while preserving perfect reconstruction of the audio signal.

Fig. 10.

Wave form of (a) Original Audios, (b) Encrypted Audios, and (c) Decrypted Audios.

Spectrogram analysis

A spectrogram visually maps audio information in three dimensions, time, frequency, and amplitude, onto a two, dimensional plane, thus allowing detailed time, frequency analysis of a signal³⁶. Here, the changes in color intensity represent the energy magnitude of a specific time, frequency point. Spectrogram analysis is an easy and potent way to measure audio encryption performance because it shows whether any structural features of the original signal can still be detected after encryption. A good encryption system results in spectrograms that have very irregular and noise, like patterns, which means that the audio content that was used as the basis has been hidden thoroughly. On the other hand, the spectrogram without any visible distortions or artificial artifacts confirming that the signal has been accurately recovered, which is a demonstration of both the security and the reliability of the encryption process. The Fig. 11 presents the spectrograms of the original audio, encrypted audio, and decrypted audio for five distinct test audio, indicated as signals 1 through 5. The rows of the figure correspond to each audio sample, and the first column (1i-5i) consists of the original signals, the second column (1ii-5ii) contains the encrypted signals, and the third column (1iii-5iii) contains the decrypted outputs. In the spectrograms of the original audio, the time-frequency representations of each audio exhibit significant variation between each audio sample. The encrypted audio signals, however, show that the time-frequency representations of the original signals have been completely distorted and effectively turned into noise, demonstrating that the spectral and temporal information of the audio has been effectively masked. The spectrograms of the decrypted audio, on the other hand, closely resemble the original audio signals, demonstrating the efficacy of the encryption-decryption framework and demonstrating its robustness and reversibility. This visual analysis of the spectrograms indicate that the encryption scheme provides superior security, while ensuring devised distortion-free flawless transmission is ensured.

Fig. 11.

Spectogram of (a) Original Audios, (b) Encrypted Audios, and (c) Decrypted Audios.

Statistical analysis

When the statistical properties of the decrypted audio are nearly the same as those of the original plain signal, it is a demonstration that the encryption-decryption system is effectively limiting information leakage and at the same time maintaining the integrity of the signal. In such a scenario, the encrypted audio has a very weak statistical relationship with the original signal, thus no visible patterns or dependencies can be used by potential adversaries. A full range of analyses is implemented to deeply assess the security level and the efficiency of the proposed encryption algorithm.

First, statistical security is gauged by entropy analysis, which evaluates the randomness and the uncertainty in the encrypted audio; higher entropy values are indicative of better diffusion characteristics and higher resistance to statistical attacks. Subsequently, correlation analysis of the adjacent samples is carried out to evaluate the linear dependence; correlation coefficients close to zero indicate that the encrypted samples are statistically independent. Histogram analysis also plays a role in examining the amplitude distributions, where a uniform histogram signals that the audio has been effectively randomized and that the structural information inherent in the original waveform has been completely removed.

Differential analysis is also used to determine how sensitive the encryption scheme is to small changes in the plaintext audio. The Number of Sample Change Rate (NSCR) and Unified Average Changing Intensity (UACI) are examples of metrics that measure how a small change in the input is spread in the encrypted signal, thus showing the resistance of the scheme to differential attacks. Moreover, the quality of the signal and the accuracy of the reconstruction are measured by Peak Signal, to, Noise Ratio (PSNR), Signal, to, Noise Ratio (SNR), Mean Squared Error (MSE), and Mean Absolute Error (MAE). High values of PSNR and SNR combined with low values of MSE and MAE mean that the decryption process reconstructs the original audio accurately and without noticeable degradation, while encryption remains secure. To sum up, these investigations serve as evidence that the proposed audio encryption scheme is dependable, robust, and effective in terms of security.

Audio signal entropy

Entropy is one of the most popular statistical indicators that is used in measuring the degree of uncertainty and randomness in signals. In the case of audio encryption, higher entropy values signify that the audio samples are so randomized that no attacker can find structures to exploit. Therefore, high entropy is indicative of a system that can hardly resist statistical analysis and information, theoretic attacks. The entropy E is given mathematically as³⁷:

where is the probability of occurrence of a particular sample value . The sample values of an ideally encrypted audio signal would be of a uniform distribution; thus the entropy would be very close to its theoretical maximum. In the case of 8, bit audio signals, the maximum entropy value is 8 and this has been used as a benchmark for strong encryption performance³⁷. In this paper, the detailed analysis of the entropy of the encrypted audio signals is used as a means to quantitatively show the effectiveness and robustness of the proposed encryption algorithm. The entropy analysis of five different audio signals is summarized in Table 1 for before encryption (Orig.), after encryption (Ency.), and after decryption (Decry.). Entropy is a measure of the randomness/uncertainty of the signal amplitude distributions, with larger values of entropy being more desirable for secure cryptographic systems as they embody a larger degree of randomness with respect to any output. Entropy values for the original audio signals ranged from 5.10 to 7.36, demonstrating the natural statistical structure of the signals. Once the audio signals were encrypted via the Chaotic Delayed–DNA Encryption Scheme examined in this study, each of the five audio signals had entropy values close to 7.99, which signifies theoretical maximum entropy values for 8-bit data sets. This demonstrates that the audio signals are now statistically random and information-uniform. Once the signals are decrypted, the entropy values of the signals returned almost exactly to their pre-encryption values, indicating that decryption is both reversible, and accurate. This demonstrates our major findings from the proposed chaotic delayed–DNA encryption scheme, significant increases in randomness with generation of ciphertext, while ensuring perfect recovery of the original audio signal.

Table 1.

Entropy analysis.

Signals	Orig.	Ency.	Decry.
1	5.8285	7.9921	5.8281
2	5.1007	7.9917	5.1003
3	6.8220	7.9964	6.8218
4	7.3581	7.9972	7.3573
5	6.3869	7.9960	6.3860

Correlation coefficient evaluation

Finding the correlation coefficients between two nearby samples prior to and following encryption is one method of examining how successful audio encryption algorithms are. This technique examines the degree to which the sound waves of two consecutive samples in the uncensored as well as encrypted audio signals exhibit a linear connection. Typically, neighboring samples in natural audio signal are statistically dependent, leading to a high level of correlation. Any safe and properly built encryption system should be able to completely break this dependence so that adjacent samples in the encrypted audio are statistically independent. The correlation coefficients are calculated based on the normal method mentioned in³⁷, where N is the total number of selected sample pairs, and m and n are two consecutive audio samples. Here, 16,000 pairs of adjacent samples are randomly taken to be statistically sufficient. The correlation scatter plots of the original (plaintext) audio signals (Audio 1-Audio 5) are given in Fig. 12(1i-5i), which visually demonstrate the strong correlation inherently present in the data. On the other hand, the corresponding encrypted audio plots in Figs. 12(1ii-5ii) show the distribution of points in a wide range, thus verifying that the sample correlation has been eliminated effectively. Also, the correlation of the decrypted audio signals, which are shown in Fig. 12(1iii-5iii), is similar to the original one, thus confirming the security of the encryption process and correctness of the decryption scheme. In the correlation plots of the original signals Fig. 12(1i–5i), we see strong linear relationships where most of the data points fall on the diagonal line, demonstrating that there is a strong correlation between samples that have been sampled at neighboring time points. The encrypted audio plots 12(1ii–5ii) show highly scattered distributions which resulted in the pattern we see, suggesting a successful destruction of statistical dependencies and introduction of strong randomness during the audio encryption process. The decrypted plots 12(1iii–5iii) closely resemble the plots we saw for the original signals, demonstrating the utility of the proposed encryption–decryption scheme in reconstructing the signal correlations that existed in the original signals.

Fig. 12.

Correlation plot for (a) Original Audios, (b) Encrypted Audios, and (c) Decrypted Audios.

Histogram evaluation

A histogram is one of the basic statistical tools to reveal how numbers are spread in a dataset. For audio signals, histogram analysis is the standard method to look at the distribution of sample amplitudes. The histogram of a well, encrypted audio signal, therefore, will generally show a shape that is close to uniform, meaning that the original signal has been mixed thoroughly and its structure cannot be recognized anymore. The uniformity implies a high degree of randomness in the sample values, hence it is practically impossible to extract any useful information about the original audio. As a result, a flat or uniformly distributed histogram can be considered as a strong signal of increased unpredictability, better confidentiality, and strengthened security of the encrypted audio data. The first five Fig. 13(1i, 5i) represent the histograms of the original (plaintext) audio files. The Fig. 13(1ii-5ii) are for the encrypted audio files, and the Fig. 13(1iii-5iii) are for the decrypted versions. The histograms of the original audio show distinct peaks in certain amplitude areas, which indicates that the characteristics of the audio are also structured and non-random, as natural sounds typically are. The characteristics of the histograms of the encrypted audio show nearly uniform characteristics indicating the encryption was successfully done and that the amplitude values were randomized, removing all statistical characteristics of the plaintext audio. This uniformity indicates strong diffusion characteristics, and suggests that the encrypted audio signals will not provide any information about the original audio content. Each histogram of the decrypted audio closely resembled the histograms of the original audio, with no distortion or loss of information. Overall, the comparison of the histograms of the original, encrypted, and decrypted audio files provided verification of the excellent performance in ensuring the audio is random and that the encryption scheme is robust enough to restore audio without losing integrity during decryption. Although the histogram of the encrypted short-duration Sample 2 visually appears less smooth, this is due to its small sample size rather than reduced diffusion strength.

Fig. 13.

Histograms plots for (a) original audios, (b) encrypted audios, and (c) decrypted audios.

Differential analysis

In order to compare the sensitivity of the encoded audio to the original audio, minimal changes are made to the two widely accepted criteria: the Unified Average Changing Intensity (UACI) and the Number of Sample Change Rate (NSCR). These indices measure how a change of one single sample in the audio signal can spread throughout the encrypted signal. High NSCR means that most samples in the ciphertext have been changed, which thus ensures that the influence of the plaintext is totally hidden and the possibility for differential attacks is considerably reduced. Likewise, high UACI values signal that sample amplitudes have changed substantially and evenly and hence, the problem of revealing plaintext patterns is alleviated. Hence, UACI in conjunction with NSCR represent a full numerical measure of diffusion performance, i.e. the proposed encryption scheme has reached a high level of security and is very efficient at hiding the relationship between the plaintext and the ciphertext.

One of the standards used to measure the strength of encryption algorithms is the NSCR. It is also the figuring out of the NSCR by which the number of audio samples decrypted per audio sample changed. The NSCR is defined as follows³⁶:

where

The metrics for each individual sample of the two coded audios are denoted by and , respectively, with N being the total number of samples.

Mathematically, UACI can expressed as³⁶.

The minimum value of the UACI is 33.33% and the optimal value of the NSCR is 100%.

Results of a differential attack analysis on five audio signals are shown in Table 2 for NPCR and UACI. The above metrics are used to determine the algorithm’s strength against differential attacks, based on the impact of a one-sample change in the plaintext on the ciphertext. It is desirable that a secure encryption scheme has NSCR and UACI values that are high; this is evidence that small changes in the plaintext are generating larger changes in the corresponding ciphertext. As can be seen from the table, all encrypted audio signals presented NSCR values greater than 99.9%, implying that nearly every audio sample has changed following a one-sample change in the plaintext. Furthermore, UACI values for all data sets were hovering around 33.3%, nearly identical to the theoretical ideal value of 33.33% for 8-bit data. Overall, the proposed chaotic delayed–DNA encryption scheme has good diffusion properties and a high avalanche effect which are proven to be strong against differential attacks as well as chosen plaintext attacks. The consistently higher NSCR and UACI values confirm that the proposed scheme provides stronger diffusion and better resistance to differential attacks compared to existing chaos-based and DNA-based encryption approaches.

Table 2.

Differential analysis and comparison with already existing schemes.

Signals	NSCR (%)	UACI (%)
1	99.94	33.37
2	99.98	33.31
3	99.96	33.33
4	99.99	33.34
5	99.98	33.32
38	99.60	33.46
39	99.71	33.50
40	99.81	33.34
23	99.62	33.50

Mean absolute error (MAE)

Mean Absolute Error (MAE) is one of the most common performance metrics that delineates the average size of the errors in two signals without taking their direction into account. When it comes to audio encryption, MAE functions as a measure of the average absolute difference between the plain audio signal and the decrypted audio signal. This measure is a direct way to show how the encrypted audio is successfully converted back to the original one. Formally, MAE can be expressed as:

where is the i-th sample of the original audio signal, stands for the corresponding sample of the decrypted audio signal, and N is the total number of audio samples. The closer the MAE value is to zero, the less the average difference between the original and the decrypted signals and, consequently, the less distortion and the higher the reconstruction fidelity. In other words, in an encrypted audio signal, after decryption, MAE should be as low as possible, implying that the security process does not affect signal quality. Hence, MAE is a faithful quantitative instrument to gauge the level of accuracy, security, and implement ability in the case of the given encryption algorithm. The MAE results of the audios that we assessed can be found in Table 3

Table 3.

Statistical analysis.

Signals	MAE	MSE	SNR (dB)	PSNR (dB)
1	2.06E-03	5.70E-06	36.18	58.92
2	1.85E-03	4.61E-06	39.04	59.10
3	1.76E-03	4.13E-06	39.69	58.90
4	2.03E-03	5.52E-06	43.16	58.95
5	1.88E-03	4.73E-06	37.40	58.91

Signal to noise ratio (SNR)

The quality of the signal can be measured by using the “Signal to Noise Ratio (SNR).” Specifically, the signal is greater than the noise when the value is greater than 0 dB. The SNR can be calculated with the knowledge of the host and audio encoded files. The following is the SNR formula³⁶:

In this case, n is the number of samples, where and indicates the host encoded audio files and their trials, respectively. The SNR results of the audio recordings that we assessed can be found in Table 3. The SNR number is negative, which indicates the effectiveness of this strategy. Table indicates that our strategy’s negative SNR is greater, show that the strategy is more favorable against adverse attacks.

Peak signal to noise ratio

PSNR is essentially a measure of the power of the clean signal relative to the power of the noise. While PSNR is mainly associated with image encryption algorithms, it can also serve as a metric for the quality of an encryption algorithm. The formula for calculating PSNR is given by³⁶.

In this case, MAX is the highest possible value of the samples, and MSE is the mean square error whose calculation is shown below³⁶:

Table 3 presents summarized statistical analysis of five audio signals after being encrypted and decrypted by the proposed chaotic delayed–DNA encryption scheme. The obtained values are compared to the original captured audio file using the MAE, MSE, SNR and PSNR parameters. The MAE and MSE values associated with each audio signal is extremely small, between the order of and respectively, indicating negligible distortion between the original audio signal and the decrypted audio signal. The SNR values generated are reported in the range of 36.18 dB and 43.16 dB indicating that the decrypted audio signals maintain a high level of fidelity regarding the original audio signals. The PSNR values also consistently hover around 59 dB confirming once again that the decrypted audio signals maintain audio reconstruction quality. In conclusion, these extensive results demonstrate a reasonable level of demonstrated strong security measures during the encryption stage while demonstrating lossless decryption and high quality audio signal recovery.

Time complexity analysis

Time complexity in encryption refers to how much time units an algorithm takes to run based on the size of its input³⁶. It is an important consideration because it contributes directly to how quickly an encryption will protect and retrieve data and especially so if one wants to deploy such applications in real-time. Therefore, time complex often uses Big-O notation to express the number of multiples that runtime increases based on input size. The Table 4 illustrates how long it takes to complete one encryption round.

Table 4.

Time analysis.

Signals	Time (s)
1	1.69
2	0.29
3	4.44
4	0.28
5	2.21

The computational framework of the suggested approach largely revolves around three main operations: (i) assessment of a chaotic delayed, system trajectory, (ii) symbolic substitution and complement operations based on DNA, and (iii) permutation of nucleotide indices. The encryption complexity is representable as:

bound of approximately O(NlogN). As a matter of fact, the chaotic key generation is only performed once and then shared among symbol blocks, while DNA substitution is carried out on fixed, length pairs and thus has a linear scaling with the input size. Experimental latency measurements (Intel, i7 CPU, MATLAB simulation) reveal encryption time varying between 0.28 and 4.44 s for audio lengths of 1, 15 s. The prototype MATLAB implementation is not strictly real, time, however, the algorithm can be inherently parallelized: chaotic key generation, DNA substitution, and permutation can be independently vectorized or launched on GPU/FPGA hardware. The estimated processing time under GPU execution reduces to < 8 ms for 1, second audio blocks and under FPGA or optoelectronic oscillator integration, key generation becomes continuous, thus streaming encryption is possible without additional latency. Hence, the system is computationally feasible for real, time secure audio communication when implemented in an optimized C/C + + or hardware, accelerated form, which is indicative of a strong potential for deployment in VoIP, tactical radio, and UAV, to, ground encrypted channels.

Key sensitivity analysis

Key sensitivity measures how small changes in either the secret key or system parameters translate to the encryption and decryption. An ideal cryptosystem should show extreme sensitivity to initial conditions and control parameters, whereby even an infinitesimal perturbation will create a completely different ciphertext. It is important to note that the proposed system inherits this property from the chaotic delayed system defined in Eq. (1) due to its strong nonlinear feedback and memory effect. In this study, to test key sensitivity, we performed the encryption step twice with the same audio input and all parameters held constant except for a very small variation of in one of the parameters that controls the chaotic system. The two ciphertexts exhibit no visual or statistical similarity: the correlation coefficient between the two was nearly zero and the MSE was greater than 0.9 indicating that even infinitesimal differences in the parameters create ciphertexts that are completely uncorrelated. Similarly, the slightly altered key set for decryption could not recover the original audio signal and instead resulted in a noisy waveform that was not intelligible with a and entropy close to that of white noise. The experiment shows that the encryption is very sensitive to every key parameter adopted, as well as the sequences of DNA mapping. This sensitivity guarantees the impossible of successfully attacking the encrypted audio with keys that are close or rounded off. Hence, the chaotic delayed–DNA encryption algorithm achieves extremely high sensitivity of keys, thus securing a high level of safety against brute-force, differential, and chosen-plaintext attacks.

Key space analysis

The key space assessment looks at the entire number of unique key combinations available in the proposed chaotic delayed-DNA audio encryption design, which will provide the resistance to brute force one would expect. The proposed scheme has multiple independent parameters for obtaining key diversity such as the chaotic control parameters, initial conditions and DNA-based dynamic sequences (). Each of these parameters are in double precision with an approximate computational accuracy of (i.e., about possible values per random parameter), therefore the chaotic subsystem alone will provide a key space order of . In addition, the chaotic time series produces three pseudo-random sequences, i.e., substitution-key , permutation-key , and complement-flag . Therefore, audio data containing N samples generates about key space (i.e., there is additional key space even when all the keys are in use, ensuring exponential growth with the length of the audio file, implying that it provides far more than is necessary for cryptographic functions). Hence, when the chaotic and DNA landscapes are combined, the total key space is definitely more than , or more than the equivalent of bits of entropy; and that is far more than the standard threshold of bits, as sufficient to protect against brute-force attacks. One more property in addition to the astounding size of the key space is that, although even extremely small changes (as little as ) can make vast differences in the key space even parameters of the chaotic delayed system can produce vastly different encrypted outcomes. Accordingly, it is not only the significant size of the key space, but also the clear sensitivity of the window to change, which demonstrates the proposed scheme is vigor, additional attributes prove its viability to remain a large, unprovoked key space, and against numerous statistical average attacks. In this study, the samples N is the number of total discrete amplitude points with the processed audio signal. For example, considering a 10-sec mono sampled at , we have . This N value increases the effective key space tremendously because of the growth of the factorial and the exponential nature of the DNA permutation and substitution operations.

Rigorous assessments of chaotic key sequence under noise, parameter drift, and sampling inaccuracies

The security of chaos, based cryptosystems heavily relies on the key sequences remaining unpredictable and irrecoverable under various conditions. This includes the environmental disturbances, noise of the implementation, or parameter mismatches. It is, therefore, an absolute necessity to examine the stability and reliability of the delayed nonlinear chaotic system under such perturbations that may arise in the real world. These may include electronic noise, component aging, timing jitter, and imprecise digitization apart from each other. Three different conditions were numerically and quantitatively simulated in order to measure the robustness to these perturbations. These are (i) noise during the system’s numerical integration of the delay, (ii) drift of the cubic nonlinear coefficient representing the physical degradation of the device, and (iii) sampling inaccuracy in order to emulate reduced, rate digital acquisition and clock jitter. is denoted as the original chaotic key and as the perturbed key sequence. The statistical degradation was measured by the correlation coefficient and mean, squared error (MSE). A perfectly secure scenario is characterized by low correlation (indicating no linear predictability) and high MSE (signifying full statistical divergence). Table 5 shows the performance of our chaotic key generator in terms of robustness to three different perturbations as well as wrong, key sensitivity verification where a microscopic parameter shift () is applied while decrypting (Table 6).

Table 5.

Robustness of chaotic key sequence under practical perturbations.

Perturbation type	Simulation description	Parameter change applied	Correlation	MSE	Security interpretation
Gaussian Noise Injection	Additive perturbation applied during DDE integration				No statistical similarity → unpredictable and irrecoverable
Parameter Drift	Aging and thermal drift effect on chaotic coefficient				Drastically different key stream → prevents attacker simulation
Sampling Inaccuracy	Down-sampling to model clock jitter or ADC loss				Missing data does not help attacker → key cannot be reconstructed
Wrong-Key Sensitivity Test	Microscopic incorrect parameter used at decrypt side				Decryption completely fails → extremely high sensitivity & strong brute-force resistance

Table 6.

Characteristics of test audio samples.

Sample ID	Duration (s)	Sampling rate (Hz)	Number of samples
Audio 1	5
Audio 2	1
Audio 3	8
Audio 4	1
Audio 5	6

The numerical outcomes explicitly show that any disturbance, even a microscopic change, results in a statistically unrelated key sequence, which is a direct confirmation of the fundamental chaotic nature of the proposed delayed system. In terms of cryptography, this provides three strong guarantees:

1. Chosen, plaintext and adaptive reconstruction attacks are impossible, as any attempt to approximate the chaotic parameters results in fully decorrelated keys (MSE > 0.85).

2. Noise at the hardware level during the implementation or jitter in sampling does not release information, which means that embedded deployment on IoT or communication modules remains secure.

3. The encryption key cannot be reproduced without the exact full, precision parameters, thus effectively providing exponential, space key hardness.

The enhanced security of the presented scheme is mainly due to the synergistic interaction between the delayed chaotic system and adaptive DNA arithmetic. In contrast to chaos, only methods that mainly encrypt audio signals through value, level transformation and DNA, only schemes that simply rely on fixed nucleotide substitution rules, our design elevates the system to two layers where the chaotic state controls the DNA rule, complement flag, and substitution index at each sample position. Therefore, the encryption procedure comes very close to perfect randomness (entropy approximately 8), completely removes the correlation between adjacent samples, and achieves extremely strong diffusion (NSCR > 99.95%, UACI 33.3%), thus, the performance of the proposed method is several times better than that of the two individual paradigms. This multilayer nonlinearity impedes attackers from reconstructing keys through differential or chosen, plaintext strategies, hence a security advantage of great magnitude is exhibited.

Audio length and dataset characteristics

The length of the audio signal is an important factor when determining the robustness and scalability of audio encryption schemes. This is because measures of statistical security, such as entropy, correlation, histogram uniformity, and differential attack metrics, are all dependent on the number of samples processed. In order to be transparent, reproducible, and allow for a fair comparison, the duration and sampling characteristics of all test audio signals used in this work are clearly given. Prior to encryption, all audio samples were resampled to a common sampling rate of 16 kHz in order to keep the security and computational performance analyses consistent. The chosen audio signals cover short, medium, and long durations, thus the proposed encryption scheme can be tested under different signal lengths that are appropriate for real, time communication and stored audio protection.

The conducted experiments demonstrate that the proposed chaos, driven DNA, based encryption scheme preserves security features with high fidelity in all cases of audio durations that were tested. The histogram plots of short, duration signals (1 s) visually show slight non, uniformity due to the very small number of samples; however, quantitative security metrics such as entropy, correlation coefficient, NSCR, and UACI are very close to their ideal values, thus, cryptographic strength has not diminished. In the case of longer audio signals (58 s), the encrypted waveforms and histograms become more visually uniform from the statistical point of view, which further confirms the scalability and robustness of the proposed method. What is more, the computational processing time is growing linearly with the length of the signal, thus the real, time implementation of secure voice communication systems is feasible. The robustness of the proposed encryption scheme as a function of signal length has been confirmed by explicitly mentioning audio durations and sample counts, thus, the scheme is also appropriate for short, frame real, time communication as well as longer audio transmission scenarios.

Conclusion and future recommendation

This article presented a comprehensive hybrid framework that combined the nonlinear dynamics of a cubic optoelectronic oscillator (OEO) along with time-delayed feedback and DNA-based computation techniques for data security and reversible audio encryption. The study obtained the OEO system, using multiple scales (MMS) and DDE-BIFTOOL methods to identify and classify Hopf – Hopf bifurcation points, periodic and quasi-periodic oscillations, and chaos regimes. These complex dynamic behaviors were then successfully used to produce complex uncorrelated sequences to use as the basis for all of the DNA-level permutation, substitution, and complementation operations imbedded for the purpose of achieving nonlinear diffusion and confusion steps for the audio encryption. This work proposed a novel chaotic delayed–DNA-based audio encryption framework that combines delayed nonlinear dynamical systems with adaptive DNA arithmetic to achieve multilayer nonlinear confusion and diffusion. The chaotic trajectory dynamically governs DNA rule switching, complement selection, and substitution, which significantly enhances randomness and security robustness compared to chaos-only and DNA-only methods. The security evaluation demonstrated near-ideal statistical performance: cipher text entropy reaching approximately 8.00, adjacent-sample correlation ≈ 0.00, NSCR > 99.95%, and UACI ≈ 33.3%. In addition, robustness tests confirmed that the generated chaotic key stream remains unpredictable under Gaussian noise, parameter drift, and sampling inaccuracies, making the system resilient to physical-layer disturbances in practical environments. Computational analysis further revealed that although MATLAB implementation incurs moderate latency, the algorithm has an overall complexity of and is feasible for real-time operation when deployed on optimized GPU, FPGA, or optoelectronic-assisted hardware.

Overall, the results affirm that the hybrid cryptographic architecture achieves strong statistical randomness, demonstrates extreme sensitivity to parameters, and preserves audio fidelity upon decryption, suggesting its suitability for encrypted streaming applications such as VoIP, satellite communication, UAV tactical communication, and digital media protection.

Future work

Future research may extend the framework in several technical directions e.g^41–48. , . First, optimization of the cryptosystem using embedded implementations such as CUDA-GPU cores or FPGA-based chaotic oscillators will be investigated to demonstrate fully real-time hardware performance. Second, machine-learning-driven adaptive security where chaotic parameters evolve based on traffic pattern prediction or intrusion signals could make the system even more attack-resilient. Third, integrating biometric-guided parameter seeding (e.g., EOG, ECG, or voiceprint-based entropy sources) may provide personalized encryption with zero-knowledge authentication. Finally, future work may also explore the application of multi-scroll or fractional-order chaotic systems and DNA-based multilevel substitution to further expand key-space dimensionality, thereby enhancing resistance against quantum-era cryptanalysis and enabling multi-user secure key-sharing protocols.

Author contributions

**M.A. and J.Y.:** Conceptualization (equal); Software (equal); Writing - original draft (equal); Writing - review & editing (equal). **A.Z.A.:** Investigation (equal); Methodology (equal); Writing - original draft (equal); Writing - review & editing (equal). **N.A.:** Methodology (equal); Supervision (equal); Visualization (equal); Writing - review & editing (equal). **N.B.:** Methodology (equal); Resources (equal); Visualization (equal); Writing - review & editing (equal). **W.H.:** Conceptualization (equal); Supervision (equal); Visualization (equal); Supervision (equal); Writing - review & editing (equal). **K.G.:** Methodology (equal); Software (equal); Supervision (equal); Writing - review & editing (equal). All the authors have agreed to publish this manuscript.

Funding

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R730), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.  The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number "NBU-FFR-2026-2933-01".

Data availability

Data underlying this study are available from Dr. Walid Hassen (email: hassen.walid@gmail.com) or Dr. Aqsa Zafar Abbasi (email: zafariqbalabbasi67@gmail.com) upon reasonable request.

Competing interests

The authors declare no competing interests.

Ethical approval and consent to participate

The study does not involve any ethical problem and data collection was completed in accordance with the ethical regulations.