Hardware Implementation of a 2D Chaotic Map-Based Audio Encryption System Using S-Box

Abstract

This paper presents a hardware-based audio encryption system using a 2D chaotic map and dynamic S-box design implemented on an Artix-7 FPGA platform. Three distinct chaotic maps—logistic–fraction (2D-LF), logistic–sine (2D-LS), and fraction–sine (2D-FS)—were investigated and implemented on an FPGA. The 2D-LF map was employed in the encryption system for its throughput and power efficiency performance. The proposed encryption system benefits from the randomness of chaotic sequences for block permutation and S-box substitution to enhance the diffusion and confusion properties of the encrypted speech signal. The system’s encryption strength is validated through performance evaluations, using the mean squared error (MSE), signal-to-noise ratio (SNR), correlation coefficients, and NIST randomness tests, which confirm the unpredictability of the encrypted speech signal. The hardware implementation results show a throughput of 2880 Mbps and power consumption of 0.13 W.

1. Introduction

The field of secure communication experienced significant improvements with the advances of chaotic maps [1,2,3]. Chaotic maps show complex, unpredictable behavior, which makes them suitable for encryption algorithms [4,5,6]. Their sensitivity to initial conditions [4,7] provided a foundation for generating pseudorandom sequences that enhance the security of transmitted information [8,9]. Chaotic maps have been effectively used in hardware-based implementations [10,11] and offer speed and power efficiency advantages. Field-Programmable Gate Arrays (FPGAs) are well suited for this purpose due to their reconfigurable architectures and parallel processing capabilities [12].

A reconfigurable FPGA implementation of three fractional-order chaotic systems was proposed in [13], based on the Grünwald–Letnikov approximation. A memristive chaotic system with transcendental nonlinearities was implemented in [14], employing a reconfigurable CORDIC (coordinate rotation digital computer) architecture on an Artix-7 FPGA. The CORDIC algorithm [15] demonstrates efficiency in computing trigonometric, hyperbolic, and logarithmic functions, while iterative shift-add operations make it ideal for hardware implementation Lakshmi2010. The reconfigurable CORDIC design presented in [14] could support all configurations of the CORDIC algorithm for three coordinate systems within a single generic design.

The FPGA realization of a fractional-order multi-scroll 2 × 2 chaotic grid system was implemented using the GL technique [16]. It was dynamically translated and rotated in two and three dimensions through the CORDIC algorithm. The system was employed as a pseudorandom number generator (PRNG) in an image encryption scheme. A fractional-order version of a memristive chaotic system with transcendental nonlinearities was implemented in [17]. This system was used in the image encryption algorithm as a PRNG with substitution and permutation operations. The fractional orders served as additional parameters, adding extra degrees of freedom to the system.

Chaotic signals can be predicted if the system is simple by identifying the mathematical models of the chaotic systems generating these signals [18]. In [19,20], the initial state control parameters were deduced. In chaos-based cryptography systems, attackers could predict the chaotic signals to determine cryptographic keys and potentially break systems without secure keys [21]. Increasing the chaotic system’s complexity can provide a solution to this. For example, the complexity of chaotic maps was improved using higher-dimensional chaotic maps [22].

The Sinusoidal Feedback Sine ICMIC Modulation Map (SF-SIMM), a high-dimensional hyperchaotic map introduced in [23], was used to generate two independent PRNG sequences. The n-dimensional chaotic model (nD-CM) framework in [24], allowed the generation of chaotic maps of any desired dimension. It uses existing one-dimensional (1D) chaotic maps as seed maps, making it a versatile tool for secure communication systems [24]. An Artificial Neural Network Chaotic PRNG (ANNC-PRNG) using the capability of an ANN to fit chaotic generators was introduced in [25]. The proposed ANN was trained on chaotic time series and employed in a simple encryption system on an FPGA. In [26], single-hidden layer feedforward neural networks with nonlinear activation functions effectively emulated chaotic systems such as the Lorenz-63 and Hénon maps for encryption systems.

Chaotic maps enhance encryption system security and data processing efficiency. In [27], a hybrid chaotic system developed using Zaslavsky and Zigzag maps generated PRNG numbers for permuting speech samples. A system based on the Discrete Wavelet Transform (DWT) and Hénon chaotic maps provided speech signal compression and encryption system in [28]. A four-level DWT transform domain used with permutation for speech encryption was implemented on an FPGA in [29].

In [30], a five-dimensional hyper-chaotic system was employed as a chaotic oscillator producing PRNG numbers for substituting scrambled speech samples. The sampling of speech signals at five levels used various types of chaotic maps, with multiple levels of permutation applied to each sampled signal. Parameters from these five chaotic maps fed into SHA-1 alongside a Blowfish key to compute a 160-bit message digest. A multiuser wireless speech encryption system was introduced in [31], using synchronized chaotic systems as key generators. It incorporated a hybrid encryption approach with XOR bitwise operations and an encryption scheme called Chaotic Matrix Operation for Randomization or Encryption (C-MORE). In [32]. The Nose Hoover chaotic generator was used with and without dynamic shift and bit permutation techniques for the speech encryption system.

Furthermore, a substitution box (S-box) and XOR gate were employed with the chaotic system to encrypt speech signals in [33]. S-boxes are FPGA-friendly compact designs [34]. They are categorized as static or dynamic [35]. Static S-boxes are fixed data for all inputs, making them susceptible to cyberattacks. In contrast, dynamic S-boxes, which change outputs based on inputs, offer a level of adaptability that could significantly enhance security, despite their requirement for more hardware resources [36]. A speech encryption method based on the Lorenz system and S-Box table was proposed in [37]. An algorithm for encrypting speech signals was developed [38] combining scrambling, DNA coding, and substitution boxes constructed using chaotic maps as PRNG generators.

This paper presents the implementation of various 2D chaotic maps, including fraction–sine, logistic–sine, and logistic–fraction maps, on an Artix-7 FPGA platform using a reconfigurable CORDIC algorithm. These 2D chaotic maps are generated from nominal chaotic maps. A robust 2D chaotic map-based audio encryption system with dynamic S-box substitution is proposed. The encryption system uses the 2D-LF chaotic map because it provides the best throughput and the highest operating frequency among the three maps. The effectiveness of the proposed system is evaluated through statistical analyses, using correlation coefficients, the mean squared error (MSE), the signal-to-noise ratio (SNR), and hardware performance metrics.

This paper is organized as follows: Section 2 discusses the implementation of three 2D chaotic maps based on the reconfigurable CORDIC algorithm. Section 3 outlines the proposed encryption system and its hardware implementation. Section 4 presents the evaluation metrics, investigating the effect of encryption on speech signals and showing the system’s effectiveness. Finally, Section 5 concludes the paper.

2. The 2D Chaotic Map System Architecture

Two-dimensional chaotic maps are systems generated from the n-dimensional maps’ definitions in [24]. They are formed by combining two one-dimensional (1D) chaotic maps, which are systems that operate on a single variable. The state of the system at time step $i+1$ is determined using the outputs of the 1D chaotic maps as follows [24]:

$$\begin{array}{c}\hfill \begin{array}{cc}{x}_{i+1}& =sin(\pi (F_1(x_i)+F_2(y_i))),\\ y_{i+1}& =sin(\pi (F_1(y_i)+F_2(x_i))),\end{array}\end{array}$$

(1)

where $F_1(.)$ and $F_2(.)$ are 1D chaotic maps, and $x_i$, and $y_i$ are the ith observation states of the chaotic model.

2.1. The 2D Logistic-Fraction Map (2D-LF)

To generate a new 2D logistic–fraction (2D-LF) map using (1), the seed chaotic maps $F_1(.)$ and $F_2(.)$ are set to the logistic and fraction maps, respectively. The mathematical equation for the 2D-LF map can be written as follows [24]:

$$\begin{array}{c}\hfill \begin{array}{c}{x}_{i+1}=sin(\pi (4p_1{x}_i(1-x_i)+{\displaystyle \frac{1}{y_i^2+0.1}}-p_2{y}_i)),\\ y_{i+1}=sin(\pi (4p_1{y}_i(1-y_i)+{\displaystyle \frac{1}{x_i^2+0.1}}-p_2{x}_i)),\end{array}\end{array}$$

(2)

The 2D-LF map has two control parameters, $p_1$ and $p_2$, from the logistic and fraction maps. Because the sine transformation is a bounded operation, parameters $p_1$ and $p_2$ can have large values. This study investigates the properties of the 2D-LF map for $p_1$ and $p_2$$ϵ[1,100]$.

2.2. The 2D Logistic–Sine (2D-LS) Map

If the fraction map is replaced with the sine map in (2), the seed chaotic maps $F_1(.)$ and $F_2(.)$ are set to the logistic and sine maps, respectively. This generates a new 2D logistic–sine (2D-LS) map, given as [24]

$$\begin{array}{c}\hfill \begin{array}{c}{x}_{i+1}=sin(\pi (4p_1{x}_i(1-x_i)+p_{2}sin(p_i{y}_i))),\\ y_{i+1}=sin(\pi (4p_1{y}_i(1-y_i)+p_{2}sin(p_i{x}_i)))),\end{array}\end{array}$$

(3)

The 2D-LS map has two control parameters, $p_1$ and $p_2$. The ranges of these parameters are the same as the two parameters in the 2D-LF map, $p_1$ and $p_2$$ϵ[1,100]$.

2.3. The 2D Fraction–Sine (2D-FS) Map

Setting the seed chaotic maps $F_1(.)$ and $F_2(.)$ in (1) to the fraction and sine maps, respectively, generates a new chaotic map, the 2D fraction–sine (2D-FS) map, written as follows [24]:

$$\begin{array}{c}\hfill \begin{array}{c}{x}_{i+1}=sin(\pi ({\displaystyle \frac{1}{x_i^2+0.1}}-p_1{x}_i+p_{2}sin(p_i{y}_i))),\\ y_{i+1}=sin(\pi ({\displaystyle \frac{1}{y_i^2+0.1}}-p_1{y}_i+p_{2}sin(p_i{x}_i)))),\end{array}\end{array}$$

(4)

Similar to the 2D-LF and 2D-LS maps, the 2D-FS map also has two control parameters, $p_1$ and $p_2$, which can have large values. In this study, the values of $p_1$ and $p_2$$ϵ[1,100]$.

Figure 1 illustrates the phase space trajectories of the three maps, with initial values $x_0=0.2$ and $y_0=-0.7$. The control parameters for all three maps were set to $p_1=3$ and $p_2=8$.

2.4. Hardware Implementation of the 2D Maps

The hardware implementation of the 2D chaotic maps was conducted on an Artix-7 FPGA platform, using key computational components such as a Reconfigurable CORDIC [14], multipliers, adders, subtractors, and comparators. Each chaotic map was implemented through a series of computational stages designed to handle complex arithmetic efficiently. The architecture is detailed in Figure 2 for the three investigated maps, where each block performs specific operations essential to the chaotic map’s functionality. The main blocks include the computation of x-y terms ($4p_1{x}_i(1-x_i)-p_2{y}_i$), angular adjustments, multiplications, and sine function evaluations.

For the hardware architecture of the 2D-LF map shown in Figure 2a, the system begins by accepting two inputs, $x_0$ and $y_0$, which are routed through a multiplexer (MUX) controlled by selection signals $se{l}_{\theta x}$ and $se{l}_{\theta y}$. These signals determine the input stream directed to the CORDIC unit, which operates iteratively, refining intermediate results through multiple iterations. The MUX provides dynamic control over the input paths, ensuring that the system can be reconfigured as necessary to suit different operational contexts.

Once the CORDIC computes its output, the result is passed to a multiplier (MUL) block, where scaling is applied based on terms such as $Term{X}_{En}$ and $Term{Y}_{En}$. The $\theta$ block manages the angular parameters, while the $O_{i+1}$ block adjusts intermediate results based on the current iteration index. This iterative process progressively updates the $x_i$ and $y_i$ values, converging toward the final outputs $x_{i+1}$ and $y_{i+1}$. The controller oversees the operation, generating the necessary control signals and managing the synchronization of the CORDIC, multipliers, and other blocks.

Each block contributes to the practical realization of the chaotic maps as follows:

• x-y Term Computation Block (See Figure 3a): This block computes the terms $4p_1{x}_i(1-x_i)-p_2{y}_i$ and $4p_1{y}_i(1-y_i)-p_2{x}_i$, which are essential to the chaotic maps. It consists of three multipliers, two subtractors, a right shifter, a multiplexer, and a register.
• Angle Calculation Block (See Figure 3b): It calculates the angle for the sine transformation, keeping it within the critical bounds of $[-\pi ,\pi ]$. If the angle exceeds these bounds, the block adjusts it accordingly.
• Multiplication Block (See Figure 3c): This block adjusts the CORDIC output range to the correct bounds, ensuring that the result is within the range [0, 2].
• Next Iteration Block (See Figure 3b): After determining the angles, this block uses a reconfigurable CORDIC unit to compute the next state values $x_{i+1}$ and $y_{i+1}$, which are stored in registers for future iterations.

Figure 4 illustrates the phase space trajectories of the fixed-point hardware design compared with the floating-point MATLAB simulations for the three chaotic maps. Using a 36-bit representation, the fixed-point implementation shows strong agreement with the floating-point MATLAB results, demonstrating the accuracy of the hardware implementation.

Table 1. FPGA utilization for 2D chaotic maps.

Resources	LF Map	LS Map	FS Map
Lookup Tables (LUTs)	8663	8209	9033
Flip Flops (FFs)	425	360	539
Digital Signal Processors (DSPs)	28	28	24
Operating Frequency (MHz)	6.024	2.376	5.208
Power (w)	0.108	0.103	0.12
Throughput (Mbps)	54.216	21.384	31.248

presents the FPGA utilization for the hardware implementation of the three chaotic maps. The 2D-LS map consumes the least resources in terms of Lookup Tables (LUTs) and Flip Flops (FFs), while the 2D-LF map achieves the highest operating frequency. Although the 2D-FS map is more computationally demanding, it delivers higher throughput than the 2D-LS map.

3. The Proposed Encryption System

Figure 5 depicts the complete block diagram of the proposed audio encryption system. The process begins with an audio input divided into 16-bit data blocks. The proposed encryption system uses the 2D-LF chaotic map because it provides the best throughput and the highest operating frequency among the three maps, essential for minimizing delays in the encryption system.

The x value, generated by the 2D-LF map, determines the order in which the data blocks within the audio file are permuted. Specifically, the permutation is performed by shuffling the positions of the 16-bit block file according to the chaotic sequence generated by the x values. This step disrupts the linear structure of the original audio data, ensuring that even minor changes in the chaotic map’s initial parameters result in entirely different permutations, thus providing strong diffusion.

The permuted data undergo S-box substitution, where each 16-bit block is divided into four 4-bit segments, and each segment is substituted using one of the four predefined S-boxes, as shown in

Table 2. The used S-boxes in the proposed audio encryption.

S-Box 1				S-Box 2
1	8	12	6	2	11	15	5
3	4	5	9	0	7	6	10
0	15	10	7	3	12	9	4
14	13	11	2	13	14	8	1
S-Box 3				S-Box 4
5	12	8	2	7	14	10	0
7	0	1	13	5	2	3	15
4	11	14	3	6	9	12	1
10	9	15	6	8	11	13	4

. The selection of which S-box to use for each segment is not arbitrary; it is controlled by the chaotic map’s y values. Each S-box is applied to the corresponding four-bit segment. After S-box substitution, the transformed four-bit segments are scrambled using an XOR operation with the y values.

The encryption process concludes with recombining the 4-bit segments into 16-bit blocks to form the fully encrypted audio data. This structure allows for the efficient transmission or storage of the encrypted speech, while the chaotic map ensures that decryption without knowledge of the initial chaotic parameters is infeasible. Algorithm 1 details the entire encryption process.

Algorithm 1 Audio Data Encryption Algorithm

Require:  Audio file, initial parameters for the chaotic map    Chaotic Map Setup and Read Audio Data:    Initialize the chaotic map parameters.    Generate x and y random numbers using the chaotic map.    Scale x values to range from 0 to the length of the audio file (number of data blocks).    Scale y values to range from 0 to $2^{16}$ (16-bit values).    At the same time, load the audio data.    Convert the audio data into a binary matrix representation.    Divide the binary data into 16-bit blocks for encryption.    Permute Data Blocks:    Permute the 16-bit data blocks according to the chaotic map’s x values. The x values determine the new positions for the data blocks, effectively shuffling them.    S-BOX Substitution:    for each 16-bit data block do     Divide the block into four 4-bit segments.     for each 4-bit segment do      Each coresponding 4 bits go to the respective S-box.      Select the S-box based on the S-box index calculated from y (e.g., if $ymod4=0$, use S-Box 1; if $ymod4=1$, use S-Box 2, etc.).      Substitute the 4-bit segment using the selected S-box.     end for    end for    XOR Operation:    for each 4-bit segment (after S-Box substitution) do     Perform an XOR operation between the substituted 4-bit segment and the corresponding x value from the chaotic map.    end for    Combine Encrypted Data:    Recombine the XORed 4-bit segments into 16-bit blocks to form the final encrypted audio data. Ensure:  Output the fully encrypted audio data.

The decryption process is designed to be the inverse of the encryption process, ensuring that the original audio data can be accurately recovered. During decryption, the encrypted audio data are first divided into 16-bit blocks and then processed in the reverse order of the encryption steps. The S-box substitutions are applied in the opposite sequence, using the inverse of the used S-boxes. Following this, the permuted data blocks are rearranged according to the inverse permutation determined by the chaotic map’s x values. Finally, the 16-bit blocks are combined to reconstruct the original audio signal.

4. Results and Discussion

Two audio files from the TIMIT-TTS dataset were selected for testing [39]. The speech data used for evaluation were sampled at 16,000 Hz, ensuring sufficient detail. The performance of the proposed 2D chaotic map-based audio encryption system was evaluated through simulations conducted using Vivado, with further analysis performed in Matlab.

Table 3. Performance metrics of the proposed encryption system.

Performance Measure	Description and Equation
MSE	Measures the difference between the original and encrypted signals. Higher MSE values indicate more robust encryption. $\mathrm{MSE}={\displaystyle \frac{1}{U\times V}}{\sum }_{p=1}^U{\sum }_{q=1}^V{(Q(p,q)-E(p,q))}^2$ Here, $U\times V$: size of the signal or data block; $Q(p,q)$: original signal value; $E(p,q)$: encrypted signal value.
SNR	Measures the noise level in the encrypted signal. A highly negative SNR value indicates high noise compared to the original. $\mathrm{SNR}=10{log}_{10}\left({\displaystyle \frac{{\sum }_{p=1}^U{\sum }_{q=1}^{V}Q{(p,q)}^2}{{\sum }_{p=1}^U{\sum }_{q=1}^V{(Q(p,q)-E(p,q))}^2}}\right)$ Here, $Q(p,q)$: original signal value; $E(p,q)$: encrypted signal value.
Correlation Coefficient	Assesses the linear relationship between the original and encrypted signals. Lower correlation coefficient values indicate better encryption. $\rho ={\displaystyle \frac{\mathrm{cov}(a,b)}{\sqrt{K(a)K(b)}}}$ Here, $a,b$: original and encrypted signal values; Z: number of signal samples; $\mathrm{cov}(a,b)$: covariance between original and encrypted signals.
Histogram	Investigates the distribution of signal values. A balanced histogram indicates a more uniform distribution of amplitudes in the encrypted signal. Not applicable.
Spectrogram	Visualizes how frequencies change over time, offering insights into the energy distribution across different frequencies in the encrypted signal. Not applicable.

summarizes the performance metrics for the two speech files employed in the encryption process, including the mean squared error (MSE), signal-to-noise ratio (SNR), and correlation coefficients.

Table 4. Time waveforms, histograms, and spectrograms for audio test files.

	Time Waveform	Histogram	Spectrogram
speech 1
speech 2

presents the original speech signals’ time waveforms, histograms, and spectrograms. After applying both the S-box and XOR operations, the corresponding encrypted signals exhibit significant distortion, as shown in

Table 5. Time waveforms, histograms, spectrograms, and correlation data for encrypted and decrypted Speeches 1 and 2.

	Speech 1		Speech 2
	Encryption	Decryption	Encryption	Decryption
Time Waveform
Histogram
Spectrogram
Correlation

. The highly scrambled signals confirm the high levels of confusion and diffusion achieved by the encryption system.

The high MSE values of $1.2156\times {10}^9$ for Speech 1 indicate substantial differences between the original and encrypted signals, highlighting the encryption’s robustness. Similarly, the negative SNR values (e.g., −26.23 dB for Speech 2) suggest that the encrypted signals differ from the original speech, making it nearly impossible to recover the original without the correct decryption key. The low correlation coefficients (e.g., 0.002948 for Speech 2) demonstrate minimal similarity between the original and encrypted signals as shown in

Table 6. Encryption system quality metrics.

Speech	MSE	Correlation Coefficient	SNR
Speech 1	$1.2109\times {10}^9$	0.003229	−25.593280
Speech 2	$1.2156\times {10}^9$	0.002948	−26.231795

.

NIST randomness tests were conducted to further assess the encryption system’s security. As summarized in

Table 7. NIST test results.

Test	p-Value	Result	Proportion	Result
Frequency	0.066882	✓	0.938	✓
BlockFrequency	0.122325	✓	1.000	✓
CumulativeSums	0.051028	✓	0.938	✓
Runs	0.017912	✓	0.938	✓
LongestRun	0.000954	✓	1.000	✓
Rank	0.122325	✓	0.938	✓
FFT	0.017912	✓	1.000	✓
NonOverlappingTemplate	0.120018	✓	0.989	✓
OverlappingTemplate	0.534146	✓	1.000	✓
Universal	0.213309	✓	1.000	✓
ApproximateEntropy	0.002043	✓	1.000	✓
RandomExcursions	0.424511	✓	1.000	✓
RandomExcursionsVariant	0.317583	✓	1.000	✓
Serial	0.111904	✓	1.000	✓
LinearComplexity	0.008879	✓	1.000	✓

, the encrypted outputs passed all tests, including frequency, block frequency, cumulative sums, and approximate entropy.

The proposed encryption system was implemented on an Artix-7 FPGA.

Table 8. Encryption system hardware resource utilization.

Resources	LUTs	FFs	DSPs	Frequency (MHz)	Power (W)	Throughput (Mbps)
Utilization	14,528	440	36	180	0.13	2880

details the hardware resource utilization, showing that the system uses 14,528 LUTs, 440 Flip Flops (FFs), and 36 DSPs. The system achieves a high throughput of 2880 Mbps with a power consumption of 0.13 W, operating at a frequency of 180 MHz.

The proposed system offers significantly higher throughput and operating frequency than other systems, as shown in

Table 9. Encryption system hardware resource comparison.

Resources	This Paper	[37]	[40]
Lookup Tables (LUTs)	14,528	1400	18,451
Flip Flops (FFs)	440	144	193
DSPs	36	NA	36
Operating Frequency (MHz)	180	50	178

, although it uses slightly more LUTs than some alternatives. For instance, [37] used a dynamic S-box based on DNA sequences and a Lorenz chaotic generator, while [40] introduced a 3D chaotic system with a capsule-shaped equilibrium curve for masking speech signals.

5. Conclusions

This paper presented a 2D chaotic map-based audio encryption system using dynamic S-box substitution implemented on an FPGA platform. Three chaotic maps were explored: the logistic–fraction (2D-LF), logistic–sine (2D-LS), and fraction–sine (2D-FS) maps. Among these, the 2D-LF map was selected for use in the encryption system due to its superior performance on hardware, achieving a throughput of 2880 Mbps, operating at a frequency of 180 MHz, and consuming only 0.13 W. This map provided the best balance of computational efficiency and encryption security, making it ideal for minimizing delays and ensuring robust encryption of audio data. The system’s effectiveness was evaluated using Speech 1 and 2, with statistical analysis showing an MSE of $1.2109\times {10}^9$ and a correlation coefficient of 0.003229 for Speech 1, and an MSE of $1.2156\times {10}^9$ and a correlation coefficient of 0.002948 for Speech 2. Additionally, the system successfully passed the NIST randomness tests. For future work, higher-dimensional chaotic maps can be investigated. Additionally, more layers can be added to the encryption process, such as DNA encoding or hashing algorithms, such as SHA-1 and SHA-2, enhancing the system’s robustness.