2.1. Various Modulation Formats

Modulation techniques like OOK with Non-Return-to-Zero, Return-to-Zero formats, Pulse Position Modulation, and Differential Pulse Interval Modulation were employed in this work. These modulation techniques offer energy efficiency, bandwidth efficiency, and transmission reliability with ease of implementation. The data train of symbols 8 and 1 for all four modulation schemes mentioned is shown in Figure 2.

On-Off Keying: OOK is the most basic modulation technique where the laser is ‘on’ when the bit is ‘1’ and the laser is off when the bit is ‘0’. OOK can be implemented using two main types of pulse formats: Non-Return-to-Zero (NRZ) and Return-to-Zero (RZ) [44]. In NRZ, the pulse duration is the complete bit period (T_b), whereas in RZ, the pulse duration is half of the bit period. i.e., Slot duration,

i.e., Slot duration,

Differential Pulse Interval Modulation: DPIM is both power efficient and bandwidth efficient as the redundant space is not present, i.e., the clock restarts every time a pulse is received, and the empty slots following it are removed. In PIM, the information is present in the number of empty slots between two pulses. A guard band can also be added right after the pulse to efficiently identify the slots; this is called Differential Pulse Interval Modulation 1 Guard Slot (DPIM 1GS), and if there is no guard band present, then it is called Differential Pulse Interval Modulation No Guard Slot (DPIM NGS) [46]. Since the symbol duration in DPIM is variable, the slot duration is chosen such that the mean symbol duration is equal to the time taken to send the same number of bits using a fixed frame length, like OOK and PPM [47].

i.e., Slot duration,

2.2. Forward Error Correction (FEC) Codes

Comparing these formats along with the encoding techniques will provide a comparative feasibility and link performance analysis of a UOWC system [48,49,50]. We have employed several forward error correction codes for which a brief explanation is presented below [51].

Repeat Codes: Repeat codes, such as Repeat (3) and Repeat (5), ensure data integrity in communication systems by transmitting each bit multiple times. For example, it is three times in Repeat (3), five times in Repeat (5), and n times in Repeat (n), the higher the repetitions more secure the data will be. These simple error-correction methods are effective in low-data-rate applications and work across all formats.

In Repeat (5), the bits ‘1101’ will be encoded as ‘11111111110000011111’. If the receiver receives a different bit sequence instead, i.e., one or two bits are flipped, leading to ‘11111101110010011111’, the decoder will take the most likely bit, i.e., the bit which is repeated a greater number of times in the 5-bit duration, leading to the decoded data as ‘1101’ which was the originally transmitted data sequence. A similar logic is applied in repeat (3) and other repeat codes.

The Hamming Code: This ensures the correction of single-bit errors by adding parity bits to the data block. Their efficiency makes them ideal for scenarios with moderate noise interference. Each block of four data bits in Hamming code is converted into a block of seven bits by adding three parity bits to the block (see Algorithm 1). The purpose of these parity bits is to detect and correct transmission errors. Parity bits are added at 2^k (k=0, 1, 2, 3…n) and are obtained by performing XOR operations on the data bits present in the remaining positions; the same is done in the receiver to decode and obtain the original data.

Algorithm 1. Hamming code parity bit placement

The parity bits are calculated as follows:

P1: Bit positions 3, 5, 7 -> P1 = 0

P2: Bit positions 3, 6, 7 -> P2 = 1

P3: Bit positions 5, 6, 7 -> P3 = 0

BCH Code: BCH code is based on Galois field theory. BCH codes correct multiple random errors, making them well-suited for high bit-error-rate environments. The number of bits that a code can correct depends on the size of the Galois Field used to perform arithmetic operations within the code. Choosing a generator polynomial—a polynomial that produces a set of code words that satisfy specific error-correction properties—is a step in the development of BCH codes. The generator polynomial is built by choosing a basic Galois field component and calculating its powers. The generator polynomial is then created by combining these powers. The data is encoded using the generator polynomial, which is derived through polynomial division.

The encoded data consists of the original data and the residual bits from the division, which are a set of parity bits. The number of parity bits depends on the generating polynomial’s degree. Data corruption may result from transmission errors. The receiver can detect errors in the received data by calculating the syndrome. The syndrome is computed by multiplying the supplied data by the generating polynomial. If the remainder is 0, the data is error-free. If the remainder is not zero, the syndrome identifies the location of the data errors. To fix the errors, the receiver uses the error position and syndrome to solve a sequence of linear equations. These equations can be solved using matrix algebra, and the solution provides the correct values for the missing bits.

Reed-Solomon Code: RS codes encode data in multi-bit symbols, effectively handling burst errors in fluctuating water conditions. While computationally intensive, they are crucial for maintaining high data integrity in deep-sea applications. Before transmission, redundant symbols are added to the original message; these redundant symbols are calculated utilizing finite field-based mathematical techniques. The number of redundant symbols used depends on the desired level of error correction and the design of the code.

By selecting a finite field and defining a generator polynomial, a Reed-Solomon code is constructed. With ‘n’ representing the number of redundant symbols, the generator polynomial has the degree ‘n’. The generator polynomial is used to generate a set of code words that satisfy particular error-correction requirements. Throughout the encoding process, the message is separated into blocks of ‘k’ symbols, where ‘k’ is the total number of information symbols. The encoder uses the generating polynomial to calculate ‘n’ redundant symbols, which are then appended to the original message to create a code word of length ‘n+k’. Once the issues have been resolved, the receiver can remove the extra symbols to restore the original message.

A flowchart illustrating the encoding algorithms is provided below in Figure 3 and also included in the supplementary information (Figures S1–S8).

Table 1 shows a comparative summary of the above FEC codes as below.

2.3. Rivest-Shamir-Adleman (RSA) Encryption

In addition to its speed, affordability, and limitless bandwidth, optical wireless communication offers excellent data security because of its Line of Sight (LOS) property. However, using an encryption technique will further solidify the protection. RSA is an asymmetric encryption technique; RSA uses a pair of keys: a public key for encryption and a private key for decryption. The encryption strength of RSA lies in the difficulty of factoring large prime numbers, making unauthorized decryption nearly impossible without the private key. This provides a robust security measure for UOWC systems, particularly useful for high-stakes applications such as military or scientific data transfer, where protection from interception is paramount. RSA’s computational demands are a trade-off for its high level of security, making it an optimal choice in applications where secure communication outweighs the need for rapid data processing. The working of the RSA algorithm is represented in Figure 4.

The proof for RSA can be obtained by using Fermat’s Little Theorem [36,52], to prove we have to show that the decryption function is the inverse of the encryption function. Fermat’s Little Theorem states,

Let us apply the decryption function to the encryption function,

The equation,

Let us consider two equations,

1. $m\equiv 0\left(modp\right)$

   Then, $p|m$ or, $m=\alpha p$ for any $\alpha \in {\mathbb{Z}}^{+}$ ,

   $\therefore {\left(m^e\right)}^d={\left(p\alpha \right)}^{de}=p·p^{ed-1}·{\alpha }^{de}$ or, $p|m^de$ or, ${\left(m^e\right)}^d\equiv m\left(modp\right)$

2. $m\not\equiv 0\left(modp\right)\Rightarrow p\nmid m$

   According to Fermat’s Little Theorem: $m^{ed-1}\equiv 1\left(modp\right)$

   Now, $ed-1-k(p-1)(q-1)$ for any integer k

   $\therefore m^{de}=m^{ed-1}·m=m^{k\left(p-1\right)\left(q-1\right)}·m\equiv {\left(m^{p-1}\right)}^{q-1·k}$

   Or, $m^{de}\equiv {\left(1\right)}^{q-1·k}·m\equiv m\left(modp\right)$

Using similar logic as shown above, it is possible to prove Equation (13) as well.

If for two distinct prime numbers p and q, i.e., $\gcd \left(p,q\right)=1$

Then we can conjecture that $a\equiv c\left(modpq\right)$

So, from Equations (12) and (13), which have been proved, we can obtain

That is the required equation to prove RSA encryption.

Together, these modulation schemes, Error Correction codes, and encryption methods form a comprehensive framework to enhance the effectiveness, reliability, and security of underwater optical communication systems.

4.1. RSA Encryption Analysis

As explained in Section 2, a public key and a private key are generated using combinations of prime numbers, and the data is transmitted in encrypted form. The received data will be decrypted only with the combination of these two keys. Figure 7 shows a graph illustrating the increasing strength of encryption, represented by the public and private keys, as the values of the prime numbers p and q increase. It can be noticed that there is an increasing trend in both; however, the combination of keys is unpredictable, making the decryption difficult, particularly at higher values of p, q. Table S1 listing the prime numbers used along with their corresponding public and private keys has been included in the supplementary information.

4.2. RSA Encryption and Performance Analysis of UWOC with Different Modulation Schemes

We have developed RSA encryption and applied it across various modulation schemes, i.e., OOK-NRZ, RZ, PPM, and DPIM. To verify the impact of RSA encryption on different modulation schemes, we have estimated the practical execution time for different sequential combinations of p, q. This results in different public keys (n) and private keys (d). Figure 8a shows the execution time versus the public key from 143 (11, 13) to 186,623 (431, 433), up to 20 data points. This experiment was performed with a data rate of 19,200 kbps. A similar experiment was performed to check the effective implementation of RSA at various speeds, and as shown in Figure 8b, the values chosen for p, q are 47, 53 for a public key value of 2491. The two experiments show that the systematic variation suggests the robustness of the RSA implementation.

After that, we compared the originally sent data and the received data from the UOWC system to estimate the error fraction without RSA encryption and with RSA encryption. This comparative study provided information on communication reliability. Figure 9a presents the error fraction versus data rate without RSA encryption applied, and Figure 9b shows the error fraction versus different modulation schemes with RSA encryption. In both cases, there is an increase in the error fraction at higher data rates. RSA encryption highlights the additional challenges introduced by encryption, due to processing overhead and synchronization issues. Together, these subfigures demonstrate the trade-offs between encryption strength and transmission reliability. This can be overcome by implementing Ethernet protocols.

4.3. Integration of Forward Error Correction codes with RSA Encryption

Despite offering higher security to the data, RSA encryption is prone to a higher error fraction. One solution to address this issue is to implement mitigation techniques such as error correction codes alongside modulation schemes to optimize or minimize errors. To investigate the same, we have employed five forward error correction codes like Repeat Codes, Hamming Codes, BCH Codes, and Reed-Solomon Codes along with encryption and modulation techniques. We choose a data rate of 19.2 kbps and p, q as 47, 53 to conduct initial experiments. Figure 10 shows the initial results of Forward Error Correction with RSA encryption across various modulation schemes. In Figure 10a, the execution time is plotted against different FEC codes, while Figure 10b illustrates the increase in data size resulting from the implementation of FEC. The findings indicate that more complex FEC codes, like Reed-Solomon and BCH, significantly increase execution times due to their computational demands, especially when paired with encryption.

Figure 10b shows the execution size with various error correction codes along with RSA encryption. There is an increase in the file size of the data with the error correction codes. It is more suitable for PPM modulation and represents a mere increase over OOK schemes, whereas the introduction of differential schemes like DPIM has further reduced the size, offering an advantage over other pulse modulation schemes.

After proving the basic RSA encryption with modulation and encoding formats, a systematic study was conducted to investigate the effect of encryption strength. For the same, we used a combination of “p, q” to generate various public keys from 143 (11, 13) to 186,623 (431, 433) up to 20 data points and estimated the execution time for all error correction codes for all modulation separately.

An in-depth analysis of the execution time required for RSA encryption when combined with different Forward Error Correction codes across various modulation schemes is provided in Figure 11. The graphs illustrate the relationship between the size of the RSA public key and execution time for various modulation schemes like OOK-NRZ, OOK-RZ, PPM, and DPIM. In the NRZ case, the execution time consistently increases with larger public key sizes. This trend shows the complexity of FEC codes like Reed-Solomon compared to simpler ones, such as Repeat Codes. Similarly, RZ modulation shows a steady rise in execution time, with its synchronization advantages somewhat mitigating the overhead introduced by FEC codes. However, more complex codes like BCH still lead to significant delays. In the case of PPM, which offers better noise immunity, the execution time remains relatively manageable, although the impact of robust FEC codes becomes more evident as public key sizes increase. Lastly, DPIM, known for its lower power requirements, also demonstrates consistent performance, balancing efficiency and error correction overhead effectively. Across all modulation schemes, the trade-offs between encryption strength, error correction, and computational efficiency are clear, emphasizing the importance of optimizing these factors for secure and reliable underwater optical communication.

4.4. Performance Analysis of UWOC System with Forward Error Correction Codes and RSA Encryption

After the successful implementation of the various FECs, modulation schemes, and RSA encryption combined, we have saved the data and measured the error fraction in the received file compared to the original sent file. This will give us a clear idea about which modulation scheme, with which error correction, will reduce the error, and where it will increase the error. For this, we have again used a data rate of 19.2 kbps and p, q as 47, 53. The results found show interesting trends and are presented in Figure 12. OOK-NRZ modulation has shown a reduction in error fraction for almost all error correction codes. OOK-RZ has also shown a reduction in error fraction for all FECs except the repeat code 5. The PPM modulation scheme did not contribute to any reduction in error; instead, it increased the error, possibly due to bandwidth inefficiency and strong channel-induced attenuation. DPIM showed minimal impact from FEC implementation.

Analysis of the link performance of various Forward Error Correction codes combined with RSA encryption across different modulation schemes is shown in Figure 12. The findings indicate that more robust FEC codes, such as Reed-Solomon and Bose-Chaudhuri-Hocquenghem, result in significantly higher execution times due to their complex algorithms, while simpler codes like Repeat Codes exhibit lower execution times but reduced error correction capability. The error correction effectiveness of these codes reveals that advanced FEC techniques provide flexibility against underwater signal disruptions, ensuring reliability even in adverse conditions, but with the additional load of increased computational overhead due to the larger data sizes due from added parity bits and encryption payloads. Modulation schemes with inherent noise immunity, such as PPM, perform better in terms of both execution time and error correction compared to conventional schemes. The results indicate that the choice of advanced modulation schemes is a good option for the UOWC system.