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Based on the scientific identification and evaluation of scenic resources, the formulation of planning and management is of great significance to the sustainable development of tourism in national parks. This paper uses GF-2 high-resolution remote sensing images as the data source, and the Yesanpo National Park as the research area. It uses pixel-based MLC, NN, and SVM three classification methods to identify scenic resources, and adopts the system sampling method evaluation 3 methods to identify the classification accuracy of scenic resources, effectively improving the objectivity and accuracy of classification accuracy evaluation. The results show that the three classification methods used in GF-2 images meet the accuracy requirements of scenic resource identification, which is an effective method to identify scenic resources. Due to different geomorphic features, the erosion Zhanggu landform-Baili Xia scenic areas uses MLC classification, and granite fracture structure The canyon landform-LongmenTianguan scenic areas uses NN classification, and the karst cave spring landform-Yugu Dong scenic areas uses SVM classification with the highest overall accuracy. The results show that the proposed method can be applied to the identification of large-scale, high-resolution scenic resources, and provide more refined data support for scenic resource analysis and sustainable development of local tourism.

We present three greedy strategies for finding primitive polynomials of large degree over GF(2). Maximal length cellular automata (CAs) are used as tools to generate the primitive polynomials. The proposed strategies greedily synthesize (linear) CAs of different sizes, which are almost always of maximal length. Since the characteristic polynomial of a maximal length cellular automaton is primitive, characteristic polynomials of the synthesized CAs are claimed as primitive.

Combinatorial t-designs have nice applications in coding theory, finite geometries and engineering areas. t-designs can be constructed from image sets of a fixed size of some special polynomials. This paper constructs t-designs from the quadratic polynomial $$x^{4}+x^{3}$$ over $$\mathrm {GF}(2^{n})$$ and determine their parameters. We yield 2- $$\left( 2^n,3\cdot 2^{n-2},3\cdot 2^{n-2}\left( 3\cdot 2^{n-2}-1 \right) \right) $$ designs for n even and 3- $$\left( 2^n,2^{n-1},2^{n-1}\left( 2^{n-2}-1 \right) \right) $$ designs for n odd.

Binary extension finite fields ${\mathrm{ GF}}(2^{m})$ have received prominent attention in the literature due to their application in many modern public-key cryptosystems and error-correcting codes. In particular, the inversion over ${\mathrm{ GF}}(2^{m})$ is crucial for current and postquantum cryptographic applications. Schemes such as Fermat’s little theorem (FLT) and the Itoh–Tsujii algorithm (ITA) have been studied to achieve better performance; however, this arithmetic operation is a complex, expensive, and time-consuming task that may require thousands of gates, increasing its vulnerability chance to natural defects. In this work, we propose efficient hardware architectures based on cyclic redundancy check (CRC) as error detection schemes for state-of-the-art finite field inversion over ${\mathrm{ GF}}(2^{m})$ for a polynomial basis. To verify the derivations of the formulations, software implementations are performed. Likewise, hardware implementations of the original finite field inversions with the proposed error detection schemes are performed over Xilinx field-programmable gate array (FPGA) verifying that the proposed schemes achieve high error coverage with acceptable overhead.

In this article, we present the design of a high-performance simplified elliptic curve integrated encryption scheme (S-ECIES) cryptoprocessor. The cryptoprocessor was designed using a Montgomery ladder scalar multiplier, which was implemented with three finite field multipliers to improve the computational time of the scalar multiplication ${kP}$ , and using random curves and Gaussian normal bases over GF(2163) and GF(2233). Also, considering the National Institute of Standards and Technology (NIST) recommendations, a true random number generator is implemented to generate a secret key $k$ , which is used during the encryption process. The S-ECIES cryptoprocessor was synthesized on field-programmable gate array (FPGA) Stratix IV EP4SGX230KF40C2, simulated in ModelSim, and verified in hardware using the DE4 board and SignalTap tool. According to the synthesis results, the scalar multiplication operation is performed in 5.31 and $8.77~\mu \text{s}$ for GF(2163) and GF(2233), respectively. Also, the encryption process is performed in 20.70 and $30.90~\mu \text{s}$ for GF(2163) and GF(2233), respectively, and the decryption process is calculated in 8.10 and $11.9~\mu \text{s}$ for GF(2163) and GF(2233), respectively. The consumption power for the S-ECIES is 921 and 935 mW for GF(2163) and GF(2233), respectively.

Edge computing, as an emerging computing paradigm, aims to reduce network bandwidth transmission overhead while storing and processing data on edge nodes. However, the storage strategies required for edge nodes are different from those for existing data centers. Erasure code (EC) strategies have been applied in some decentralized storage systems to ensure the privacy and security of data storage. Product-matrix (PM) regenerating codes (RGCs) as a state-of-the-art EC family are designed to minimize the repair bandwidth overhead or minimize the storage overhead. Nevertheless, the high complexity of the PM framework contains more finite-domain multiplication operations than classical ECs, which heavily consumes computational resources at the edge nodes. In this paper, a theoretical derivation of each step of the PM minimum storage regeneration (PM-MSR) and PM minimum bandwidth regeneration (PM-MBR) codes is performed and the XOR complexity over finite fields is analyzed. On this basis, a new construct called product bitmatrix (PB) is designed to reduce the complexity of XOR operations in the PM framework, and two heuristics are used to further reduce the XOR numbers of the PB-MSR and PB-MBR codes, respectively. The evaluation results show that the PB construction significantly reduces the XOR number compared to the PM-MSR, PM-MBR, Reed–Solomon (RS), and Cauchy RS codes while retaining optimal performance and reliability.

Let G be isomorphic to GL n ⁢ ( q ) {\mathrm{GL}_{n}(q)} , SL n ⁢ ( q ) {\mathrm{SL}_{n}(q)} , PGL n ⁢ ( q ) {\mathrm{PGL}_{n}(q)} or PSL n ⁢ ( q ) {\mathrm{PSL}_{n}(q)} , where q = 2 a {q=2^{a}} . If t is an involution lying in a G-conjugacy class X, then, for arbitrary n, we show that, as q becomes large, the proportion of elements of X which have odd order product with t tends to 1. Furthermore, for n at most 4, we give formulae for the number of elements in X which have odd order product with t, in terms of q.

This article presents a compact and efficient bit-parallel systolic array structure for multiplication over the extended binary field, GF(2 $^{m}$ ). The systolic array has a regular arrangement with local connections, making it more suitable for VLSI implementations. Also, it has the merits of having hardware complexity of order $\mathcal{O}(m)$ that distinguishes it from the previously reported bit-parallel designs having hardware complexity of order $\mathcal{O}(m^{2})$ . The achieved results exhibited that the suggested parallel architecture realizes a significant reduction in hardware complexity and the area-delay complexity over the competitor architectures previously published in the literature. Therefore, it is more suitable for usage in constrained hardware environments, having more restrictions on space, such as portable devices and smart cards.

Surface water is the most important and common water resource on earth. Accurate and effective mapping and detecting of surface water have been made possible by remote sensing technology, highresolution satellite data, playing an important role in surface water monitoring and mapping, which has become the current hot research for water information extraction in recent decades. Therefore, in this paper, we tested and analysed four models to extract water bodies using China’s GF-2 HD satellite (GF- 2) image, including Normalized Difference Water Index (NDWI), Modified Shadow Water Index (MSWI), Support Vector Machine (SVM) and Object-Oriented Method (OOM). The results showed applying water extraction models can map surface water with an overall accuracy of 0.8935, 0.9256, 0.9467 and 0.9357, respectively. SVM owns the highest overall accuracy value of 0.9467, followed by OOM. SVM performed significantly better at surface water extraction with kappa coefficients improved by 9.00%, 5.00%, and 2.00%, respectively, which yielded the best results and used to map surfaces water bodies in the study region, while index methods (NDWI and MSWI) are mostly classified into the water and non-water information based on a threshold value, with higher total omission and commission errors at 12.45%, 25.64%, 6.38% and12.87%, respectively. Therefore, we proposed SVM as the best algorithm to identify water body and effectively detect surface water from the GF-2 image.

With the successful launch of China’s high spatial resolution satellite Gaofen-2 (GF-2), the use of high spatial resolution satellite images for land change detection has high research potential. B...

Scan chains provide increased controllability and observability for testing digital circuits. The increased testability, however, can also be a source of information leakage for sensitive designs. The state-of-the-art defenses to secure scan chains apply dynamic keys to pseudo-randomly invert the scan vectors. In this paper, we pinpoint an algebraic vulnerability of these dynamic defenses that involves creating and solving a system of linear equations over the finite field GF(2). In particular, we propose a novel GF(2)-based flush attack that breaks even the most rigorous version of state-of-the-art dynamic defenses. Our experimental results demonstrate that our attack recovers the key as long as 500 bits in less than 7 seconds. Our attack times are, on average, over 4300x faster than state-of-the-art SAT based attacks on the same defenses and circumvent any obfuscation on the combinational logic portions of the design. We then demonstrate how our attacks can be extended to scan chains compressed with Multiple- Input Signature Registers (MISRs).

Recently, moderate-resolution remote sensing images have been widely used in extracting coastal aquaculture areas that are usually used to obtain marine fishery resources. However, in moderate-reso...

Recently, new multiplication schemes over the binary extension field $GF(2^{m})$ based on an $n$ -term Karatsuba algorithm have been proposed for irreducible trinomials. In this paper, we extend these schemes for trinomials to any irreducible polynomials. We introduce some new types of pentanomials and propose multipliers for those pentanomials utilizing the extended schemes. We evaluate the rigorous space and time complexities of the proposed multipliers, and compare those with similar bit-parallel multipliers for pentanomials. As a main contribution, the best space complexities of our multipliers are $\frac {1}2m^{2}+O\left({m^{\frac {3}2}}\right)$ AND gates and $\frac {1}2m^{2}+O\left({m^{\frac {3}2}}\right)$ XOR gates, which nearly correspond to the best results for trinomials. Also, specific comparisons for three fields $GF(2^{163})$ , $GF(2^{283})$ , and $GF(2^{571})$ recommended by NIST show that the proposed multiplier has roughly 40% reduced space complexity compared to the fastest multipliers, while it costs a few more XOR gate delay. It is noticed that our space complexity gain is much greater than the time complexity loss. Moreover, the proposed multiplier has about 21% reduced space complexity than the best-known space efficient multipliers, while having the same time complexity. The results show that the proposed multipliers are the best space optimized multipliers.

A custom hardware coprocessor is described to improve the efficiency of applications using arithmetic in the Galois field GF(2) and the extension fields GF( $2^{m}$ ). The evaluation of such processor enhancements requires a variety of test exercises based on GF(2) arithmetic. Various test procedures are presented, which leverage the properties of finite fields to exercise the GF(2) arithmetic coprocessor incorporated with a 32-bit field-programmable gate array (FPGA)-based soft processor (Nios II). Up to 60 times speed improvement was achieved in typical calculations using the coprocessor. Resume —Un coprocesseur personnalise a ete decrit pour ameliorer l’efficacite des applications utilisant l’arithmetique en extension de Galois GF(2) et l’extension de corps GF(2m). L’evaluation des ameliorations d’un tel processeur necessite une variete d’exercices tests qui se basent sur l’arithmetique de GF(2). De nombreux tests de procedures ont ete presentes, ils exploitent les proprietes des corps finis afin d’exercer le coprocesseur arithmetique GF(2). Ce dernier a ete incorpore sur une matrice de portes programmables par l’utilisateur (FPGA) a 32 bits en se basant sur un processeur embarque (Nios II). La vitesse des calculs a ete amelioree jusqu’a 60 fois en utilisant le processeur.

We give an elementary proof that by multiplication on left and right by affine permutations A, B ∈ AGL(3, 2) each permutation π : GF(2)3 → GF(2)3 may be reduced to one of the 4 permutations for which the 3 × 3-matrices consisting of the coefficients of quadratic terms of coordinate functions have as an invariant the rank, which is either 3, or 2, or 1, or 0, respectively. For comparison, we evaluate the number of classes of affine equivalence by the Pólya enumerative theory.

Security in embedded systems are presented in this work based on Elliptic Curve crypto-processor (ECCP). Comparison with state-of-the-art algorithms that have been used to implement ECCP are also presented. It proves that Montgomery ladder algorithm is preferred one when the speed is needed. Optimization problem for general ECCP architecture are started from the selected algorithm and ends with liveness analysis and forward path. The inversion operation based on Itoh-Tsujii GF(2409) is discussed in this work. The proposed ECCP is implemented for GF(2163) and GF(2409) where the execution time are 8 μs and 61.2 μs respectively in sequentially design. The design and results are implemented using Xilinx ISE Virtex6.

Efficient implementation of finite field multipliers based on a reordered normal basis (RNB) is highly desirable in the current/emerging cryptosystems since it offers almost free realization of squaring operation. Therefore, in this paper, we propose novel bit-parallel and digit-serial finite field multipliers over $GF(2^{m})$ based on RNB. By efficient transformation of the core multiplication algorithm using a unique circular shifting feature, we have derived an efficient algorithm for low-complexity systolic mapping. Both bit-parallel and digit-serial structures of the multipliers are then obtained and optimized to enhance the area–time efficiency. We have also utilized the unique feature of the proposed multiplication algorithm to obtain the systolic multipliers by Karatsubalike decomposition. Detailed analysis and comparison show the superior performance of the proposed implementation. For example, the proposed regular and Karatsuba-based bit-parallel designs involve at least 48.4% less area-delay product (ADP) and 42.2% less power-delay product (PDP) than the best existing ones (37.7% and 55.3% less ADP and PDP on field-programmable gate array platform), respectively. The proposed multipliers, because of their lower area–time complexities, can be used for efficient realization of cryptographic applications.

Mangrove forests are important constitutions for sustainable development of coastal ecosystems, and they are often mapped and monitored with remote sensing approaches. Satellite images allow detail...

The newly launched GF-2 satellite is now the most advanced civil satellite in China to collect high spatial resolution remote sensing data. This study investigated the capability and strategy of GF-2 multispectral data for land use and land cover (LULC) classification in a region of the North China Plain. The pixel-based and object-based classifications using maximum likelihood (MLC) and support vector machine (SVM) classifiers were evaluated to determine the classification strategy that was suitable for GF-2 multispectral data. The validation results indicated that GF-2 multispectral data achieved satisfactory LULC classification performance, and object-based classification using the SVM classifier achieved the best classification accuracy with an overall classification accuracy of 94.33% and kappa coefficient of 0.911. Therefore, considering the LULC classification performance and data characteristics, GF-2 satellite data could serve as a valuable and reliable high-resolution data source for land surface monitoring. Future works should focus on improving LULC classification accuracy by exploring more classification features and exploring the potential applications of GF-2 data in related applications.

During the past decade, elliptic curve cryptography (ECC) has been widely deployed in different scenarios as the main asymmetric cryptosystem due to its smaller key length and relatively higher speed compared with other asymmetric cryptosystems. The most critical operation in ECC computation is point multiplication. In some popular applications such as signature verification schemes, the double point multiplication can be exploited. In this paper, we propose an algorithm and its corresponding architecture to speed up the double point multiplication using a modified binary differential addition chain. The proposed method is highly parallelizable and has been implemented on Virtex-4, Virtex-5, and Virtex-7 over ${{GF}}(2^{163})$ , ${{GF}}(2^{233})$ , and ${{GF}}(2^{283})$ , respectively. Experimental results using hardware implementation on Virtex-4 indicate that the proposed architecture achieves 63% and 16% improvements compared with the previous double point multiplication implementation in terms of required time and efficiency over ${ {GF}}(2^{233})$ , respectively. Additionally, the proposed architecture shows time reduction compared with twice the execution time of the best previous single point multiplication by 39% while achieving 258% higher efficiency. The proposed architecture has also been implemented on ASIC, and the results show that the proposed work improves time and energy consumption compared with the previous work.

This paper presents how the steps that used to determine a multiplicative inverse by method based on the Euclidean algorithm, can be used to find a greatest common divisor for polynomials in the Galois field (2^n ).

Accurate measurement of forest growing stem volume (GSV) is important for forest resource management and ecosystem dynamics monitoring. Optical remote sensing imagery has great application prospects in forest GSV estimation on regional and global scales as it is easily accessible, has a wide coverage, and mature technology. However, their application is limited by cloud coverage, data stripes, atmospheric effects, and satellite sensor errors. Combining multi-sensor data can reduce such limitations as it increases the data availability, but also causes the multi-dimensional problem that increases the difficulty of feature selection. In this study, GaoFen-2 (GF-2) and Sentinel-2 images were integrated, and feature variables and data scenarios were derived by a proposed adaptive feature variable combination optimization (AFCO) program for estimating the GSV of coniferous plantations. The AFCO algorithm was compared to four traditional feature variable selection methods, namely, random forest (RF), stepwise random forest (SRF), fast iterative feature selection method for k-nearest neighbors (KNN-FIFS), and the feature variable screening and combination optimization procedure based on the distance correlation coefficient and k-nearest neighbors (DC-FSCK). The comparison indicated that the AFCO program not only considered the combination effect of feature variables, but also optimized the selection of the first feature variable, error threshold, and selection of the estimation model. Furthermore, we selected feature variables from three datasets (GF-2, Sentinel-2, and the integrated data) following the AFCO and four other feature selection methods and used the k-nearest neighbors (KNN) and random forest regression (RFR) to estimate the GSV of coniferous plantations in northern China. The results indicated that the integrated data improved the GSV estimation accuracy of coniferous plantations, with relative root mean square errors (RMSErs) of 15.0% and 19.6%, which were lower than those of GF-2 and Sentinel-2 data, respectively. In particular, the texture feature variables derived from GF-2 red band image have a significant impact on GSV estimation performance of the integrated dataset. For most data scenarios, the AFCO algorithm gained more accurate GSV estimates, as the RMSErs were 30.0%, 23.7%, 17.7%, and 17.5% lower than those of RF, SRF, KNN-FIFS, and DC-FSCK, respectively. The GSV distribution map obtained by the AFCO method and RFR model matched the field observations well. This study provides some insight into the application of optical images, optimization of the feature variable combination, and modeling algorithm selection for estimating the GSV of coniferous plantations.

The higher number of iterations and longer pipeline delays are the major bottlenecks in the performance of Elliptic Curve Cryptography (ECC). Being the core of the arithmetic unit, the design of the high-performance multiplier is an important aspect that directly affects the design of the encryption circuit. In this paper, a full parallel LSB modular multiplication operator for pipelined Karatsuba-Offman Algorithm (KOA) structure is presented. Moreover, an n-term high-performance architecture for scalable modular multiplication operator is also proposed by combining the (b,2)way KA method, which is highly suitable for generalized polynomials. In comparison to existing designs, the proposed multiplier accomplishes lower complexity in time and space that leads to faster encryption with lesser area consumption. The proposed scalable multiplier achieves a 44.73% reduction in time, 36.40% savings in area consumption, and attains 43.98% efficiency in energy consumption.

Cryptography hardware processors enable fine-tuned designs supporting secure communications in extreme environments such as resource-constrained applications. Secure communications of resource-constrained wireless devices require efficient cryptographic arithmetic circuits such as finite field multipliers and inverters. Gaussian Normal Basis (GNB) provides cost-free squaring operation which is advantageous in repeated square-multiply cryptographic algorithms. In this paper, we realize the Itoh-Tsuji inverter based on the parallel-in-serial-out multiplier, for the first time to the best of our knowledge. We also report the design complexity of the GNB multipliers and inverters using the same technology and provide a comprehensive analysis of existing GNB multipliers and inverters in current literature. We conduct performance analysis of different multiplier architectures with varying digit sizes ( $d\geq 1$ ) and bit-serial ( $d=1$ ) inverter architectures. We also analyze the power variation of multiplier/inverter architectures for controlled input variations. The area, time complexities, and power analysis are reported using the standard 65nm CMOS and the open-source 45nm NangateOpenCellLibrary technology. The results show that the new inverter design has a competitive area and power consumptions compared to IT inverter based on Parallel-In-Parallel-Out multiplier when implemented using 65nm CMOS Standard cell library.

With the development of remote sensing technology and the differences in remote sensing image classification, it is particularly important to be able to accurately use classification methods to classify images and to compare classification algorithms. In this paper, taking Yangshuo County as the research area, five common supervised classifications, namely support vector machine (SVM), maximum likelihood classification (MLC), neural network (NN), spectral angle mapping (SAM) and spectral information divergence (SID), are used to classify the land cover of remote sensing image data of GF- 2、Landsat8 and its fusion in the same area. The classification results are obtained and compared. Moreover, the overall classification accuracy (OA) and Kappa coefficient are used to evaluate the performance of the image classification algorithm. The results show that both MLC and SVM perform best on these three data sets. For higher spatial resolution GF-2 and fusion data, the OA and Kappa coefficients of both image data classifiers is 10% higher than those of Landsat8 data with higher spectral resolution.

The quasi-circular vegetation patches (QVPs) are better objects for studying the ecosystem evolution, functioning, and maintenance in the Yellow River Delta, China. Remote sensing with linear change detection technique is an effective approach for mapping the vegetation dynamics. In this paper, the multitemporal Kauth-Thomas transformation (MKT) change detection technique with the decision tree classifier was used to map the QVPs based on the modified intensity-hue-saturation pansharpened April and August Gaofen-2 images. Results indicated that mapping the QVPs could be performed well using the approaches used in this paper. The precision, recall rate, and F-measure were 66.7%, 52.9%, and 59.0%, respectively. In the future, patch splitting techniques and more test areas should be tested for improving the detection accuracy of the QVPs. In addition, an assessment on the possibility of change in brightness and greenness between multitemporal images for mapping the dominant communities of the QVPs should be performed, which were important for the establishment, evolution, and disappearance of the QVPs.

To realize the elliptic curve cryptography (ECC) effectively, the design method of modular multiplication based on optimized binary finite filed algorithm was presented. By the study of the binary finite fields, paralleled modular multiplication algorithm and inversion algorithm which was based on Euclidean algorithm were presented. The two algorithms were optimized during the process and then realized the fast evaluation of point multiplication by adopting Montgomery algorithm. ECC hardware implementation design was proposed based on the algorithm, and converted to logic designs using Verilog RTL, and finally, it worked on the XC7A100T FPGA platform of Xilinx. By pre-simulation, synthetical verification, and analyzing the results of post-simulation, the clock frequency of the designed circuit could reach up to 110 MHz and the operating rate attained to 2.92 ms which demonstrated the feasibility and effectiveness of the project.

Through theoretical analysis, the calculation method of dual basis multiplication in GF (28) field based on CCSDS Berlekamp is given. Based on this calculation method, a VLSI architecture for parallel multiplication and serial operation in circuits is proposed. At the same time, the hardware resource occupation and the timing performance of each VLSI architecture are analyzed in detail.

In this paper, we propose a class of binary sequences induced by monomial permutation polynomials over \(\mathrm {GF}(2^{p})\) and study the period property and the shift-equivalence of these binary sequences. In particularly, we give a necessary and sufficient condition for such a sequence to have maximal period. Moreover, we also give a necessary and sufficient condition for two such sequences to be shift equivalent.

This paper proposes how to accomplish encryption and decryption in conic curves cryptosystem over finite field GF(\(2^n\)) using tile self-assembly. Two parameters in ciphertext could be obtained by two separate models of point-multiplication, one of which changes the seed configuration with inputs to adapt the demands. The decryption process is fulfilled by a new designed tile assembly model containing three sub-models that respectively perform the operation of point-multiplication, the operation of negative point and the operation of point-addition. Assembly time complexity of the model to decrypt is \(\varTheta (n^3)\) and the space complexity is \( \varTheta (n^6)\).

In this article, a new architecture of bit-serial polynomial basis (PB) multipliers over the binary extension field $GF(2^m)$ G F ( 2 m ) generated by irreducible trinomials is presented. Bit-serial $GF(2^m)$ G F ( 2 m ) PB multiplication offers a performance/area trade-off that is very useful in resource constrained applications. The architecture here proposed is based on LFSR ( Linear-Feedback Shift Register ) and can perform a multiplication in $m$ m clock cycles with a constant propagation delay of $T_{A} + T_{X}$ T A + T X . These values match the best time results found in the literature for bit-serial PB multipliers with a slight reduction of the space complexity. Furthermore, the proposed architecture can perform the multiplication of two operands for $t$ t different finite fields $GF(2^m)$ G F ( 2 m ) generated by $t$ t irreducible trinomials simultaneously in $m$ m clock cycles with the inclusion of $t(m-1)$ t ( m - 1 ) flipflops and $tm$ t m XOR gates.

Finite field GF(2\(^m\)) multipliers are employed in practical applications such as elliptic curve cryptography (ECC) and Reed-Solomon encoders. Digit-level finite field multipliers are best suitable for applications that require low-hardware implementation while operating at speeds that conform to today’s high data rates. With the emergence of Internet of Things (IoT), many resource-constrained devices such as IoT edge devices came into proliferate usage. To secure these constrained devices, ECC must be implemented in these devices with low-hardware complexity. Hence, it requires to design efficient digit-serial finite field multipliers since the performance of ECC greatly depends on the performance of the finite field multiplier employed. Many efficient designs for digit-serial finite field multipliers are presented in the literature to achieve better area and time complexities. In this paper, we present an area-efficient sequential digit-serial finite field multiplier for trinomials. The hardware and time complexities of the proposed multiplier are estimated for GF(2\(^{409}\)) and compared with the similar multipliers available in the literature. The comparison shows that the proposed multiplier achieves lower hardware complexity. Therefore, the proposed multiplier is attractive for cost-effective high-speed applications such as IoT edge devices.

Gaofen 2 (GF-2) high spatial resolution imagery has been recognized as an important data source for mapping vegetation pattern dynamics. The Kauth-Thomas (KT) indices of brightness, greenness, and wetness derived from single-date imagery has also been proved to be better than the common vegetation indices in monitoring the vegetation, mapping land desertification, and classifying land covers. However, for change detection of vegetation status, multitemporal Kauth-Thomas (MKT) transformation is found to be a more effective approach. In this paper, the parameters defining the KT dimensions for single-date mIHS pansharpeniing GF-2 data (acquired on April 30, 2016) were derived. Then, the MKT transformation matrix was proposed using a linear transformation process, which could be applied to GF-2 digital number image data for detecting vegetation pattern dynamics in the future.

At SODA 2017 Lokshtanov et al. presented the first worst-case algorithms with exponential speedup over exhaustive search for solving polynomial equation systems of degree d in n variables over finite fields. These algorithms were based on the polynomial method in circuit complexity which is a technique for proving circuit lower bounds that has recently been applied in algorithm design. Subsequent works further improved the asymptotic complexity of polynomial method-based algorithms for solving equations over the field \(\mathbb {F}_2\). However, the asymptotic complexity formulas of these algorithms hide significant low-order terms, and hence they outperform exhaustive search only for very large values of n.

There are various crypt-analytic techniques where solving a large dense or sparse system of linear equations over finite field becomes a challenge due to high computation. For instance, the problem like NFS for factorization of large integers, symmetric ciphers for crypt-analytic problem, discrete log problem, and algebraic attacks involves solving large sparse or dense linear systems over finite field. Here, we consider GF(2) finite field. Gaussian elimination is the popular and relevant method for solving large dense systems while Block Lanczos and Block Wiedemann algorithms are well known for solving large sparse systems. However, the time complexity of such popular method makes it reluctant and hence, the concept of parallelism is made compulsory for such methods. In addition, the availability of high end parallel processors and accelerators such as general-purpose graphics processing units (GPGPUs) solves computationally intensive problems in reasonable time. The accelerators with thousand of cores available today explore the bandwidth of memory and take advantage of multi-level parallelism on multi-node and multi-GPU units. Here, we consider Nvidia GPUs like Keplar, Pascal, and Volta along CUDA and MPI. Also, CUDA-aware MPI leverages GPU-Direct RDMA and P2P for inter- and intranode communication.

GF(2)-grammars are a recently introduced grammar family that has some unusual algebraic properties and is closely connected to the family of unambiguous grammars. By using the method of formal power series, we establish strong conditions that are necessary for subsets of \(a_1^* a_2^* \cdots a_k^*\) to be described by some GF(2)-grammar. By further applying the established results, we settle the long-standing open question of proving the inherent ambiguity of the language \(\{ \, a^n b^m c^\ell \mid n \ne m \text { or } m \ne \ell \, \}\), as well as give a new, purely algebraic, proof of the inherent ambiguity of the language \(\{ \, a^n b^m c^\ell \mid n = m \text { or } m = \ell \, \}\).

This paper presents a low complexity lossy compression algorithm with protograph LDPC codes (Lossy P-LDPC Codes) over GF(2), for binary symmetric sources. The proposed scheme takes a shuffled reinforced belief propagation (SRBP) algorithm for encoding, which achieves asymptotically the theoretical rate-distortion function. Additionally, the proposed coding algorithm converges fast and performs very close to the counterparts using ultra sparse LDPC codes over GF(256). Simulation results also reveal that the code structures need to be further optimized, in order to adapt to different source statistics.