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A six-degree-of-freedom measurement system possessing a pure algebraic and error-free calculation algorithm is proposed based on the rotation/translation matrix decomposition approach. Generally, such complicated mathematical models are simplified through linearization. However, this inevitably leads to calculation errors. Accordingly, in the method proposed in this study, the object motion behavior in three-dimensional space is described instead using a pose-change homogeneous coordinate transformation matrix consisting of two separate matrices, namely a rotation matrix which describes the nonlinear component of the object motion and a translation matrix which describes the linear component. In the solution process, the image-orientation-change method is first used to determine the rotation matrix. The translation matrix is then obtained using the singular-value-decomposition least-squares method. Finally, the two matrices are multiplied together to obtain the desired pose-change matrix. The validity of the proposed approach is demonstrated by means of an illustrative numerical example and a simple experimental trial. It is shown that, since the calculation algorithm is a purely analytical and algebraic method and involves no approximations or omissions (i.e., the relevant equations are not simplified through linearization and thus retain their original nonlinear integrity), the derived solutions are error-free if the errors introduced by the measurement technology are ignored.

An online template attack (OTA) is a powerful technique previously used to attack elliptic curve scalar multiplication algorithms. This attack has only been analyzed in the realm of power consumption and EM side channels, where the signals leak related to the value being processed. However, microarchitecture signals have no such feature, invalidating some assumptions from previous OTA works. In this paper, we revisit previous OTA descriptions, proposing a generic framework and evaluation metrics for any side-channel signal. Our analysis reveals OTA features not previously considered, increasing its application scenarios and requiring a fresh countermeasure analysis to prevent it. In this regard, we demonstrate that OTAs can work in the backward direction, allowing to mount an augmented projective coordinates attack with respect to the proposal by Naccache, Smart and Stern (Eurocrypt 2004). This demonstrates that randomizing the initial targeted algorithm state does not prevent the attack as believed in previous works. We analyze three libraries libgcrypt, mbedTLS, and wolfSSL using two microarchitecture side channels. For the libgcrypt case, we target its EdDSA implementation using Curve25519 twist curve. We obtain similar results for mbedTLS and wolfSSL with curve secp256r1. For each library, we execute extensive attack instances that are able to recover the complete scalar in all cases using a single trace. This work demonstrates that microarchitecture online template attacks are also very powerful in this scenario, recovering secret information without knowing a leakage model. This highlights the importance of developing secure-by-default implementations, instead of fix-on-demand ones. publishedVersion

The Montgomery kP algorithm i.e. the Montgomery ladder is reported in literature as resistant against simple SCA due to the fact that the processing of each key bit value of the scalar k is done using the same sequence of operations. We implemented the Montgomery kP algorithm using Lopez-Dahab projective coordinates for the NIST elliptic curve B-233. We instantiated the same VHDL code for a wide range of clock frequencies for the same target FPGA and using the same compiler options. We measured electromagnetic traces of the kP executions using the same input data, i.e. scalar k and elliptic curve point P, and measurement setup. Additionally, we synthesized the same VHDL code for two IHP CMOS technologies, for a broad spectrum of frequencies. We simulated the power consumption of each synthesized design during an execution of the kP operation, always using the same scalar k and elliptic curve point P as inputs. Our experiments clearly show that the success of simple electromagnetic analysis attacks against FPGA implementations as well as the one of simple power analysis attacks against synthesized ASIC designs depends on the target frequency for which the design was implemented and at which it is executed significantly. In our experiments the scalar k was successfully revealed via simple visual inspection of the electromagnetic traces of the FPGA for frequencies from 40 to 100 MHz when standard compile options were used as well as from 50 MHz up to 240 MHz when performance optimizing compile options were used. We obtained similar results attacking the power traces simulated for the ASIC. Despite the significant differences of the here investigated technologies the designs’ resistance against the attacks performed is similar: only a few points in the traces represent strong leakage sources allowing to reveal the key at very low and very high frequencies. For the “middle” frequencies the number of points which allow to successfully reveal the key increases when increasing the frequency.

Machine Learning techniques have proven effective in Side Channel Analysis (SCA), enabling multiple improvements over the already-established profiling process of Template Attacks. Focusing on the need to mitigate their impact on embedded devices, a design model and strategy is proposed that can effectively be used as a backbone for introducing SCA countermeasures on Elliptic Curve Cryptography (ECC) scalar multipliers. The proposed design strategy is based on the decomposition of the round calculations of the Montgomery Power Ladder (MPL) algorithm and the Scalar Multiplication (SM) algorithm into the underlined finite field operations, and their restructuring into parallel-processed operation sets. Having as a basis the proposed design strategy, we showcase how advanced SCA countermeasures can be easily introduced, focusing on randomizing the projective coordinates of the MPL round’s ECC point results. To evaluate the design approach and its SCA countermeasures, several simple ML-based SCAs are performed, and an attack roadmap is provided. The proposed roadmap assumes attackers that do not have access to a huge number of leakage traces, and that have limited resources with which to mount Deep Learning attacks. The trained models’ performance reveals a high level of resistance against ML-based SCAs when including SCA countermeasures in the proposed design strategy.

In order to quickly and conveniently identify the synchronous motion errors of dual-five-axis CNC machine tool, this paper proposes an effective error measurement and identification method, which consists of an error identification model and a new error measurement trajectory for the dual-five-axis CNC machine tool based on R-test measuring instrument. According to the theory of multi-body system and the method of homogeneous coordinate transformation, this paper developed the kinematic model of machine tool to identify the synchronous motion error of machine tool and a new error measuring track of spherical “8” shape. Thus, three kinds of synchronous motion errors of A1 / A2 axis, A1 / B2 axis, and A2 / B2 / C1 axis can be identified respectively. In addition, a series of milling experiment have been performed on the dual-five-axis CNC machine tool to verify the feasibility and effectiveness of the propsoed model. The experimental results show that the proposed model and method are more efficient.

We present an efficient implementation of the supersingular isogeny Diffie-Hellman (SIDH) key exchange protocol on 64-bit ARMv8 processors for 125- and 160-bit post-quantum security levels. We analyze the use of both affine and projective SIDH formulas and provide a comprehensive analysis of both approaches based on the inversion-to-multiplication ratio. Implementation results show that regardless of security concerns, affine SIDH is competitive with the projective coordinates implementation, and even outperforms projective implementation in the final round of SIDH; however, projective SIDH shows better overall performance for the whole key exchange protocol. Notably, over larger finite fields, using optimized field multiplication leads to the much better performance of projective compared to affine formulas. We integrate our optimized software into the open quantum-safe OpenSSL library and compare our software with other available post-quantum primitives. The benchmark results on ARMv8 demonstrate speedup of up to 5X over the generic version of SIDH implementation which is available inside the OQS library for the same quantum security level. We observe that our highly-optimized implementation still suffers from a large number of operations for computing isogenies of elliptic curves. However, in terms of communication overhead, supersingular isogeny-based cryptosystem provides significantly smaller key size compared to its counterparts.

Strict requirements for concentricity of the multistage high pressure rotor of an aero-engine are employed to guarantee performances such as vibration. Tedious and time-wasting trial assembly by adjusting the installation angles of stages is needed to meet the requirements due to the lack of effective analysis methods. Furthermore, there is no quick way to find out where the problem is and how to repair the parts when the installation-angle-adjusting method fails. This article focuses on a solution to optimize the installation angle of each stage and to make repair decisions in the assembly process. The run-out data are processed by least square method to get the spatial positions and attitudes of flanges and a deviation propagation analysis model is built by virtue of homogeneous coordinate transformation theory to predict the accumulative errors of each stage. The eccentricities of stages are evaluated with reference to the common axis and the installation angles of stages are optimized by minimizing the sum of eccentricities. Sensitivities of eccentricity, eccentric angle and parallelism of each stage are analyzed and repair decisions for parts are made to meet more strict requirements. An example of a three-stage subassembly is presented to demonstrate the solution.

The public key cryptographic algorithms considered unsuitable for Internet of Things due to use of longer keys, stressful calculations and consumption of battery resources for 16bit processers. This paper presents a novel algorithm based on elliptical curve for securing communication between internet of things by use of single coordinate system of Elliptical curve cryptography. Though lengdre symbol or single coordinate system exist in mathematics literature, it’s use for securing IoT devices is fairly new. In the proposed system IoT device transfer one coordinate of public key with extra bit allowing to save 50 % of bandwidth and storage for IoT devices.

Positioning accuracy is crucial to servo system, therefore, study error identification and compensation methods is significant. Composed of a linear and a rotary axis, the CNC servo positioning platform studied, and kinematics modeling carried out on the DH (Denavit-Hartenberg homogeneous coordinate model) method. Supposing that modeling errors change with driving instructions, in order to obtain the relationship among them, identifying the actual axis equations of the linear and rotating shaft, then the state matrix of the actual coordinated system of each link was re-established. Based on this, the relationship curves between errors of each matrix and driving constructions fitted, the ideal homogeneous coordinate transformation matrix compensated to realize the positioning of the CNC servo platform under different driving instructions. Further, the simulation for compensation strategy carried out, which verify that the results are consistent with the theoretical assumptions and positioning error can be compensated by the compensation method.

In the traditional cultural relic image digital repair system, there are obvious color aberrations and structural deviations between the restored part and the surrounding part, which weakens the visual connectivity of the restored result. In view of this, this research designs a digital repair system of damaged cultural relics image in professional training of restoration. In the hardware of the system, an integrated fast spherical camera is designed, and the transformation between image and pixel coordinate system is expressed by homogeneous coordinate system according to its parameters. In the system software, the damaged areas of cultural relics images are segmented and marked first, and the optimization algorithm of texture synthesis and restoration based on samples is designed. The experimental test results show that the system in this paper can ensure the use of the known information of the original image to the greatest extent, and make the repair result more reasonable on the basis of the minimum modification of the original image, so as to achieve the effect of professional practical training of repair.

In the current situation of the development of science and technology, virtual technology arises at the historic moment. As a kind of computer simulation technology, it has the simulation ahead of the real world. Because of its many advantages, virtual technology is already being used in fields such as education and medicine. However, there are still some shortcomings in the current virtual technology. Although the development tools are equipped, there are still some disadvantages in the process of using them, which hinders the developers to improve their work efficiency significantly. Among the prospects, this technology is the most widely used in education, and this paper aims to develop a virtual reality application engine that is convenient for party school teaching. In the education and learning of the party school, the design and r&d of VR products should be completed under the support of the model and theory of education, and should respond to the characteristics of the discipline. This paper expounds a series of processes from design to r&d of VR products. At the beginning, I analyses the subject and investigated the market prospect of VR education, so as to get a product with the same learning efficiency and teaching efficiency. Secondly, the depth plane analysis is carried out on the mathematical knowledge needed in the construction of 3d model, and the emphasis is put on the introduction of quaternions, transformation matrix, homogeneous coordinates and the functions of these knowledge in 3d application.

At EUROCRYPT 2004, Naccache et al. showed that the projective coordinates representation of the resulting point of an elliptic curve scalar multiplication potentially allows to recover some bits of the scalar. However, this attack has received little attention by the scientific community, and the status of deployed mitigations to prevent it in widely adopted cryptography libraries is unknown. In this paper, we aim to fill this gap, by analyzing several cryptography libraries in this context. To demonstrate the applicability of the attack, we use a side-channel attack to exploit this vulnerability within libgcrypt in the context of ECDSA. To the best of our knowledge, this is the first practical attack instance. It targets the insecure binary extended Euclidean algorithm implementation using a microarchitectural side-channel attack that allows recovering the projective representation of the output point of scalar multiplication during ECDSA signature generation. We captured 100k traces to estimate the number of traces an attacker would need to compromise the libgcrypt ECDSA implementation, resulting in less than 2k for commonly used elliptic curve secp256r1, demonstrating the attack feasibility. During exploitation, we found two additional vulnerabilities. However, we remark the purpose of this paper is not merely exploiting a library but about providing an analysis on the projective coordinates vulnerability status in widely deployed open-source libraries, filling a gap between its original description in the academic literature and the adoption of countermeasures to thwart it in real-world applications., IACR Transactions on Cryptographic Hardware and Embedded Systems, Volume 2020, Issue 3

As the payload of the reconnaissance aircraft, the positioning accuracy of the airborne photoelectric platform is the key to whether the weapon can achieve accurate strike and whether the battlefield information can be obtained accurately. This paper introduces the structure of the photoelectric measurement pod, establishes the auxiliary coordinate system of the positioning model. According to the homogeneous coordinate transformation method, rotates and transforms the parallax vector of two machines into the unified coordinate system, and completes the solution of the target according to the principle of space intersection. Combined with the disadvantages of traditional rendezvous positioning algorithm, an improved algorithm is proposed. Based on the model, the error factors affecting the positioning accuracy are determined, which provides theoretical guidance for engineering application. The simulation results show that the improved algorithm is better than the traditional algorithm.

A photoelectric detection system is a typical type of device widely used for detecting purposes based on unmanned aerial vehicles (UAV). Stability accuracy is the key performance index. Compared to traditional analysis methods aimed at unpredictable error-causing sources, assembly errors can be easily controlled during the manufacturing processes. In this research, an analysis method of assembly error effect on stability accuracy is proposed. First, by using kinematics analysis of homogeneous coordinate transformation, stability accuracy is comprehensively modeled and simulated. Then, by analyzing the manufacturing process, assembly errors of axis perpendicularities, run-outs and gyroscopes are defined and modeled. By simulating different carrier movements, the effects caused by assembly errors under various environments are studied. Finally, error sensitivity is proposed by using standard deviation analysis. Results show that the most sensitive assembly errors are identified, and ranked in order of sensitivity as follows: x-component of pitch axis perpendicularity, y-component of the azimuth gyroscope assembly, and z-component of the pitch gyroscope assembly. In conclusion, the results can be used as standards of manufacturing process improvements, and the proposed methods can be used to provide valuable references for real application scenarios.

The Montgomery kP algorithm is reported in literature as resistant against simple SCA due to the fact that the processing of each key bit value of the scalar k is done using the same sequence of operations. We implemented the Montgomery kP algorithm using Lopez-Dahab projective coordinates for the NIST elliptic curve B-233. We instantiated the same VHDL code for wide range of clock frequencies for the same target FPGA and using the same compiler options. We measured electromagnetic traces of the kP executions using the same input data, i.e. scalar k and elliptic curve point P, and measurement setup. Our experiments clearly show that the success of simple electromagnetic analysis depends on the target frequency for which the design was implemented and at which it is executed significantly. In our experiments the used scalar k was successfully revealed via simple visual inspection of the trace for frequencies from 40 MHz to 100 MHz for standard compile options as well as from 50 MHz up to 240 MHz for performance optimizing compile options.

Based on Montgomery modular multiplication structure, this paper proposed and realized SM2 algorithm core module for IC cards, that is, scalar multiplication hardware structure of elliptic curves in prime field. Modular addition operation was improved so as to gain faster operation and anti-SPA capacity. Point addition and multiple point algorithm of Jacobi projective coordinates and Montgomery modular inversion algorithm were adopted in the structure. Experimental results indicated that, the structure had better performance and met the specific design indicators of the IC cards. Contact IC card communication protocol module (ISO7816 Protocol) was designed, and serving as the interface, it was tested by the contact IC card reader. In previous scalar multiplication designs, designers tended to focus on certain performance improvement of scalar multiplication, rather than a fixed product. This paper designed scalar multiplication encryption module, which was specific to the design of the interface of the contact IC cards widely used in life, so as to make the IC cards work more effectively.

Geometric errors are inevitably introduced into any multiaxis measuring system, and the geometric error is one of the main factors that seriously affects the measurement accuracy. The present work investigates the error distribution of the prototype of a 5-axis measuring machine based on sensitivity analysis of geometric errors. The measurement error modeling of the 5-axis measuring machine is first established via the homogeneous coordinate transformation, and the Sobol global sensitivity analysis method is then employed to quantify the influence of geometric errors on the measurement result with the sensitivity index. The result shows that most of the angular errors are the crucial geometric errors seriously affecting the measurement result. These errors are supposed to be fully considered in the accuracy design and manufacturing stages. The error levels of the crucial geometric errors were distributed and readjusted according to the sensitivity analysis result. Some practical approaches to distribute and improve the crucial geometric errors have been given in detail. The error distribution method is effective to equalize the influence of the crucial geometric errors on the measurement result as possible. The findings of this study provide significant meanings for the optimal design and accurate manufacturing of the 5-axis measuring machine, and the proposed method can be used to improve the measurement accuracy of the 5-axis measuring machine.

The work presents an elliptic curve crypto accelerator to implement scalar multiplication on standardized NIST curve over Galois binary fields by using polynomial basis. The proposed design provides efficient hardware utilization and low power consumption. For this purpose, hybrid finite field multiplier is proposed using standard Karatsuba multiplier and shlft-and-add multiplication algorithms. The proposed hybrid finite field multiplier can perform one finite field multiplication in $\frac{m}{2}$ clock cycles, with m being the key-length. Montgomery algorithm with projective coordinates (LopezDahab) is used for the computation of scalar multiplication. An FSM based control unit is developed, governing the overall operations of proposed accelerator. The proposed architecture has been modeled in Verilog-HDL and implemented up to place and route level using the Xilinx Vivado and ISE design software for various FPGA devices. The achieved results demonstrate the convenience of proposed ECC accelerator for resource constrained embedded systems applications requiring low area and low power hardware.

In this article, we present a high-performance hardware architecture for Elliptic curve based (authenticated) key agreement protocol “Elliptic Curve Menezes, Qu and Vanstone” (ECMQV) over Binary Edwards Curve (BEC). We begin by analyzing inversion module on a 251-bit binary field. Subsequently, we present Field Programmable Gate Array (FPGA) implementations of the unified formula for computing elliptic curve point addition on BEC in affine and projective coordinates and investigate the relative performance of these two coordinates. Then, we implement the w -coordinate based differential addition formulae suitable for usage in Montgomery ladder. Next, we present a novel hardware architecture of BEC point multiplication using mixed w -coordinates of the Montgomery laddering algorithm and analyze it in terms of resistance to Simple Power Analysis (SPA) attack. In order to improve the performance, the architecture utilizes registers efficiently and uses efficient scheduling mechanisms for the BEC arithmetic implementations. Our implementation results show that the proposed architecture is resistant against SPA attack and yields a better performance when compared to the existing state-of-the-art BEC designs for computing point multiplication (PM). Finally, we present an FPGA design of ECMQV key agreement protocol using BEC defined over GF(2 251 ). The execution of ECMQV protocol takes 66.47μs using 32,479 slices on Virtex-4 FPGA and 52.34μs using 15,988 slices on Virtex-5 FPGA. To the best of our knowledge, this is the first FPGA design of the ECMQV protocol using BEC.

This work presents a new solution for coordinated search with a team of heterogeneous robots executing a time-critical mission. It is challenging to specify and represent search locations (targets) in known but dynamic environments as well as to find robotic paths to visit the locations. We propose a technique to construct an information map that includes locations of uncertain targets, and generate optimal paths. We especially focus on combining a satellite map that has global coordinates with local images gathered from an aerial robot. Specific targets are represented on a homogeneous coordinate system, so that different types of robots, capable to gather necessary information, may cooperatively conduct a mission. Once a homogeneous map is constructed, a centralized pathfinding algorithm can be applied. Our path-finding algorithm is to choose a set of paths, suggesting a proper number of robots along with their initial locations. In our work, robots can independently travel search locations, which may have dynamics or changes, but collaboratively cover all target locations. Through the experiments with real robotic platforms, we validate the generation of a map including targets and a choice of paths, and compare with existing algorithms.

The automation of tolerance analysis is the ambitious goal of the researcher. In order to achieve this aim, the homogeneous coordinate transformation is a usual method for the position calculation of geometric features and parts in the assembly tolerance analysis method. The hierarchy definition and the automatic creation of the coordinate system are the key work for the automation of tolerance analysis. In this paper, the position calculation equations of the geometric feature in assembly model are described firstly, then the error propagation relation graph for geometric features in the part model is analyzed based on the datum–target mechanism of geometric tolerance. The assembly errors propagation graph of the part in the machine model is described based on the assembly relation and assembly precedence. The controlling points variation model (CPVM) of a geometric feature is used to generate and represent the position of the substitute geometry of the geometric feature, and the position parameters of substitute geometry are generated using Monte Carlo simulations. Based on the CPVM model and datum–target mechanism, the hierarchy of the coordinate system for a geometric feature is defined on the part level. Based on the assembly relations and the real part model, the assembly coordinate system between the locating part and the assembling part is established. According to these two errors propagation graphs, the homogeneous coordinate transformation matrix between the related coordinate systems is generated automatically. Finally, a case study is provided to illustrate the proposed method.

Purpose The purpose of this paper is to provide a method with convenient modelling as well as precise computation to the research of complex multi-body system, such as robot arms and solar power satellite. Classical modelling method does not always fit these two requirements. Design/methodology/approach In this paper, tensor coordinates (TC) and homogeneous tensor coordinates (HTC) method with gradient components are developed, which also have a convenient interface with classical theory. Findings The HTC proved its precision and effectiveness by two examples. In HTC model, equations have a more convenient form as matrix and the results coincide well with classical one. Research limitations/implications There is no plenty detailed operations supported in mathematics yet, which may be developed in further research. Practical implications With TC/HTC method, the research work can be separated more thoroughly: a simpler modelling work is left to scientists, when more computing work is handed to the computers. It may ease scientists’ brains in multibody modelling. Originality/value The HTC method has the advantages of absolute nodal coordinate formulations, tensor and homogeneous coordinate (HC) and it may be used in multibody mechanics, or other related engineerings.

This paper presents an EM analysis for extracting a signature generation key from ECDSA hardware. In particular, we presents a horizontal side-channel attack using EM waves radiated from compact ECDSA hardware based on Jacobian projective coordinates. The validity of the attack is demonstrated through an experiment using a prototype LSI. We also mention an efficient algorithm-level countermeasure against the attack.

The requirements of video communication make people to use the Internet daily in their life. The security also plays a vital role to achieve smooth communication in multimedia. Multimedia encryption provides elegant solution to fulfill all these requirements. Among all public key encryption algorithms, elliptic curve cryptography (ECC) is attracted toward encryption of multimedia for better security with less bandwidth. We proposed a method named as perceptual encryption, which performs encryption on selective bitplanes of a grayscale image. The proposed method uses ECC on Koblitz curves with Lopez–Dahab projective coordinates to improve the efficiency of multimedia encryption and decryption. Moreover, scalar multiplication dominates the performance of ECC over Koblitz curves. The performance of Koblitz curves can be further improved by reducing number of field operations required at the time of point addition using Lopez–Dahab projective coordinates, anyway doubling needs less cost due to Frobenius endomorphism. Two parameters such as peak signal-to-noise ratio (PSNR) and structural similarity measure (SSIM) are considered to measure the quality of an image. The requirements of multimedia are analyzed based on the results obtained.

In this paper we describe our flexible ECDSA design for elliptic curve over binary extended fields GF(2l). We investigated its resistance against Horizontal Collision Correlation Attacks (HCCA). Due to the fact that our design is based on the Montgomery kP algorithm using Lopez-Dahab projective coordinates the scalar k cannot be successful revealed using HCCA, but this kind of attacks can be helpful to divide the measured traces into parts that correspond to processing of a single bit of the scalar k. The most important contribution of this paper is that our flexible field multiplier is resistant against horizontal attacks. This inherent resistance makes it a valuable building block for designing unified field multipliers., Author's version accepted for ReConFig-2018; the final publication is available at https://ieeexplore.ieee.org/abstract/document/8641730

This contribution describes a new approach to a solution of multidimensional dynamical systems using the Laplace transform and geometrical product, i.e. using inner product (dot product, scalar product) and outer product (extended cross-product). It leads to a linear system of equations Ax=0 or Ax=b which is equivalent to the outer product if the projective extension of the Euclidean system and the principle of duality are used. The paper explores property of the geometrical product in the frame of multidimensional dynamical systems. The proposed approach enables to avoid division operation and extents numerical precision as well. It also offers applications of matrix-vector and vector-vector operations in symbolic manipulation, which can lead to new algorithms and/or new formula. The proposed approach can be applied also for stability evaluation of dynamical systems. In the case of numerical computation, it supports vector operation and SSE instructions or GPU can be used efficiently. Tento příspěvek popisuje nový přístup k řešení vícerozměrných dynamických systémů pomocí Laplaceovy transformace a geometrického produktu, tj. Pomocí vnitřního produktu (dot produkt, skalární produkt) a vnějšího produktu (rozšířený křížový produkt). To vede k lineární soustavě rovnic Ax = 0 nebo Ax = b, která je ekvivalentní vnějšímu produktu, pokud je použito projektivní rozšíření euklidovského systému a princip duality. Článek zkoumá vlastnosti geometrického produktu v rámci vícerozměrných dynamických systémů. Navržený přístup umožňuje vyhnout se dělení operací a také rozsah numerické přesnosti. Nabízí také aplikace maticových a vektorových operací při symbolické manipulaci, což může vést k novým algoritmům nebo novému vzorci. Navrhovaný přístup lze použít také pro hodnocení stability dynamických systémů. V případě numerického výpočtu podporuje vektorovou operaci a lze efektivně využívat instrukce SSE nebo GPU.

Our skin is the first line of defense from external environment. It covers whole body and protects the tissues and organs from various external assaults. Skin also keeps a stable internal environment for body’s metabolism, like regulating temperature and saving necessary water. Furthermore the nerve of skin gives us tactile sensation, which makes us to better adapt to our living environment. When skin diseases occur, the partial functionality of skin cannot be accomplished. One of fatal diseases is skin cancer. With skin cancer progression, the color of skin will change on different stage. Paper proposes an algorithm to detect and quantize this kind of visual change. The program tries to assist people to diagnosis skin cancer, even self-exam. The proposed algorithm mainly contains three parts. Initially paper segments the skin area from images. Then two main components of skin color are retrieved by ICA. Finally paper estimates if there is pigmentation disorder through comparing the ratio of two main components, and a novel comparing model, projective coordinates is used in this process.

Energy consumption is an important factor for any cryptoscheme, implemented on devices having limited energy resource. In this paper, we analyze the energy consumption during the Diffie-Hellman key exchange implemented on both supersingular and ordinary elliptic curves. The former protocol is Supersingular Isogeny based Diffie-Hellman (SIDH) and the latter is Elliptic Curve based Diffie-Hellman (ECDH) respectively. Implementations are executed on 64 bit Intel Skylake processor. The energy consumption is analysed for the classical bit security levels of 128 and 192. In this paper, a detailed comparison of power and energy consumption by elliptic curve point addition and doubling operations is presented for affine and standard projective coordinates. Projective coordinates based elliptic curve operations are found to be around 50 to 60 times more energy efficient. We then analyze SIDH and ECDH, implemented using projective coordinates. Our results show that, SIDH consumes around 37 to 47 times more energy in comparison to ECDH for the above mentioned bit security levels.

Tento příspěvek představuje novou formulaci Plückerových souřadnic pomocí geometrické algebry a standardní lineární algebry s projektivní reprezentací. Plückerovy souřadnice se obvykle používají pro zobrazení čar v prostoru, které je dáno dvěma body. Čára však může být také dána průnikem dvou rovin ve vesmíru. Princip duality vede k jednoduché formulaci pro oba případy. Prezentovaný přístup využívá homogenní souřadnice s aplikací principu duality. Je vhodný pro aplikaci také na GPU. Plückerovy souřadnice se používají v mnoha aplikacích, např. v robotice, počítačově podporovaném návrhu a algoritmech počítačové grafiky atd. This contribution presents a new formulation of Plücker coordinates using geometric algebra and standard linear algebra with projective representation. The Plücker coordinates are usually used for a line representation in space, which is given by two points. However, the line can be also given as an intersection of two planes in space. The principle of duality leads to a simple formulation for both cases. The presented approach uses homogenous coordinates with the duality principle application. It is convenient for application on GPU as well. The Plücker coordinates are used in many applications, e.g. in robotics, computer aided design and computer graphics algorithms etc.

This paper presents an EM analysis for extracting a signature generation key from ECDSA hardware. In particular, we show a possibility of a horizontal side-channel attack using EM waves radiated from compact ECDSA hardware based on Jacobian projective coordinates. We also mention a possible countermeasure against the attack.

This paper presents a fast scalar point multiplier for an elliptic curve crypto-processor in the field \( {\text{GF}}\left( {2^{163} } \right) \). Elliptic curve-based cryptographic algorithms have been in wide use since the early 2000s after being introduced in 1986. With the ever-increasing need for information security, it is essential for systems to perform the required operations in a fast and efficient manner. In this work, a hybrid type Karatsuba multiplier has been used for fast field multiplications and a dedicated inverter module based on the extended Euclidean algorithm is used for fast field inversions. The point multiplication is performed using the standard double-and-add algorithm for which the point doubling and point addition are done using standard projective coordinates. The use of the fast multipliers and field inverters makes the implementation a fast one as compared to other high-performance implementations that have been reported over the years, at the cost of increased resource usage. The results obtained, however, justify this increased resource usage as the point multiplication is the most time-intensive operation in the encryption and decryption process of elliptic curve cryptography.

GCD attack depends on modifying the cipher text and then get an access to the decryption of the modified cipher text that is discarded identifying as due to bad implementation. In this paper we mount a GCD attack on Demytko’s cryptosystem on elliptic curves. In this we implement the attack by point addition with projective coordinates using a fast computation method. As this involves working only with x -coordinates. We start with developing the formulas for the projective coordinates ] : [ Z X generalizing the ideas of Montgomery and propose to use these formulas to generate the polynomials for the GCD attack.

In this paper we describe a design to compute an inversion in F p as well as division. Inversion can be used in Elliptic Curve Cryptography systems and pairing-based cryptography, which are becoming popular for Public Key Cryptosystems. For the same level of security, ECC and pairing use much smaller key length than RSA but need modular inversion. In ECC when points are represented in so-called affine coordinates, the addition of two points involves a field inversion. Some pairing require one inversion over F p in order to perform the final exponentiation. Usually, inversions are avoided in Elliptic Curve Cryptography as they are expensive. For example, inversions in affine coordinates are transform into multiplication in Jacobian or projective coordinates. In order to improve performance of Public Key Cryptosystems, we present in this paper an improved algorithm for prime field modular inversion. We demonstrate that affine coordinates can be more efficient than projective or jacobian for the scalar multiplication.

In order to obtain all the geometric errors of five-axis CNC machine tool, a measuring strategy with cross grid encoder and double ball-bar (DBB) is proposed. According to the theory of multi-body systems, by using the homogeneous coordinate transformation method the model of geometric errors is established. The cross grid encoder is used to detect the translation geometric errors and the DBB is used to measure the rotation ones. Based on the kinematic chain and the properties of geometric errors, the translation geometric errors are measured before the rotation ones. For each axis, both PIGEs (position-independent errors) and PDGEs (position-dependent errors) are under consideration. Then a complete compensated model can be established.

The Montgomery kP algorithm using Lopez-Dahab projective coordinates is a time and energy efficient way to perform the elliptic curve point multiplication. It even can provide resistance against simple power analysis attacks. Nevertheless, straight forward implementations of this algorithm are not resistant against differential power analysis attacks. But highly-efficient designs i.e. designs in which different functional units are operating in parallel can increase the resistance of ECC implementations against DPA. In this paper we demonstrate this fact on the examples of our 6 implementations of the Montgomery kP algorithm. We perform horizontal DPA attacks to demonstrate the fact that the resistance of the kP algorithm against side channel analysis attacks can be increased through its implementation as an efficient design.

The machining accuracy of NC machine can be improved significantly through the error compensation. The kernels of (multi-body system)MBS theories are that the number arrays of low-order body are used to describe the topological structures which are taken to generalize and refine MBS, and the characteristic matrix are employed to represent the relative positions and gestures between any two bodies in MBS. Based on the theory of rigid body dynamics and homogeneous coordinate transformation techniques, the space error model of five-axis gantry NC machine can be established, the modified NC instructions can be obtained and it prepares well ready for the software error compensation and the practical application. Gyratory motion of five-axis NC machine will generate the coupling of translational motion, so it is different from the three-axis NC machine in the process of error compensation and we need to decoupling.

Several hardware implementations of elliptic curve cryptography have been recently proposed for resource-constrained applications but few of them considered the power analysis countermeasures. In this paper, a new modular multiplication with zero-value attack countermeasure is proposed. The Montgomery ladder algorithm and randomized projective coordinates method are used to resist simple power analysis and differential power analysis. The circular shift register has been adopted to reduce the area. The overall design has been implemented in binary field. 16.5K Gate Equivalent (GE) area is needed and 10.8ms is required for one scalar point multiplication at 13.56MHz. The implementation result shows that the proposed ECC processor achieves power analysis resistance with low resource cost.

Cryptographic algorithms in smart cards and other constrained environments increasingly rely on Elliptic Curves and thus it is desirable to have fast algorithms for elliptic curve arithmetic. In this paper, we provide (i) faster differential addition formulae for elliptic curve arithmetic on Generalized Edwards’ Curves improving upon the currently known formulae in the literature, proposed by Justus and Loebenberger at IWSEC 2010, (ii) more efficient affine differential addition formulae for a new model of Binary Edwards Curves proposed by Wu, Tang and Feng at INDOCRYPT 2012 and (iii) an algorithm for point doubling on Twisted Edwards Curves with a smaller footprint when the implementation is desired to work across Homogeneous Projective, Inverted and Extended Homogeneous Projective Coordinates.

Many problems, not only in computer vision and visualization, lead to a system of linear equations Ax = 0 or Ax = b and fast and robust solution is required. A vast majority of computational problems in computer vision, visualization and computer graphics are three dimensional in principle. This paper presents equivalence of the cross–product operation and solution of a system of linear equations Ax = 0 or Ax = b using projective space representation and homogeneous coordinates. This leads to a conclusion that division operation for a solution of a system of linear equations is not required, if projective representation and homogeneous coordinates are used. An efficient solution on CPU and GPU based architectures is presented with an application to barycentric coordinates computation as well.

For a proper way of modelling and simulation, as well as the precise and completeness, and a new multi-level method named homogeneous tensor coordinate(HTC) was developed by improving the absolute nodal coordinate formulation(ANCF) and the homogeneous coordinates(HC). In HTC, a new form of 2×2 matrix was employed instead of angles, when the capacity of expression and calculation was increased at the same time. HTC has good performance in traditional rigid body mechanics, which especially suitable for multibody modelling. Therefore, the robotic engineering may benefit from this finding, and a new formulation for scientific research may also affect the physics although some mathematical problems are raised in matrix theory.

An efficient iterative pose estimation method was proposed to improve the accuracy and robustness of monocular vision measurement systems.By using homogeneous coordinates,it avoids the limit of existing algorithms on the selection of reference points for a paraperspective camera model.First,a method to compute the pose of an object under the paraperspective camera model was researched using homogeneous coordinates,and its geometric meaning was expatiated in detail.Then,this method was used to estimate the pose of an object in an iterative way under the full perspective camera model with high precision.Simulation results indicate that the proposed method improves the accuracy,speed and robustness of the paraperspective camera model based iterative pose estimation.Real measurement results show that the translation measurement precision and rotation measurement precision of the proposed method are better than 0.1 mm and 0.1°,respectively,and it can satisfy the requirements of various vision detection systems.Furthermore,the accuracy of the proposed method can be improved further by using the land mark point method and sub-pixel feature locating techniques.

In order to test large and off-axis aspheric surfaces without the aid of other null optics,a new method subaperture stitching interferometry(SSI) is introduced.The basic principle of the technique is analysed,and the synthetical optimization stitching model and effective stitching algorithm are established based on homogeneous coordinates transformation and simultaneous least-squares method.The stitching software and prototype for testing of large aspheres by SSI are devised and developed.An off-axis asphere with the aperture of 376×188 mm2 is tested by this method.For the comparison and validation,the asphere is also tested by null compensation,the synthesized surface map is consistent to the entire surface map from the null test,and the difference of PV and RMS error between them is 0.047λ and 0.006λ,respectively.So it provides another quantitive measurement for testing large aspheric surfaces and off-axis aspheres besides null-compensation.

The natural homogeneous coordinate system is the analog of the Cartesian coordinate system for projective geometry. Roughly speaking a projective geometry adds an axiom that parallel lines meet at a point at infinity. This removes the impediment to line-point duality that is found in traditional Euclidean geometry. The natural homogeneous coordinate system is surprisingly useful in a number of applications including computer graphics and statistical data visualization. In this article, we describe the axioms of projective geometry, introduce the formalism of natural homogeneous coordinates, and illustrate their use with four applications. WIREs Comp Stat 2010 2 678–685 DOI: 10.1002/wics.122 For further resources related to this article, please visit the WIREs website.

Optical methods of contouring that utilise image recordings by cameras are based on a fundamental discipline, projective geometry. The 3-D world is projected in 2-D utilising a camera modelled in the technical literature by the pinhole camera. To get back 3-D information, the fundamental property measured is parallax. Parallax is a vector resulting from the difference of the projective coordinates of a point in space when projected onto a plane from two different points. The oldest method used for measuring parallax is photogrammetry. It is assumed to be the most precise technique, with the capability of obtaining 10−5 of the largest dimension of the measured object. This study summarises the state-of-the-art methods based on projecting a spatial carrier. Starting with the concept of moire as a form of photogrammetry, the different optical techniques for parallax determination are discussed. Although the moire method has reached 1 μm accuracy in laboratory work, a question remains: can moire become a standardised contouring technique yielding 10−6 m accuracy? This study is devoted to the analysis of high accuracy contour measurements, through both theoretical derivations and experimental verifications.

Kinematics and dynamics calculation is meaningful both in the design stage and the movement control stage of a manipulator. After the comparison among different methods of manipulator kinematics and dynamics，this paper adopts homogeneous coordinate transformation to analyze and research positive solutions of the space 5-DOF manipulator kinematics, applies mathematics in the approximate calculation to solve the equations’ inverse problems because of the explicit difficult nature to derive analytic solution; conducts dynamics analysis to the manipulator and adopts the Lagrange dynamics method to establish an ideal model, i.e., a dynamics model integrating with corresponding computer algorithm without considering the influences of friction force, to complete the research on the simulation calculation of dynamics inverse solution.

Computer graphics is ostensibly based on projective geometry. The graphics pipeline—the sequence of functions applied to 3D geometric primitives to determine a 2D image—is described in the graphics literature as taking the primitives from Euclidean to projective space, and then back to Euclidean space. This is a weak foundation for computer graphics. An instructor is at a loss: one day entering the classroom and invoking the established and venerable theory of projective geometry while asserting that projective spaces are not separable, and then entering the classroom the following week to tell the students that the standard graphics pipeline performs clipping not in Euclidean, but in projective space—precisely the operation (deciding sidedness, which depends on separability) that was deemed nonsensical. But there is no need to present Blinn and Newell’s algorithm (Comput. Graph. 12, 245–251, 1978; Commun. ACM 17, 32–42, 1974)—the crucial clipping step in the graphics pipeline and, perhaps, the most original knowledge a student learns in a fourth-year computer graphics class—as a clever trick that just works. Jorge Stolfi described in 1991 oriented projective geometry. By declaring the two vectors $(x,y,z,w)^{{\ensuremath {\mathsf {T}}}}$and $(-x,-y,-z,-w)^{{\ensuremath {\mathsf {T}}}}$distinct, Blinn and Newell were already unknowingly working in oriented projective space. This paper presents the graphics pipeline on this stronger foundation.

Methods for simultaneous refining the outer orientation parameters of satellite cameras in synchronized shooting from two spacecrafts moving in the same orbit are proposed and investigated. The methods are based on taking into account perspective relations between corresponding coordinates of projections of region elements in a synchronized series of stereograms under assumption that the common aiming point of the cameras does not change during shooting. A method for space shooting simulation is described and used for investigating the dependences of the errors of estimating the orientation angles of the cameras on the number of stereograms and the noise level in projective coordinates of region element images.

High-dimensional data visualization is a changing task with many applications in a various fields of sciences. Parallel coordinates is one of the most widely used information visualization technique for multivariate data analysis and high-dimensional geometry. The dimension ordering is an original problem for exploring structures in a high-dimensional data space. In this paper, we propose a new metric for measuring distance between two line-segment on the parallel coordinates. The metric is suitable and effective on the parallel coordinates. We use our metric distance for finding an optimal dimension ordering on the parallel coordinates. Finally, we demonstrate our method can be applied to visualize clusters in high-dimensional data on the parallel coordinates.