Relevant bibliographies by topics / Extended Euclidean algorithm

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Academic literature on the topic 'Extended Euclidean algorithm'

Author: Grafiati
Published: 4 June 2021
Last updated: 11 February 2022

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Journal articles on the topic "Extended Euclidean algorithm"

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Okazaki, Hiroyuki, Yosiki Aoki, and Yasunari Shidama. "Extended Euclidean Algorithm and CRT Algorithm." Formalized Mathematics 20, no. 2 (December 1, 2012): 175–79. http://dx.doi.org/10.2478/v10037-012-0020-2.

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Summary In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based on the source code of the NZMATH, a number theory oriented calculation system developed by Tokyo Metropolitan University [8].
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Levrie, Paul, and Rudi Penne. "The extended Euclidean Algorithm made easy." Mathematical Gazette 100, no. 547 (March 2016): 147–49. http://dx.doi.org/10.1017/mag.2016.25.

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Kim, Daehak, and Kwang Sik Oh. "Computer intensive method for extended Euclidean algorithm." Journal of the Korean Data and Information Science Society 25, no. 6 (November 30, 2014): 1467–74. http://dx.doi.org/10.7465/jkdi.2014.25.6.1467.

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Havas, George. "On the complexity of the extended euclidean algorithm (Extended Abstract)." Electronic Notes in Theoretical Computer Science 78 (April 2003): 1–4. http://dx.doi.org/10.1016/s1571-0661(04)81002-8.

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Aldaya, Alejandro Cabrera, Alejandro J. Cabrera Sarmiento, and Santiago Sánchez-Solano. "SPA vulnerabilities of the binary extended Euclidean algorithm." Journal of Cryptographic Engineering 7, no. 4 (July 8, 2016): 273–85. http://dx.doi.org/10.1007/s13389-016-0135-4.

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Luo, Zhikun, Huafei Sun, and Xiaomin Duan. "The Extended Hamiltonian Algorithm for the Solution of the Algebraic Riccati Equation." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/693659.

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We use a second-order learning algorithm for numerically solving a class of the algebraic Riccati equations. Specifically, the extended Hamiltonian algorithm based on manifold of positive definite symmetric matrices is provided. Furthermore, this algorithm is compared with the Euclidean gradient algorithm, the Riemannian gradient algorithm, and the new subspace iteration method. Simulation examples show that the convergence speed of the extended Hamiltonian algorithm is the fastest one among these algorithms.
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Liu, Weihua, and Andrew Klapper. "AFSRs synthesis with the extended Euclidean rational approximation algorithm." Advances in Mathematics of Communications 11, no. 1 (2017): 139–50. http://dx.doi.org/10.3934/amc.2017008.

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KAIHARA, M. E. "A Hardware Algorithm for Modular Multiplication/Division Based on the Extended Euclidean Algorithm." IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E88-A, no. 12 (December 1, 2005): 3610–17. http://dx.doi.org/10.1093/ietfec/e88-a.12.3610.

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Yi, Jin, Shiqiang Zhang, Yueqi Cao, Erchuan Zhang, and Huafei Sun. "Rigid Shape Registration Based on Extended Hamiltonian Learning." Entropy 22, no. 5 (May 12, 2020): 539. http://dx.doi.org/10.3390/e22050539.

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Shape registration, finding the correct alignment of two sets of data, plays a significant role in computer vision such as objection recognition and image analysis. The iterative closest point (ICP) algorithm is one of well known and widely used algorithms in this area. The main purpose of this paper is to incorporate ICP with the fast convergent extended Hamiltonian learning (EHL), so called EHL-ICP algorithm, to perform planar and spatial rigid shape registration. By treating the registration error as the potential for the extended Hamiltonian system, the rigid shape registration is modelled as an optimization problem on the special Euclidean group S E ( n ) ( n = 2 , 3 ) . Our method is robust to initial values and parameters. Compared with some state-of-art methods, our approach shows better efficiency and accuracy by simulation experiments.
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Al-Hiaja, Qasem Abu, Abdullah AlShuaibi, and Ahmad Al Badawi. "Frequency Analysis of 32-bit Modular Divider Based on Extended GCD Algorithm for Different FPGA chips." INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY 17, no. 1 (January 16, 2018): 7133–39. http://dx.doi.org/10.24297/ijct.v17i1.6992.

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Modular inversion with large integers and modulus is a fundamental operation in many public-key cryptosystems. Extended Euclidean algorithm (XGCD) is an extension of Euclidean algorithm (GCD) used to compute the modular multiplicative inverse of two coprime numbers. In this paper, we propose a Frequency Analysis study of 32-bit modular divider based on extended-GCD algorithm targeting different chips of field-programmable gate array (FPGA). The experimental results showed that the design recorded the best performance results when implemented using Kintex7 (xc7k70t-2-fbg676) FPGA kit with a minimum delay period of 50.63 ns and maximum operating frequency of 19.5 MHz. Therefore, the proposed work can be embedded with many FPGA based cryptographic applications.
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Dissertations / Theses on the topic "Extended Euclidean algorithm"

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Daneshbeh, Amir. "Bit Serial Systolic Architectures for Multiplicative Inversion and Division over GF(2m)." Thesis, University of Waterloo, 2005. http://hdl.handle.net/10012/776.

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Systolic architectures are capable of achieving high throughput by maximizing pipelining and by eliminating global data interconnects. Recursive algorithms with regular data flows are suitable for systolization. The computation of multiplicative inversion using algorithms based on EEA (Extended Euclidean Algorithm) are particularly suitable for systolization. Implementations based on EEA present a high degree of parallelism and pipelinability at bit level which can be easily optimized to achieve local data flow and to eliminate the global interconnects which represent most important bottleneck in todays sub-micron design process. The net result is to have high clock rate and performance based on efficient systolic architectures. This thesis examines high performance but also scalable implementations of multiplicative inversion or field division over Galois fields GF(2m) in the specific case of cryptographic applications where field dimension m may be very large (greater than 400) and either m or defining irreducible polynomial may vary. For this purpose, many inversion schemes with different basis representation are studied and most importantly variants of EEA and binary (Stein's) GCD computation implementations are reviewed. A set of common as well as contrasting characteristics of these variants are discussed. As a result a generalized and optimized variant of EEA is proposed which can compute division, and multiplicative inversion as its subset, with divisor in either polynomial or triangular basis representation. Further results regarding Hankel matrix formation for double-basis inversion is provided. The validity of using the same architecture to compute field division with polynomial or triangular basis representation is proved. Next, a scalable unidirectional bit serial systolic array implementation of this proposed variant of EEA is implemented. Its complexity measures are defined and these are compared against the best known architectures. It is shown that assuming the requirements specified above, this proposed architecture may achieve a higher clock rate performance w. r. t. other designs while being more flexible, reliable and with minimum number of inter-cell interconnects. The main contribution at system level architecture is the substitution of all counter or adder/subtractor elements with a simpler distributed and free of carry propagation delays structure. Further a novel restoring mechanism for result sequences of EEA is proposed using a double delay element implementation. Finally, using this systolic architecture a CMD (Combined Multiplier Divider) datapath is designed which is used as the core of a novel systolic elliptic curve processor. This EC processor uses affine coordinates to compute scalar point multiplication which results in having a very small control unit and negligible with respect to the datapath for all practical values of m. The throughput of this EC based on this bit serial systolic architecture is comparable with designs many times larger than itself reported previously.
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Abu-Mahfouz, Adnan Mohammed. "Elliptic curve cryptosystem over optimal extension fields for computationally constrained devices." Diss., University of Pretoria, 2004. http://hdl.handle.net/2263/25330.

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Data security will play a central role in the design of future IT systems. The PC has been a major driver of the digital economy. Recently, there has been a shift towards IT applications realized as embedded systems, because they have proved to be good solutions for many applications, especially those which require data processing in real time. Examples include security for wireless phones, wireless computing, pay-TV, and copy protection schemes for audio/video consumer products and digital cinemas. Most of these embedded applications will be wireless, which makes the communication channel vulnerable. The implementation of cryptographic systems presents several requirements and challenges. For example, the performance of algorithms is often crucial, and guaranteeing security is a formidable challenge. One needs encryption algorithms to run at the transmission rates of the communication links at speeds that are achieved through custom hardware devices. Public-key cryptosystems such as RSA, DSA and DSS have traditionally been used to accomplish secure communication via insecure channels. Elliptic curves are the basis for a relatively new class of public-key schemes. It is predicted that elliptic curve cryptosystems (ECCs) will replace many existing schemes in the near future. The main reason for the attractiveness of ECC is the fact that significantly smaller parameters can be used in ECC than in other competitive system, but with equivalent levels of security. The benefits of having smaller key size include faster computations, and reduction in processing power, storage space and bandwidth. This makes ECC ideal for constrained environments where resources such as power, processing time and memory are limited. The implementation of ECC requires several choices, such as the type of the underlying finite field, algorithms for implementing the finite field arithmetic, the type of the elliptic curve, algorithms for implementing the elliptic curve group operation, and elliptic curve protocols. Many of these selections may have a major impact on overall performance. In this dissertation a finite field from a special class called the Optimal Extension Field (OEF) is chosen as the underlying finite field of implementing ECC. OEFs utilize the fast integer arithmetic available on modern microcontrollers to produce very efficient results without resorting to multiprecision operations or arithmetic using polynomials of large degree. This dissertation discusses the theoretical and implementation issues associated with the development of this finite field in a low end embedded system. It also presents various improvement techniques for OEF arithmetic. The main objectives of this dissertation are to --Implement the functions required to perform the finite field arithmetic operations. -- Implement the functions required to generate an elliptic curve and to embed data on that elliptic curve. -- Implement the functions required to perform the elliptic curve group operation. All of these functions constitute a library that could be used to implement any elliptic curve cryptosystem. In this dissertation this library is implemented in an 8-bit AVR Atmel microcontroller.
Dissertation (MEng (Computer Engineering))--University of Pretoria, 2006.
Electrical, Electronic and Computer Engineering
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Vestin, Albin, and Gustav Strandberg. "Evaluation of Target Tracking Using Multiple Sensors and Non-Causal Algorithms." Thesis, Linköpings universitet, Reglerteknik, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-160020.

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Today, the main research field for the automotive industry is to find solutions for active safety. In order to perceive the surrounding environment, tracking nearby traffic objects plays an important role. Validation of the tracking performance is often done in staged traffic scenarios, where additional sensors, mounted on the vehicles, are used to obtain their true positions and velocities. The difficulty of evaluating the tracking performance complicates its development. An alternative approach studied in this thesis, is to record sequences and use non-causal algorithms, such as smoothing, instead of filtering to estimate the true target states. With this method, validation data for online, causal, target tracking algorithms can be obtained for all traffic scenarios without the need of extra sensors. We investigate how non-causal algorithms affects the target tracking performance using multiple sensors and dynamic models of different complexity. This is done to evaluate real-time methods against estimates obtained from non-causal filtering. Two different measurement units, a monocular camera and a LIDAR sensor, and two dynamic models are evaluated and compared using both causal and non-causal methods. The system is tested in two single object scenarios where ground truth is available and in three multi object scenarios without ground truth. Results from the two single object scenarios shows that tracking using only a monocular camera performs poorly since it is unable to measure the distance to objects. Here, a complementary LIDAR sensor improves the tracking performance significantly. The dynamic models are shown to have a small impact on the tracking performance, while the non-causal application gives a distinct improvement when tracking objects at large distances. Since the sequence can be reversed, the non-causal estimates are propagated from more certain states when the target is closer to the ego vehicle. For multiple object tracking, we find that correct associations between measurements and tracks are crucial for improving the tracking performance with non-causal algorithms.
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Liu, Hsin-Yu, and 劉欣諭. "A Variation on Extended Euclidean Algorithm." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/19364081886415380592.

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碩士
國立中興大學
資訊科學研究所
93
Finite field arithmetic is more and more important in cryptography, such examples as in the RSA Cryptosystem [14], Digital Signature Algorithm and Elliptic Curve Cryptosystem [10][12] etc. Amount the basic operations; the multiplicative inversion is the most complex one. As we know, there are some methods to compute multiplicative inverses; such as Fermat’s little theorem, extended Euclid algorithm (EEA) and Mongromery algorithm[4][11][16]. In the software implement, EEA based scheme is the most efficiently and widely used algorithm for inversion in large number of elements finite field. But, there are variable size counters, which need many addition/ substation, multiplication operations to trace the inverse. To reduce those operations, there are many variant EEA for computing the multiplicative inversion over GF(2^m) with field elements with polynomial basis. Such as, Calvez et al. proposed a variation Euclid algorithm, which calculate the greatest common divisor (GCD) and inverse for polynomials [5]. They reduce many multiplication and division operations in their scheme. In 2004, Goupil and Palicot introduce another variation to reduce more multiplication and quotient [6]. The question that we must consider next is an integer version. Although Goupil and Palicot had pointed out the integer version is straight forwardly accessible from their variation, we think there is a clearer and efficient method that can solve this question. For this reason, we find a method, which is not only compute modular inverse but also find a solution with Bezout's identity. This variation algorithm modifies the initial value and the determine statement of basic Euclid algorithm (BEA), which used only simpler modular operation and omitted addition, multiplication and division. Furthermore, we have proven that our variation can work very well.
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Hazmi, Ibrahim. "Systolic design space exploration of EEA-based inversion over binary and ternary fields." Thesis, 2018. https://dspace.library.uvic.ca//handle/1828/9981.

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Cryptographic protocols are implemented in hardware to ensure low-area, high speed and reduced power consumption especially for mobile devices. Elliptic Curve Cryptography (ECC) is the most commonly used public-key cryptosystem and its performance depends heavily on efficient finite field arithmetic hardware. Finding the multiplicative inverse (inversion) is the most expensive finite field operation in ECC. The two predominant algorithms for computing finite field inversion are Fermat’s Little Theorem (FLT) and Extended Euclidean Algorithm (EEA). EEA is reported to be the most efficient inversion algorithm in terms of performance and power consumption. This dissertation presents a new reformulation of EEA algorithm, which allows for speedup and optimization techniques such as concurrency and resource sharing. Modular arithmetic operations over GF(p) are introduced for small values of p, observing interesting figures, particularly for modular division. Whereas, polynomial arithmetic operations over GF(pm) are discussed adequately in order to examine the potential for processes concurrency. In particular, polynomial division and multiplication are revisited in order to derive their iterative equations, which are suitable for systolic array implementation. Consequently, several designs are proposed for each individual process and their complexities are analyzed and compared. Subsequently, a concurrent divider/multiplier-accumulator is developed, while the resulting systolic architecture is utilized to build the EEA-based inverter. The processing elements of our systolic architectures are created accordingly, and enhanced to accommodate data management throughout our reformulated EEA algorithm. Meanwhile, accurate models for the complexity analysis of the proposed inverters are developed. Finally, a novel, fast, and compact inverter over binary fields is proposed and implemented on FPGA. The proposed design outperforms the reported inverters in terms of area and speed. Correspondingly, an EEA-based inverter over ternary fields is built, showing the lowest area-time complexity among the reported inverters.
Graduate
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Book chapters on the topic "Extended Euclidean algorithm"

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Ito, Takehiro, Yuichiro Shibata, and Kiyoshi Oguri. "Implementation of the Extended Euclidean Algorithm for the Tate Pairing on FPGA." In Field Programmable Logic and Application, 911–16. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-540-30117-2_98.

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Peinado, Alberto, and Andrés Ortiz. "Cryptanalysis of Multicast Protocols with Key Refreshment Based on the Extended Euclidean Algorithm." In Computational Intelligence in Security for Information Systems, 177–82. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-21323-6_22.

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Ramachandran, Parthasarathy. "Use of Extended Euclidean Algorithm in Solving a System of Linear Diophantine Equations with Bounded Variables." In Lecture Notes in Computer Science, 182–92. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006. http://dx.doi.org/10.1007/11792086_14.

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"Extended Euclidean Algorithm." In Encyclopedia of Cryptography and Security, 436. Boston, MA: Springer US, 2011. http://dx.doi.org/10.1007/978-1-4419-5906-5_1073.

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Prajapat, Gopal Krishan, and Rakesh Kumar. "A Hybrid Approach for Facial Expression Recognition Using Extended Local Binary Patterns and Principal Component Analysis." In Research Anthology on Rehabilitation Practices and Therapy, 517–44. IGI Global, 2021. http://dx.doi.org/10.4018/978-1-7998-3432-8.ch026.

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Facial feature extraction and recognition plays a prominent role in human non-verbal interaction and it is one of the crucial factors among pose, speech, facial expression, behaviour and actions which are used in conveying information about the intentions and emotions of a human being. In this article an extended local binary pattern is used for the feature extraction process and a principal component analysis (PCA) is used for dimensionality reduction. The projections of the sample and model images are calculated and compared by Euclidean distance method. The combination of extended local binary pattern and PCA (ELBP+PCA) improves the accuracy of the recognition rate and also diminishes the evaluation complexity. The evaluation of proposed facial expression recognition approach will focus on the performance of the recognition rate. A series of tests are performed for the validation of algorithms and to compare the accuracy of the methods on the JAFFE, Extended Cohn-Kanade images database.
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Conference papers on the topic "Extended Euclidean algorithm"

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Phiamphu, Dongmuanthang, and Prabir Saha. "Redesigned the Architecture of Extended-Euclidean Algorithm for Modular Multiplicative Inverse and Jacobi Symbol." In 2018 2nd International Conference on Trends in Electronics and Informatics (ICOEI). IEEE, 2018. http://dx.doi.org/10.1109/icoei.2018.8553922.

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Fournaris, Apostolos P., and Odysseas Koufopavlou. "One Dimensional Systolic Inversion Architecture Based on Modified GF(2^k) Extended Euclidean Algorithm." In 2009 12th Euromicro Conference on Digital System Design, Architectures, Methods and Tools (DSD). IEEE, 2009. http://dx.doi.org/10.1109/dsd.2009.161.

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Fournaris, A. P., and O. Koufopavlou. "A Systolic Inversion Architecture Based on Modified Extended Euclidean Algorithm for GF(2K) Fields." In 2006 13th IEEE International Conference on Electronics, Circuits and Systems. IEEE, 2006. http://dx.doi.org/10.1109/icecs.2006.379626.

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Muhammad, Khairun Nisak, and Hailiza Kamarulhaili. "On the sequences ri, si, ti ∈ ℤ related to extended Euclidean algorithm and continued fractions." In INNOVATIONS THROUGH MATHEMATICAL AND STATISTICAL RESEARCH: Proceedings of the 2nd International Conference on Mathematical Sciences and Statistics (ICMSS2016). Author(s), 2016. http://dx.doi.org/10.1063/1.4952482.

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Kampf, Sabine, and Martin Bossert. "A fast Generalized Minimum Distance decoder for Reed-Solomon codes based on the extended Euclidean algorithm." In 2010 IEEE International Symposium on Information Theory - ISIT. IEEE, 2010. http://dx.doi.org/10.1109/isit.2010.5513705.

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Gasiorek, Marcin, Daniel Simson, and Katarzyna Zajac. "On Corank Two Edge-Bipartite Graphs and Simply Extended Euclidean Diagrams." In 2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). IEEE, 2014. http://dx.doi.org/10.1109/synasc.2014.17.

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Khan, Nadeem S., Jami J. Shah, and Joseph K. Davidson. "Probability Tolerance Maps: A New Statistical Model for Non-Linear Tolerance Analysis Applied to Rectangular Faces." In ASME 2010 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/detc2010-28937.

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A new statistical model for the tolerance analysis based upon joint probability distribution of the trivariate normal distributed variables involved in the construction of Tolerance-maps (T-Maps) for rectangular face is presented. Central to the new model is a Tolerance-Map (T-Map) (Patent No. 69638242). It is the range of points resulting from a one-to-one mapping from all the variational possibilities of a perfect-form feature, within its tolerance-zone, to a specially designed Euclidean point-space. The model is fully compatible with the ASME/ANSI/ISO Standard for geometric tolerances. In this research, 4D probability T-Maps (prob T-Maps) have been developed in which the probability value of a point in space is represented by the size of the marker and the associated color. Additionally, 3D prob T-Maps (3D cross sections of the 4D prob T-Maps at pre specified values) are used to represent the probability values of two variables at a time for a constant value of the third variable on a plane. Superposition of the probability point cloud with the T-Map clearly identifies which points are inside and which are outside the T-Map. This represents the pass percentage for parts manufactured with the statistical parameters such as mean and standard deviation as of the assumed trivariate probability distribution. The effect of refinement with form and orientation tolerance is highlighted by calculating the change in pass percentage with the pass percentage for size only. Delaunay triangulation and ray tracing algorithms have been used to automate the process of identifying the points inside and outside the T-Map. Proof of concept software has been implemented to demonstrate this model and to determine pass percentages for various cases. The model is further extended to assemblies by employing convolution algorithms on two trivariate statistical distributions to arrive at the statistical distribution of the assembly. Accumulation T-Maps generated by using Minkowski Sum techniques on the T-Maps of the individual parts is superimposed on the probability point cloud resulting from convolution. Delaunay triangulation and ray tracing algorithms are employed to determine the assemleability percentages for the assembly.
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