• Journal of KIISE:Computer Systems and Theory
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• v.36 no.6
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• pp.502-510
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• 2009

Multiprecision squaring is one of the most significant algorithms in the core public key cryptography operation. The aim of this work is to present a new improved squaring algorithm compared with the MIRACL's multi precision squaring algorithm in which the previous work [1] on multiprecision multiplication is implemented. First, previous works on multiprecision multiplication and standard squaring are analyzed. Then, our new Lazy Doubling squaring algorithm is introduced. In MIRACLE library [3], Scott's Carry-Catcher Hybrid multiplication technique [1] is applied to implementation of multiprecision multiplication and squaring. Experimental results of the Carry-Catcher hybrid squaring algorithm and the proposed Lazy Doubling squaring algorithm both of which are tested on Atmega128 CPU show that proposed idea has achieved significant performance improvements. The proposed Lazy Doubling Squaring algorithm reduces addition instructions by the fact $a_0\;{\ast}\;2\;+\;a_1\;{\ast}\;2\;+\;...\;+\;a_{n-1}\;{\ast}\;2\;+\;a_n\;{\ast}\;2\;=\;(a_0\;+\;a_1\;+\;...\;+\;a_{n-1}\;+\;a_n)\;{\ast}\;2$ while the standard squaring algorithm reduces multiplication instructions by the fact $S_{ij}\;=\;x_i\;{\ast}\;x_j\;=\;S_{ij}$. Experimental results show that the proposed squaring method is 25% faster than that in MIRACL.

• Journal of Institute of Control, Robotics and Systems
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• v.10 no.1
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• pp.96-102
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• 2004

For sensitivity enhancement, the general assisted-GPS acquisition method adopts not only the coherent accumulation technique but also the non-coherent accumulation technique since the long coherent accumulation period increases the number of frequency search cells. But the non-coherent accumulation technique causes tile squaring loss, which is a dominant factor among the acquisition losses of assisted GPS dealing with weak GPS signals. This paper derives the squaring loss of the previous assisted-GPS acquisition method and proposes an assisted-GPS acquisition method for solving the problem of squaring loss in weak signal environment. In this paper, it is explained that the proposed assisted-GPS acquisition method prevents the squaring loss using a coupled coherent accumulation technique and the number of search cells of the proposed assisted-GPS acquisition method is much smaller than that of the previous assisted-GPS acquisition method. Finally, through the simulation by the GPS simulator, the acquisition success rate of the proposed assisted-GPS acquisition method is compared with that of the previous assisted-GPS acquisition method and the acquisition improvements are shown in weak signal environment.

• Journal of the Semiconductor & Display Technology
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• v.16 no.3
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• pp.47-52
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• 2017

Solar cell industry requires high technologies to stabilize apparatuses for the wafer manufacturing. Vibrations of squaring & grinding machines are one of the most critical factors for causing residual stresses of ingots, which are the main reasons of the breakage in the following processes such as wire sawing, cleaning, and modularity. In this study, the structure of a squaring & grinding machine has been analyzed through experiments and computer simulations to figure out the ways to suppress the vibrations effectively, and further to minimize the breakage of wafers. The result shows that simple design changes of applying a few ribs can improve the stability of the machine.

• Journal for History of Mathematics
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• v.30 no.4
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• pp.221-232
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• 2017

In this paper, we discuss how ancient Egyptians find out the area of the circle based on $\ll$Ahmose Papyrus$\gg$. Vogel and Engels studied the quadrature of the circle, one of the basic concepts of ancient Egyptian mathematics. We look closely at the interpretation based on the approximate right triangle of Robins and Shute. As circumstantial evidence for Robbins and Shute's hypothesis, Egyptians prior to the 12th dynasty considered the perception of a right triangle as examples of 'simultaneous equation', 'unit of length', 'unit of slope', 'Egyptian triple', and 'right triangles transfer to Greece'. Finally, we present a method to utilize the squaring the circle by ancient Egyptians interpreted by Robbins and Shute as the dynamic symmetry of Hambidge.

• Journal of Mechanical Science and Technology
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• v.19 no.11
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• pp.1975-1987
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• 2005

An input time delay always exists in practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via digital computers. In this paper a new scheme for the discretization of nonlinear systems using Taylor series expansion and the zero-order hold assumption is proposed. The mathematical structure of the new discretization method is analyzed. On the basis of this structure the sampled-data representation of nonlinear systems with time-delayed multi-input is presented. The delayed multi-input general equation has been derived. In particular, the effect of the time-discretization method on key properties of nonlinear control systems, such as equilibrium properties and asymptotic stability, is examined. Additionally, hybrid discretization schemes that result from a combination of the scaling and squaring technique (SST) with the Taylor series expansion are also proposed, especially under conditions of very low sampling rates. Practical issues associated with the selection of the method's parameters to meet CPU time and accuracy requirements, are examined as well. A performance of the proposed method is evaluated using a nonlinear system with time delay maneuvering an automobile.

• International Journal of Control, Automation, and Systems
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• v.4 no.3
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• pp.293-301
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• 2006

A new discretization method for calculating a sampled-data representation of a nonlinear continuous-time system is proposed. The proposed method is based on the well-known Taylor series expansion and zero-order hold (ZOH) assumption. The mathematical structure of the new discretization method is analyzed. On the basis of this structure, a sampled-data representation of a nonlinear system with a time-delayed input is derived. This method is applied to obtain a sampled-data representation of a non-affine nonlinear system, with a constant input time delay. In particular, the effect of the time discretization method on key properties of nonlinear control systems, such as equilibrium properties and asymptotic stability, is examined. 'Hybrid' discretization schemes that result from a combination of the 'scaling and squaring' technique with the Taylor method are also proposed, especially under conditions of very low sampling rates. Practical issues associated with the selection of the method parameters to meet CPU time and accuracy requirements are examined as well. The performance of the proposed method is evaluated using a nonlinear system with a time-delayed non-affine input.

• Journal of the Institute of Electronics Engineers of Korea SC
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• v.39 no.3
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• pp.211-219
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• 2002

In this paper, a new architecture that can simultaneously process modular multiplication and squaring on GF(2$^{m}$ ) in m clock cycles by using the cellular automata is presented. This can be used efficiently for the design of the modular exponentiation on the finite field which is the basic computation in most public key crypto systems such as Diffie-Hellman key exchange, EIGamal, etc. Also, the cellular automata architecture is simple, regular, modular, cascadable and therefore, can be utilized efficiently for the implementation of VLSI.

• Proceedings of the KIEE Conference
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• 1987.07b
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• pp.1191-1194
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• 1987

This paper presents a method for computing the powers and inverse of an element in GF($2^m$). This method is based on the squaring algorithm $A^2=\sum\limits_{i=0}^{2m-2}P_i$, where $Pi={\alpha}_{i/2}$ if i is even, Pi=0 otherwise, derived from the multiplication algorithm for two elements in GF($2^m$). The powers and inverses in GF($2^m$) for m=2, 3, 4,5 were obtained using computer program, and used in circuit realization of Galois switching function. The squaring and inverse generating circuits are also shown.

• 제어로봇시스템학회:학술대회논문집
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• 2002.10a
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• pp.33.6-33
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• 2002

For RF sensitivity enhancement, the previous assisted GPS acquisition architecture adopts not only the coherent integration technique but also the non-coherent integration technique since the long coherent integration time increases the number of the frequency search cells. But the non-coherent integration technique induces the squaring loss, which is the dominant factor among the acquisition losses of assisted GPS dealing with weak GPS signals. This paper proposes an efficient architecture for weak signal acquisition in assisted GPS. In this paper, it is explained that the proposed architecture reduces the squaring loss using a modified non-coherent integration technique. Furthermore, it is..

• Journal of Mechanical Science and Technology
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• v.20 no.7
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• pp.950-960
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• 2006

An output time delay always exists in practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via a digital computer. A new method for the discretization of nonlinear systems using Taylor series expansion and the zero-order hold assumption is proposed in this paper. This method is applied to the sampled-data representation of a nonlinear system with a constant output time-delay. In particular, the effect of the time-discretization method on key properties of nonlinear control systems, such as equilibrium properties and asymptotic stability, is examined. In addition, 'hybrid' discretization schemes resulting from a combination of the 'scaling and squaring' technique with the Taylor method are also proposed, especially under conditions of very low sampling rates. A performance of the proposed method is evaluated using two nonlinear systems with time-delay output.