• Journal of the Korean School Mathematics Society
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• v.11 no.4
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• pp.609-627
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• 2008

The purpose of this research was to investigate the relationship between the students' strategy types and the recognition for proportional situations. The students' strategy types which were based on the results of ratio and proportion tests were divided into an additive type, a multiplicative type, and a formal type. This research analyzed the students' activities of categorization when were given the proportional problems and nonproportional problems to the students. And it also explored how to develop students' recognizing for the discrimination between the proportional situations and nonproportional situations. The results was the following. First, the students didn't discriminate the proportional situations and the nonproportional situations in the initial state but they came to discriminate little by little. Secondly, the students didn't discriminate the direct proportions and the inverse proportions until the last stage. Third, the multiplicative type was outperformed more than the formal type in solving the ratio and proportion problems but the formal type was outperformed more than the multiplicative type in discriminating between proportional situations and nonproportional situations. These results are interpreted as showing that solving ratio and proportion tasks and recognizing proportional situations are different aspects of proportional reasoning and it is necessary to understand multiplicative strategy with formal strategy in recognizing proportional situations.

• Journal of Elementary Mathematics Education in Korea
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• v.22 no.4
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• pp.385-403
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• 2018

Fraction division can be categorized as partitive division, measurement division, and the inverse of a Cartesian product. In the contexts of quotitive division and the inverse of a Cartesian product, the multiply-by-the-reciprocal algorithm is drawn well out. In this study, I analyze the potential and significance of the method of using $1{\div}$(divisor) as an alternative way of developing the multiply-by-the-reciprocal algorithm in the context of quotitive division. The method of using $1{\div}$(divisor) in quotitive division has the following advantages. First, by this method we can draw the multiply-by-the-reciprocal algorithm keeping connection with the context of quotitive division. Second, as in other contexts, this method focuses on the multiplicative relationship between the divisor and 1. Third, as in other contexts, this method investigates the multiplicative relationship between the divisor and 1 by two kinds of reasoning that use either ${\frac{1}{the\;denominator\;of\;the\;divisor}}$ or the numerator of the divisor as a stepping stone. These advantages indicates the potential of this method in understanding the multiply-by-the-reciprocal algorithm as the common structure of fraction division. This method is based on the dual meaning of a fraction as a quantity and the composition of times which the current elementary mathematics textbook does not focus on. It is necessary to pay attention to how to form this basis when developing teaching materials for fraction division.

• Communications of Mathematical Education
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• v.24 no.2
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• pp.345-363
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• 2010

The purpose of this study is to investigate and analyze informal knowledge of students who do not learn the conception of proportion and to identify how the informal knowledge can be used for teaching the conception of proportion in order to present an effective method of teaching the conception. For doing this, proportion was classified into direct and inverse proportion, and 'What are the informal knowledge of students?' were researched. The subjects of this study were 117 sixth-graders who did not have prior learning on direct and inverse proportion. A total eleven problems including seven for direct proportion and four for inverse proportion, all of them related to daily life. The result are as follows; Even though students didn't learn about proportion, they solve the problems of proportion using informal knowledge such as multiplicative reasoning, proportion reasoning, single-unit strategy etc. This result implies mathematics education emphasizes student's informal knowledge for improving their mathematical ability.

• Journal of Electrical Engineering and Technology
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• v.10 no.2
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• pp.487-495
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• 2015

This paper proposes a new robust decentralized power oscillation dampers (POD) design of doubly-fed induction generator (DFIG) wind turbine for damping of low frequency electromechanical oscillations in an interconnected power system. The POD structure is based on the practical $2^{nd}$-order lead/lag compensator with single input. Without exact mathematical model, the inverse output multiplicative perturbation is applied to represent system uncertainties such as system parameters variation, various loading conditions etc. The parameters optimization of decentralized PODs is carried out so that the stabilizing performance and robust stability margin against system uncertainties are guaranteed. The improved firefly algorithm is applied to tune the optimal POD parameters automatically. Simulation study in two-area four-machine interconnected system shows that the proposed robust POD is much superior to the conventional POD in terms of stabilizing effect and robustness.

• The Journal of Information Technology
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• v.7 no.2
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• pp.19-28
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• 2004

Many kind of operators using Residue Number System are used to design the special purpose processor for many merits in Digital Signal Processing, Computer Graphics, etc. But It get demerits for general division and the magnitude comparison. In this paper, general division operator for divisor with a small number in RNS is proposed. If the result of division using the multiplicative inverse has remainder, the quotient of this is larger than maximum quotient of division that has the same divisor to dividend of the maximum size. This condition is used for the ending condition of the recursive operation. And, the divisor is substitute for the compared value of quotients. So, the proposed division operator has a small size and fine operation speed, but with the limitation of divisor.

• IEMEK Journal of Embedded Systems and Applications
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• v.14 no.6
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• pp.313-319
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• 2019

Finite field operations have played an important role in error correcting codes and cryptosystems. Recently, the necessity of efficient computation processing is increasing for security in cyber physics systems. Therefore, efficient implementation of finite field arithmetics is more urgently needed. These operations include addition, multiplication, division and inversion. Addition is very simple and can be implemented with XOR operation. The others are somewhat more complicated than addition. Among these operations, multiplication is the most important, since time-consuming operations, such as exponentiation, division, and computing multiplicative inverse, can be performed through iterative multiplications. In this paper, we propose a multiplexer based parallel computation algorithm that performs Montgomery multiplication over finite field using redundant basis. Then we propose an efficient multiplexer based semi-systolic multiplier over finite field using redundant basis. The proposed multiplier has less area-time (AT) complexity than related multipliers. In detail, the AT complexity of the proposed multiplier is improved by approximately 19% and 65% compared to the multipliers of Kim-Han and Choi-Lee, respectively. Therefore, our multiplier is suitable for VLSI implementation and can be easily applied as the basic building block for various applications.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.15 no.12
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• pp.981-989
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• 1990

Thos paper summarized previously proposed several public key distribution systems and proposes a new public key distribution system to generate an common secret conference key for public key distribution systems three or more user. The now system is based on discrete exponentiation, that is all operations involve reduction modulo p for large prime p and we study some novel characteristics for computins multiplicative inverse in GF(p). We use one-way communication to distribute work keys, while the other uses two-way communication. The security of the new system is based on the difficulty of determining logarithms in a finite field GF(p) and stronger than Diffie-Hellman public key distribution system.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.12 no.2
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• pp.53-64
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• 2002

This paper describes a design of cryptographic processor that implements the AES (Advanced Encryption Standard) block cipher algorithm, "Rijndael". An iterative looping architecture using a single round block is adopted to minimize the hardware required. To achieve high throughput rate, a sub-pipeline stage is added by dividing the round function into two blocks, resulting that the second half of current round function and the first half of next round function are being simultaneously operated. The round block is implemented using 32-bit data path, so each sub-pipeline stage is executed for four clock cycles. The S-box, which is the dominant element of the round block in terms of required hardware resources, is designed using arithmetic circuit computing multiplicative inverse in GF($2^8$) rather than look-up table method, so that encryption and decryption can share the S-boxes. The round keys are generated by on-the-fly key scheduler. The crypto-processor designed in Verilog-HDL and synthesized using 0.25-$\mu\textrm{m}$ CMOS cell library consists of about 23,000 gates. Simulation results show that the critical path delay is about 8-ns and it can operate up to 120-MHz clock Sequency at 2.5-V supply. The designed core was verified using Xilinx FPGA board and test system.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.31 no.3
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• pp.527-532
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• 2021

In this paper, we propose an efficient finite field computation method for Rainbow algorithm, which is the only multivariate quadratic-equation based digital signature among the current US NIST PQC standardization Final List algorithms. Recently, Chou et al. proposed a new efficient implementation method for Rainbow on the Cortex-M4 environment. This paper proposes a new multiplication method over the finite field that can reduce the number of XOR operations by more than 13.7% compared to the Chou et al. method. In addition, a multiplicative inversion over that can be performed by a 4x4 matrix inverse instead of the table lookup method is presented. In addition, the performance is measured by porting the software to which the new method was applied onto RaspberryPI 3B+.

• The Mathematical Education
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• v.47 no.4
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• pp.519-543
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• 2008

The study has been conducted with 29 children from 4th to 6th grades to realize how they understand addition, subtraction, multiplication, and division of whole numbers, and how their understanding influences solving of one-step word problems. Children's understanding of operations was categorized into "adding" and "combination" for additions, "taking away" and "comparison" for subtractions, "equal groups," "rectangular arrange," "ratio," and "Cartesian product" for multiplications, and "sharing," "measuring," "comparison," "ratio," "multiplicative inverse," and "repeated subtraction" for divisions. Overall, additions were mostly understood additions as "adding"(86.2%), subtractions as "taking away"(86.2%), multiplications as "equal groups"(100%), and divisions as "sharing"(82.8%). This result consisted with the Fischbein's intuitive models except for additions. Most children tended to solve the word problems based on their conceptual structure of the four arithmetic operations. Even though their conceptual structure of arithmetic operations helps to better solve problems, this tendency resulted in wrong solutions when problem situations were not related to their conceptual structure. Children in the same category of understanding for each operations showed some common features while solving the word problems. As children's understanding of operations significantly influences their solutions to word problems, they needs to be exposed to many different problem situations of the four arithmetic operations. Furthermore, the focus of teaching needs to be the meaning of each operations rather than computational algorithm.