• Journal of the Korea Institute of Information and Communication Engineering
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• v.21 no.7
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• pp.1267-1275
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• 2017

This paper describes a design of cryptographic processor supporting 233-bit elliptic curves over binary field defined by NIST. Scalar point multiplication that is core arithmetic in elliptic curve cryptography(ECC) was implemented by adopting modified Montgomery ladder algorithm, making it robust against simple power analysis attack. Point addition and point doubling operations on elliptic curve were implemented by finite field multiplication, squaring, and division operations over $GF(2^{233})$, which is based on affine coordinates. Finite field multiplier and divider were implemented by applying shift-and-add algorithm and extended Euclidean algorithm, respectively, resulting in reduced gate counts. The ECC processor was verified by FPGA implementation using Virtex5 device. The ECC processor synthesized using a 0.18 um CMOS cell library occupies 49,271 gate equivalents (GEs), and the estimated maximum clock frequency is 345 MHz. One scalar point multiplication takes 490,699 clock cycles, and the computation time is 1.4 msec at the maximum clock frequency.

• The Journal of the Korea institute of electronic communication sciences
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• v.12 no.4
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• pp.621-628
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• 2017

The efficiency of implementation of the arithmetic operations in finite fields depends on the choice representation of elements of the field. It seems that from this point of view normal bases are the most appropriate, since raising to the power 2 in $GF(2^n)$ of characteristic 2 is reduced in these bases to a cyclic shift of the coordinates. We, in this paper, introduce our algorithm to transform fastly the conventional bases to normal bases and present the result of H/W implementation using the algorithm. We also propose our algorithm to calculate the multiplication and inverse of elements with respect to normal bases in $GF(2^n)$ and present the programs and the results of H/W implementations using the algorithm.

• Nuclear Engineering and Technology
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• v.48 no.5
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• pp.1126-1139
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• 2016

In this research, for the first time, a new optimization method, i.e., strength Pareto evolutionary algorithm II (SPEA-II), is developed for the burnable poison placement (BPP) optimization of a nuclear reactor core. In the BPP problem, an optimized placement map of fuel assemblies with burnable poison is searched for a given core loading pattern according to defined objectives. In this work, SPEA-II coupled with a nodal expansion code is used for solving the BPP problem of Kraftwerk Union AG (KWU) pressurized water reactor. Our optimization goal for the BPP is to achieve a greater multiplication factor ($K_{eff}$) for gaining possible longer operation cycles along with more flattening of fuel assembly relative power distribution, considering a safety constraint on the radial power peaking factor. For appraising the proposed methodology, the basic approach, i.e., SPEA, is also developed in order to compare obtained results. In general, results reveal the acceptance performance and high strength of SPEA, particularly its new version, i.e., SPEA-II, in achieving a semioptimized loading pattern for the BPP optimization of KWU pressurized water reactor.

• Proceedings of the Korea Institutes of Information Security and Cryptology Conference
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• 1995.11a
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• pp.173-182
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• 1995

모듈라 멱승(modular exponentiation) 연산은 암호학에서 기본적이고 중요한 연산이다. 그러나, 이는 다정도 정수(multiple precision integer)들을 다루기 때문에 그 연산시 간이 무척 많이 걸리므로 이를 단축시킬 필요가 있다. 모듈라 멱승 연산은 모듈라 곱셈(modular multiplication)의 반복으로서, 전체 연산시간을 단축시키기 위해서는 모듈라 곱셈의 수행시간을 단축시키거나, 모듈라 곱셈의 반복횟수를 줄이는 것이 필요하다. 본 논문에서는 모듈라 곱셈을 빠르게 수행하기 위한 알고리즘 두 개를 제안한다. 하나는 서로 다른 두 수의 모듈라 곱셈 알고리즘이고, 다른 하나는 모듈라 제곱을 빠르게 수행하는 알고리즘이다. 이 둘은 기존의 모듈라 곱셈 알고리즘들에 비해 각각 절반과, l/3가량의 단정도 곱셈(single-precision multiplication)만을 필요로 한다. 실제로 PC상에서 구현한 결과 각각 100%와 30%의 속도향상을 보인다.

• Proceedings of the IEEK Conference
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• 2002.07a
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• pp.500-503
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• 2002

Biomedical signals are ubiquitously contaminated and degraded by background noise which span nearly all frequency bandwidths. This paper proposes the MADF (multiplication free adaptive digital filter) algorithm to cancel the noise. And the convergence characteristics of the algorithm is analyzed. In the experimental results, the MADF algorithm has the advantage in which has superior to a condition of low-frequency and slow data speed. This application gives an important significance in ensuring the objectivity of clinical information and in promoting the representation and the disease diagnosis.

• Journal of the Korean Institute of Telematics and Electronics B
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• v.32B no.1
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• pp.180-188
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• 1995

A modified Kohonen's self-organizing feature map (SOFM) algorithm which has binary reinforcement function and a constant learning rate is proposed. In contrast to the time-varing adaptaion gain of the original Kohonen's SOFM algorithm, the proposed algorithm uses a constant adaptation gain, and adds a binary reinforcement function in order to compensate for the lowered learning ability of SOFM due to the constant learning rate. Since the proposed algorithm does not have the complicated multiplication, it's digital hardware implementation is much easier than that of the original SOFM.

• Education of Primary School Mathematics
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• v.22 no.4
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• pp.281-294
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• 2019

The purpose of this study is to analyze errors and treatment behavior during the course of mathematics learning of academic achievement by applying reflective activities in the second semester of the third year of elementary school. The study participants are students from two classes, 21 from the third-grade S elementary school in Seoul and 20 from the comparative class. In the case of the experiment group, the multiplication unit was reconstructed into a mathematics class that applied reflective activities. They were pre-post-test to examine the changes in students' mathematics performance, and mathematical communication was recorded and analyzed for the focus group to analyze the patterns of learners' error handling in the reflective activities. In addition, they recorded and analyzed students' activities and conversations for error type and error handling. As a result of the study, the student's mathematics achievement was increased using reflective activities. When learning double digit multiplication, the error types varied. It was also confirmed that the reflective activities helped learners reflect on the multiplication algorithm and analyze the error-ridden calculations to reflect on and deal with their errors.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.15 no.7
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• pp.2610-2630
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• 2021

The aim of this paper is to propose the alternative algorithm to finish the process in public key cryptography. In general, the proposed method can be selected to finish both of modular exponentiation and point multiplication. Although this method is not the best method in all cases, it may be the most efficient method when the condition responds well to this approach. Assuming that the binary system of the exponent or the multiplier is considered and it is divided into groups, the binary system is in excellent condition when the number of groups is small. Each group is generated from a number of 0 that is adjacent to each other. The main idea behind the proposed method is to convert the exponent or the multiplier as the subtraction between two integers. For these integers, it is impossible that the bit which is equal to 1 will be assigned in the same position. The experiment is split into two sections. The first section is an experiment to examine the modular exponentiation. The results demonstrate that the cost of completing the modular multiplication is decreased if the number of groups is very small. In tables 7 - 9, four modular multiplications are required when there is one group, although number of bits which are equal to 0 in each table is different. The second component is the experiment to examine the point multiplication process in Elliptic Curves Cryptography. The findings demonstrate that if the number of groups is small, the costs to compute point additions are low. In tables 10 - 12, assigning one group is appeared, number of point addition is one when the multiplier of a point is an even number. However, three-point additions are required when the multiplier is an odd number. As a result, the proposed method is an alternative way that should be used when the number of groups is minimal in order to save the costs.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.30 no.2
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• pp.179-187
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• 2020

The FBC(Fixed-Base Comb) is a method to efficiently operate scalar multiplication, a core operation for signature generations of the ECDSA(Elliptic Curve Digital Signature Algorithm), utilizing precomputed lookup tables. Since the FBC refers to the table depending on the secret information and the values of the table are publicly known, an adversary can perform HCA(Horizontal Correlation Analysis), one of the single trace side channel attacks, to reveal the secret. However, HCA is a statistical analysis that requires a sufficient number of unit operation traces extracted from one scalar multiplication trace for a successful attack. In the case of the scalar multiplication for signature generations of ECDSA, the number of unit operation traces available for HCA is significantly fewer than the case of the RSA exponentiation, possibly resulting in an unsuccessful attack. In this paper, we propose an improved HCA on lookup table based scalar multiplication algorithms such as FBC. The proposed attack improves HCA by increasing the number of unit operation traces by determining such traces for the same intermediate value through collision analysis. The performance of the proposed attack increases as more secure elliptic curve parameters are used.

• Journal of the Chungcheong Mathematical Society
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• v.15 no.2
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• pp.85-96
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• 2003

An efficient algorithm for the multiplication in a binary finite filed using a normal basis representation of $F_{2^m}$ is discussed and proposed for software implementation of elliptic curve cryptography. The algorithm is developed by using the storage scheme of sparse matrices.