• Journal of Elementary Mathematics Education in Korea
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• v.22 no.4
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• pp.369-384
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• 2018

Multiplication is able to be described by using repeated addition, a Cartesian product, a scalar operation, rectangular array and area in many various context. Multiplication in various problem situations is learned by various of the teaching method and the order of teaching more than any other mathematical concepts and operations in elementary school. Nevertheless, the context of multiplication leaves further room for improvement. The purpose of this study is to examine the similarities and differences between the conceptual aspects of multiplication through the literature and to analyze the appropriateness of the teaching method and the order of teaching through textbook analysis. As a result of the study, it was found that multiplication of a scalar operation was introduced too early and did not properly reflect of meaning of multiplication as a scalar operation. There is also a need to use the concept of the rectangular array or area as a meaning of multiplication two quantities.

• Proceedings of the IEEK Conference
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• 2004.06b
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• pp.529-532
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• 2004

The ECC(Elliptic Curve Cryptogrphics), one of the representative Public Key encryption algorithms, is used in Digital Signature, Encryption, Decryption and Key exchange etc. The key operation of an Elliptic curve cryptosystem is a scalar multiplication, hence the design of a scalar multiplier is the core of this paper. Although an Integer operation is computed in infinite field, the scalar multiplication is computed in finite field through adding points on Elliptic curve. In this paper, we implemented scalar multiplier in Elliptic curve based on the finite field GF($2^{163}$). And we verified it on the Embedded digital system using Xilinx FPGA connected to an EISC MCU. If my design is made as a chip, the performance of scalar multiplier applied to Samsung $0.35 {\mu}m$ Phantom Cell Library is expected to process at the rate of 8kbps and satisfy to make up an encryption processor for the Embedded digital doorphone.

• Proceedings of the IEEK Conference
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• 2002.06b
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• pp.345-348
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• 2002

This paper presents new fast scalar multiplier of elliptic curve cryptosystem that is regarded as next generation public-key crypto processor. For fast operation of scalar multiplication a finite field multiplier is designed with LFSR type of bit serial structure and a finite field inversion operator uses extended binary euclidean algorithm for reducing one multiplying operation on point operation. Also the use of the window non-adjacent form (WNAF) method can reduce addition operation of each other different points.

• 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.

• Proceedings of the Korea Information Processing Society Conference
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• 2005.05a
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• pp.1071-1074
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• 2005

The ECC(Elliptic Curve Cryptogrphics), one of the representative Public Key encryption algorithms, is used in Digital Signature, Encryption, Decryption and Key exchange etc. The key operation of an Elliptic curve cryptosystem is a scalar multiplication, hence the design of a scalar multiplier is the core of this paper. Although an Integer operation is computed in infinite field, the scalar multiplication is computed in finite field through adding points on Elliptic curve. In this paper, we implemented scalar multiplier in Elliptic curve based on the finite field $GF(2^{163})$. And we verified it on the Embedded digital system using Xilinx FPGA connected to an EISC MCU(Agent 2000). If my design is made as a chip, the performance of scalar multiplier applied to Samsung $0.35\;{\mu}m$ Phantom Cell Library is expected to process at the rate of 8kbps and satisfy to make up an encryption processor for the Embedded digital information home system.

• The Journal of the Korea Contents Association
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• v.20 no.1
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• pp.356-363
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• 2020

As a becoming era of Internet-of-Things, various devices are connected via wire or wirless networks. Although every day life is more convenient, security problems are also increasing such as privacy, information leak, denial of services. Since ECC, a kind of public key cryptosystem, has a smaller key size compared to RSA, it is widely used for environmentally constrained devices. The key of ECC in constrained devices can be exposed to power analysis attacks during scalar multiplication operation. In this paper, a key-randomization method is suggested for scalar multiplication on SECG parameters. It is against differential power analysis and has operational efficiency. In order to increase of operational efficiency, the proposed method uses the property 2^lP=∓cP where the constant c is small compared to the order n of SECG parameters and n=2^l±c. The number of operation for the Coron's key-randomization scalar multiplication algorithm is 21, but the number of operation for the proposed method in this paper is (3/2)l. It has efficiency about 25% compared to the Coron's method using full random numbers.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.26 no.4
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• pp.548-555
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• 2022

This paper describes the hardware implementation of elliptic curve cryptography (ECC) used as a core operation in elliptic curve digital signature algorithm (ECDSA). The ECC processor supports eight operation modes (four point operations, four modular operations) on the NIST P-521 curve. In order to minimize computation complexity required for point scalar multiplication (PSM), the radix-4 Booth encoding scheme and modified Jacobian coordinate system were adopted, which was based on the complexity analysis for five PSM algorithms and four different coordinate systems. Modular multiplication was implemented using a modified 3-Way Toom-Cook multiplication and a modified fast reduction algorithm. The ECC processor was implemented on xczu7ev FPGA device to verify hardware operation. Hardware resources of 101,921 LUTs, 18,357 flip-flops and 101 DSP blocks were used, and it was evaluated that about 370 PSM operations per second were achieved at a maximum operation clock frequency of 45 MHz.

• JSTS:Journal of Semiconductor Technology and Science
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• v.16 no.1
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• pp.118-125
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• 2016

This paper presents a new high-efficient algorithm and architecture for an elliptic curve cryptographic processor. To reduce the computational complexity, novel modified Lopez-Dahab scalar point multiplication and left-to-right algorithms are proposed for point multiplication operation. Moreover, bit-serial Galois-field multiplication is used in order to decrease hardware complexity. The field multiplication operations are performed in parallel to improve system latency. As a result, our approach can reduce hardware costs, while the total time required for point multiplication is kept to a reasonable amount. The results on a Xilinx Virtex-5, Virtex-7 FPGAs and VLSI implementation show that the proposed architecture has less hardware complexity, number of clock cycles and higher efficiency than the previous works.

• Journal of IKEEE
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• v.25 no.1
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• pp.174-179
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• 2021

An Edwards curve cryptography (EdCC) core supporting point scalar multiplication (PSM) on Edwards curves of Edwards25519 and Edwards448 was designed. For area-efficient implementation, finite field multiplier based on word-based Montgomery multiplication algorithm was designed, and the extended twisted Edwards coordinates system was adopted to implement point operations without division operation. As a result of synthesizing the EdCC core with 100 MHz clock, it was implemented with 24,073 equivalent gates and 11 kbits RAM, and the maximum operating frequency was estimated to be 285 MHz. The evaluation results show that the EdCC core can compute 299 and 66 PSMs per second on Edwards25519 and Edwards448 curves, respectively. Compared to the ECC core with similar structure, the number of clock cycles required for 256-bit PSM was reduced by about 60%, resulting in 7.3 times improvement in computational performance.

• Journal of the Korean School Mathematics Society
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• v.22 no.3
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• pp.303-327
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• 2019

This study proposed an integrated approach to multiplication as a way to help students understand multiplication in elementary mathematics. The integrated approach to multiplication is to give students a broad understanding of multiplication by solving a situation of multiplication in a variety of ways in mathematics classes, exploring and discussing each other's methods. The integrated approach to multiplication was derived from a number of previous studies that emphasized various approaches, a consistent approach, and a specific approach to multiplication. As results, the integrated approach of multiplication can be interpreted in four ways as a situation of multiplication, and each method is connected to important characteristics of multiplication emphasized in previous studies. In addition, this study has theoretically confirmed that the integrated approach to multiplication is important not only for multiplication but also for division, fraction and operation of fractions, ratios, rates, and proportions. This study is expected to provide some implications for teachers with regard to multiplication in elementary school mathematics.