• Proceedings of the Korean Information Science Society Conference
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• 2000.10a
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• pp.593-595
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• 2000

본 논문에서는 최적확장체(Optimal Extension Field; OEF)에서 정의되는 타원곡선 상에서 효율적인 스칼라 곱셈 알고리즘을 제안한다. 이 스칼라 곱셈 알고리즘은 프로비니어스 사상(Frobenius map)을 이용하여 스칼라 값을 Horner의 방법으로 Base-Ф 전개하고, 이 전개된 수식을 일괄처리 기법(batch-processing technique)을 사용하여 연산한다. 이 알고리즘을 적용할 경우, Kobayashi 등이 제안한 스칼라 곱셈 알고리즘보다 40% 정도의 성능향상을 보인다.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.13 no.5
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• pp.169-177
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• 2003

We establish the security requirements and derive a generic condition of elliptic curve scalar multiplication to resist against DPA and Goubin’s attack. Also we show that if a scalar multiplication algorithm satisfies our generic condition, then both attacks are infeasible. Showing that the randomized signed scalar multiplication using Ha-Moon's receding algorithm satisfies the generic condition, we recommend the randomized signed scalar multiplication using Ha-Moon's receding algorithm to be protective against both attacks. Also we newly design a random recoding method to Prevent two attacks. Finally, in efficiency comparison, it is shown that the recommended method is a bit faster than Izu-Takagi’s method which uses Montgomery-ladder without computing y-coordinate combined with randomized projective coordinates and base point blinding or isogeny method. Moreover. Izu-Takagi’s method uses additional storage, but it is not the case of ours.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.13 no.3
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• pp.65-70
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• 2003

In this paper, we propose a scalar multiplication method and its hardware architecture which is resistant to SPA while its computation speed is faster than Colon's. There were SPA-resistant scalar multiplication method which has performance problem. Due to this reason, the research about an efficient SPA-resistant scalar multiplication is one of important topics. The proposed architecture resists to SPA and is faster than Colon's method under the assumption that Colon's and the proposed method use same fmite field arithmetic units(multiplier and inverter). With n-bit scalar multiple, the computation cycle of the proposed is 2n·(Inversion cycle)+3(Aultiplication cycle).

• Journal of KIISE:Computer Systems and Theory
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• v.31 no.10
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• pp.588-592
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• 2004

Scalar multiplication is to compute $textsc{k}$P when an integer $textsc{k}$ and an elliptic curve point f are given. As a general method to accelerate scalar multiplication, Agnew, Mullin and Vanstone proposed to use $textsc{k}$'s with fixed Hamming weights. We suggest a new method that uses $textsc{k}$'s with fixed signed Hamming weights and show that this method is more secure.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.17 no.2
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• pp.115-123
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• 2007

In cryptographic devices like a smart-card whose computing ability and memory are limited, cryptographic algorithms should be performed efficiently. Scalar multiplication is very important operation in Elliptic Curve Cryptosystems, and so must be constructed in safety against side channel attack(SCA). But several countermeasures proposed against SCA are exposed weaknesses by new un-dreamed analysis. 'Double-and-add always scalar multiplication' algorithm adding dummy operation being known to secure against SPA is exposed weakness by Doubling Attack. But Doubling Attack cannot apply to sABS receding proposed by Hedabou, that is another countermeasure against SPA. Our paper proposes new strengthened Doubling Attacks that can break sABS receding SPA-countermeasure and a detailed method of our attacks through experimental result.

• Proceedings of the Korea Institutes of Information Security and Cryptology Conference
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• 2001.11a
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• pp.132-138
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• 2001

본 논문에서는 타원곡선 스칼라 곱셈에 대하여 새로운 형태의 hardware fault cryptanalysis를 적용해 보고, 이에 대한 대응방법으로서 비밀키 blinding방법을 제안하고 있다. 또한 비밀키 blinding 방법을 사용함으로써 늘어나는 연산량을 기존의 대응 방법과 비교하고, 이러한 비밀키 blinding방법이 사용될 수 있는 범위에 대해 다루고 있다.

• 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 Security & Cryptology
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• v.22 no.5
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• pp.1019-1025
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• 2012

Fault attacks manipulate the computation of an algorithm and get information about the private key from the erroneous result. It is the most powerful attack for the cryptographic device. Currently, the research on error detection methods and fault attacks have been studied actively. S. Pontarelli et al. introduced an error detection method in 2009. It can detect an error that occurs during Elliptic Curve Scalar Multiplication (ECSM). In this paper, we present a new fault attack. Our attack can avoid the error detection method introduced by S. Pontarelli et al. We inject a bit flip error in the Euclidean Addition Chain (EAC) on the private key in ECSM and retrieve the private key.

• 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.

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.41 no.9
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• pp.73-84
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• 2004

This paper suggests a new scalar multiplication algerian to resist SPA which threatens the security of cryptographic primitive on the hardware recently, and discusses how to apply this algerian Our algorithm is better than other SPA countermeasure algorithms aspect to computational efficiency. Since known SPA countermeasure algorithms have dependency of computation. these are difficult to construct parallel architecture efficiently. To solve this problem our algorithm removes dependency and computes a multiplication and a squaring during inversion with parallel architecture in order to minimize loss of performance. We implement hardware logic with VHDL(VHSIC Hardware Description Language) to verify performance. Synthesis tool is Synplify Pro 7.0 and target chip is Xillinx VirtexE XCV2000EFGl156. Total equivalent gate is 60,508 and maximum frequency is 30Mhz. Our scalar multiplier can be applied to digital signature, encryption and decryption, key exchange, etc. It is applied to a embedded-micom it protects SPA and provides efficient computation.