• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2004.05b
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• pp.784-787
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• 2004

The most time-consuming back-bone operation in an elliptic curve cryptosystem is scalar multiplication. In this paper, we propose a method of inducing a GF operation named point quadruple operation to be used in the quad-and-add algorithm, whith was achieved by refining the traditional double-and-add algorithm. Induced expression of the algorithm was verified and proven by C program in a real model of calculation. The point quadruple operation can be used in fast and efficient implementation of scalar multiplication operation.

• Journal of KIISE:Computer Systems and Theory
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• v.34 no.3
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• pp.93-100
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• 2007

Since Koblitz and Miller suggested the use of elliptic curves in cryptography, there has been an extensive literature on elliptic curve cryptosystem (ECC). The use of ECC is based on the observation that the points on an elliptic curve form an additive group under point addition operation. To realize secure cryptosystems using these groups, it is very important to find an elliptic curve whose group order is divisible by a large prime, and also to find a base point whose order equals this prime. While there have been many dramatic improvements on finding an elliptic curve and computing its group order efficiently, there are not many results on finding an adequate base point for a given curve. In this paper, we propose an efficient method to find a random base point on an elliptic curve defined over $GF(p^m)$. We first show that the critical operation in finding a base point is exponentiation. Then we present efficient algorithms to accelerate exponentiation in $GF(p^m)$. Finally, we implement our algorithms and give experimental results on various practical elliptic curves, which show that the new algorithms make the process of searching for a base point 1.62-6.55 times faster, compared to the searching algorithm based on the binary exponentiation.

• Proceedings of the Korean Society of Broadcast Engineers Conference
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• 2008.02a
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• pp.69-72
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• 2008

최근 PKI-less 공개키 암호 시스템에 대한 연구가 진척되면서, 페어링(Pairing) 기반의 암호 시스템이 주목을 받고 있다. 페어링 기반의 암호 시스템은 두 개의 타원 곡선 상의 점을 유한체의 값으로 보내는 양방향 선형성(Bilinearity)을 가지는 페어링 함수를 기반으로 구성되는 암호 시스템이다. 페어링 기반의 암호 시스템 구현을 위해서는 페어링 연산 알고리즘이 필수적이며, 효율적인 페어링 연산을 위한 많은 연구가 진행되고 있다. 이러한 페어링 알고리즘에도 기존의 타원곡선 스칼라곱 알고리즘에서 야기되었던 부채널 공격이 동일하게 적용되기 때문에, 안전한 페어링 알고리즘을 위해서는 부채널 공격에 대한 저항성을 갖는 알고리즘이 필요하다. 이에 본 논문에서는 부채널 공격에도 안전하면서 비교적 효율적인 이진 필드 상의 페어링 알고리즘을 제시한다. 본 페어링 알고리즘은 기존의 부체널 공격 저항성을 갖는 페어링 알고리즘 중 가장 효율적인 알고리즘에 비해 효율성이 17% 정도 향상되었다.

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.40 no.12
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• pp.80-86
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• 2003

Elliptic curve cryptosystem(ECC) offers the highest security per bit among the known publick key system. The benefit of smaller key size makes ECC particularly attractive for embedded applications since its implementation requires less memory and processing power. In this paper, we propose a new multiplier structure with configurable output sizes and operation cycles. The number of output bits can be freely chosen in the new architecture with the performance-area trade-off depending on the application. Using the architecture, a 193-bit normal basis multiplier and inversion unit are designed in GF(2$^{m}$ ). It is implemented using HDL and 0.35${\mu}{\textrm}{m}$ CMOS technology and the operation is verified by simulation.

• 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.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2008.05a
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• pp.670-673
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• 2008

Optimized algorithms and numerical expressions which had been verified through C program simulation, should be analyzed again with HDL (hardware description language) such as Verilog, so that the verified ones could be modified to be applied directly to hardware implementation. The reason is that the characteristics of C programming language design is intrinsically different from the hardware design structure. The hardware IP verified doubly in view of hardware structure together with algorithmic verification, was implemented on the Altera Excalibur FPGA device equipped with ARM9 microprocessor core, to a real chip prototype, using Altera embedded system development tool kit. The implemented finite field calculation IPs can be used as library modules as Elliptic Curve Cryptography finite field operations which has more than 193 bit key length.

• Proceedings of the Korea Institutes of Information Security and Cryptology Conference
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• 1998.12a
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• pp.455-468
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• 1998

암호해독법은 암호시스템의 안전성을 논하는데 필수적이다. 본 논문에서는 ECDLP 공격법인 Pollard-$\rho$와 그 변형들간의 성능을 유한체 GF(2$^{19}$ ) ~ GF(2$^{41}$ ) 상의 타원곡선에서 측정 비교하였다. 또한 이 공격법을 네트웍을 통해 10대의 컴퓨터로 병렬처리해 공격시간을 1/10로 단축시켰으며 실험 데이타를 토대로 GF(2$^{163}$ )상에서 공격시간 및 저장용량을 예측하였다.

• Journal of the Korea Society of Computer and Information
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• v.8 no.4
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• pp.104-110
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• 2003

In this paper, we proposed hybrid cryptosystem of Diffie-Hellman base in Elliptic Curve, and explained for specific protocol design. The proposed system is efficient hybrid cryptosystems system that offer implicit key authentication about sender and receiver unlike existing hybrid system. This system increased safety generating session key using pseudo-random number generator by cryptographic. Because the system is hybrid system, it is more efficient in calculation amount aspect supplementing merit and fault of public key system and secret key system. Also, the system can not get right plaintext except receiver even if sender's secret key is revealed and impersonation attack is impossible. And the system offers security on known keys without influencing in safety of other session's cryptogram even if session key is exposed. And the system is provided safety about mutual entity authentication and replay attack.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.14 no.3
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• pp.1439-1444
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• 2013

Recently, as the use of the applications like online banking and stock trading is increasing by the rapid development of the network, security of data content is becoming more and more important. Accordingly, public key or symmetric key encryption algorithm is widely used in open networks such as the internet for the protection of data. Generally, public key cryptographic systems is based on two famous number theoretic problems namely factoring or discrete logarithm problem. So, public key cryptographic systems is relatively slow compared to symmetric key cryptography systems. Among public key cryptographic systems, the advantage of ECC compared to RSA is that it offers equal security for a far smaller key. For this reason, ECC is faster than RSA. In this paper, we propose a efficient key generation method for elliptic curve cryptography system based on the real number field.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.8 no.6
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• pp.1212-1217
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• 2004

The main back-bone operation in elliptic curve cryptosystems is scalar point multiplication. The most frequently used method implementing the scalar point multiplication, which is performed in the upper level of GF multiplication and GF division, has been the double-and-add algorithm, which is recently challenged by NAF(Non-Adjacent Format) algorithm. In this paper, we propose a more efficient and novel scalar multiplication method than existing double-and-add by applying redundant receding which originates from radix-4 Booth's algorithm. After deriving the novel quad-and-add algorithm, we created a new operation, named point quadruple, and verified with real application calculation to utilize it. Derived numerical expressions were verified using both C programs and HDL (Hardware Description Language) in real applications. Proposed method of elliptic curve scalar point multiplication can be utilized in many elliptic curve security applications for handling efficient and fast calculations.