• Journal of the Korea Institute of Information Security & Cryptology
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• v.12 no.5
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• pp.45-51
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• 2002

As Koblitz curve, the Frobenius endomorphism is know to be useful in efficient implementation of multiplication on non-supersingular elliptic cures defined on small finite fields of characteristic two. In this paper a method using the extended Frobenius endomorphism to speed up scalar multiplication is introduced. It will be shown that the proposed method is more efficient than Muller's block method in ［5］ because the number of point addition for precomputation is small but on the other hand the expansion length is almost same.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.5 no.1
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• pp.17-24
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• 1995

In spite of the good security of the cryptosystem on an elliptic curve defined over finite field, the cryptosystem on an elliptic curve is slower than that on a finite field. To be practical, we need a better method to improve a speed of the cryptosystem on an elliptic curve defined over a finite field. In 1991, Koblitz suggested to use an anomalous curve over $F_2$, which is an elliptic curve with Frobenious map whose trace is 1, and reduced a speed of computation of mP. In this paper, we consider an elliptic curve defined over $F_4$ with Frobenious map whose trace is 3 and suggest an efficient algorithm to compute mP. On the proposed elliptic curve, we can compute multiples mP with ${\frac{3}{2}}log_2m$+1 addition in worst case.

• Journal of the Korean Mathematical Society
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• v.58 no.1
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• pp.123-132
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• 2021

Let K be a number field and L a finite abelian extension of K. Let E be an elliptic curve defined over K. The restriction of scalars ResKLE decomposes (up to isogeny) into abelian varieties over K $$Res^L_KE{\sim}{\bigoplus_{F{\in}S}}A_F,$$ where S is the set of cyclic extensions of K in L. It is known that if L is a quadratic extension, then AL is the quadratic twist of E. In this paper, we consider the case that K is a number field containing a primitive third root of unity, $L=K({\sqrt[3]{D}})$ is the cyclic cubic extension of K for some D ∈ K×/(K×)3, E = Ea : y2 = x3 + a is an elliptic curve with j-invariant 0 defined over K, and EaD : y2 = x3 + aD2 is the cubic twist of Ea. In this case, we prove AL is isogenous over K to $E_a^D{\times}E_a^{D^2}$ and a property of the Selmer rank of AL, which is a cubic analogue of a theorem of Mazur and Rubin on quadratic twists.

• Review of KIISC
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• v.10 no.2
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• pp.21-32
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• 2000

본 고에서는 타원곡선 알고리즘의 표준화 동향을 국내외 표준들을 바탕으로 살펴보았다 먼저 ISO/IEC JTC1/SC27/WG2 Information technology-Security techniques-Cryptographic techniques based on elliptic curves 문서를 바탕으로 국제표준에 대해서 자세히 살펴보았으며 IEEE P1363 ANSI X9.62/X9.63 에 대해서 간략히 살펴보았다 또한 타원곡선 알고리즘과 관련된 국내 표준화 활동에 대해서도 살펴보았다.

• Communications of the Korean Mathematical Society
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• v.20 no.1
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• pp.107-116
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• 2005

We show an algorithm to draw the famous picture of Kleinian modular group PSL(2, $\mathbb{Z}$), which appears in describing the moduli space of elliptic curves.

• Honam Mathematical Journal
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• v.30 no.1
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• pp.47-51
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• 2008

By finding two characters of subgroups of the Heisenberg group $H_{27}$ whose induced characters are same, we prove that the Birch and Swinnerton-Dyer conjectures for two elliptic curves corresponding to the characters are equivalent.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.2 no.1
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• pp.45-48
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• 1992

제목에 주어진 타원 곡선들 $Y^{2}$=X$^{3}$+DX 은 허수 승법을 가진다. 이곡선을 소수 증명과 소인수 분해에 응용하는 방법을 이 논문에서 고찰해 본다.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.24 no.11
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• pp.1470-1476
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• 2020

This paper describes a design of a lightweight security system-on-chip (SoC) suitable for the implementation of security protocols for IoT and mobile devices. The security SoC using Cortex-M0 as a CPU integrates hardware crypto engines including an elliptic curve cryptography (ECC) core, a SHA3 hash core, an ARIA-AES block cipher core and a true random number generator (TRNG) core. The ECC core was designed to support twenty elliptic curves over both prime field and binary field defined in the SEC2, and was based on a word-based Montgomery multiplier in which the partial product generations/additions and modular reductions are processed in a sub-pipelining manner. The H/W-S/W co-operation for elliptic curve digital signature algorithm (EC-DSA) protocol was demonstrated by implementing the security SoC on a Cyclone-5 FPGA device. The security SoC, synthesized with a 65-nm CMOS cell library, occupies 193,312 gate equivalents (GEs) and 84 kbytes of RAM.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.31 no.1
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• pp.19-30
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• 2021

In this paper, when SIDH is instantiated using only 3- and 5-isogeny, we demonstrate which curve is more efficient among the Montgomery, Edwards, and Huff curves. To this end, we present the computational cost of the building blocks of SIDH on Montgomery, Edwards, and Huff curves. We also present the prime we used and parameter settings for implementation. The result of our work shows that the performance of SIDH on Montgomery and Huff curves is almost the same and they are 0.8% faster than Edwards curves. With the possibility of using isogeny of degree other than 3 and 4, the performance of 5-isogeny became even more essential. In this regard, this paper can provide guidelines on the selection of the form of elliptic curves for implementation.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2022.05a
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• pp.246-248
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• 2022

Binary Edwards curves (BEdC), a new form of elliptic curves proposed by Bernstein, satisfy the complete addition law without exceptions. This paper describes an efficient hardware implementation of point scalar multiplication on BEdC using projective coordinates. Modified Montgomery ladder algorithm was adopted for point scalar multiplication, and binary field arithmetic operations were implemented using 257-bit binary adder, 257-bit binary squarer, and 32-bit binary multiplier. The hardware operation of the BEdC crypto-core was verified using Zynq UltraScale+ MPSoC device. It takes 521,535 clock cycles to compute point scalar multiplication.