• 대한수학회보
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• 제44권1호
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• pp.47-59
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• 2007

We find the uniformizers of modular curves $X_{1}(N)\;(N=2,3)$ and explore the relationship with Thompson series and number theoretic property.

• 대한수학회논문집
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• 제23권4호
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• pp.479-485
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• 2008

In this note, we study arithmetic properties for the exponents of modular forms on ${\Gamma}_0(p)$ for primes p. Our aim is to refine the result of [4] by using the geometric property of the modular curve of ${\Gamma}_0(p)$.

• 전기전자학회논문지
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• 제24권4호
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• pp.961-968
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• 2020

타원곡선 암호에 필수적으로 사용되는 모듈러 곱셈의 고성능 하드웨어 설계에 대해 기술한다. 본 논문의 모듈러 곱셈기는 NIST FIPS 186-2에 정의된 소수체 상의 5가지 체 크기(192, 224, 256, 384, 521 비트)의 모듈러 곱셈을 지원하며, 정수 곱셈과 축약의 두 단계 과정으로 모듈러 곱셈을 연산한다. 고속 정수 곱셈을 위해 카라추바-오프만 곱셈 알고리듬이 사용되었고, 축약 연산을 위해 Lazy 축약 알고리듬이 사용되었다. 또한, Lazy 축약에 포함된 나눗셈 연산을 위해 Nikhilam 나눗셈 알고리듬이 사용되었으며, 나눗셈 연산은 주어진 모듈러 값에 대해 처음 한 번만 연산되고, 모듈로 값이 고정된 상태로 연속적인 모듈러 곱셈이 수행되는 경우에는 나눗셈을 거치지 않도록 하였다. 설계된 모듈러 곱셈기는 32 MHz의 클록 주파수로 동작하는 경우에 초당 640만번의 모듈러 곱셈을 연산할 수 있는 것으로 평가되었으며, 180-nm CMOS 셀 라이브러리로 합성한 결과, 67 MHz의 클록 주파수로 동작이 가능하며, 456,400 등가 게이트로 구현되었다.

• 충청수학회지
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• 제26권3호
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• pp.641-645
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• 2013

In this article, we show that the modular equation ${\Phi}_p^{T_q}(X,Y)$ of Thompson series $T_q$ corresponding to ${\Gamma}_0(q)+q$ gives an affine model of a modular curve $X_0(pq)+q$.

• JSTS:Journal of Semiconductor Technology and Science
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• 제17권3호
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• pp.425-437
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• 2017

For efficient hardware (HW) implementation of elliptic curve cryptography (ECC), various sub-modules for the underlying finite field operations should be implemented efficiently. Among these sub-modules, modular inversion (MI) requires the most computation; therefore, its performance might be a dominant factor of the overall performance of an ECC module. To determine the most efficient MI algorithm for an HW ECC module, we implement various classes of MI algorithms and analyze their performance. In contrast to the common belief in previous research, our results show that the right-shift binary inversion (RS) algorithm performs well when implemented in hardware. In addition, we present optimization methods to reduce the area overhead and improve the speed of the RS algorithm. By applying these methods, we propose a new RS-variant that is both fast and compact. The proposed MI module is more than twice as fast as the other two classes of MI: shifting Euclidean (SE) and left-shift binary inversion (LS) algorithms. It consumes only 15% more area and even 5% less area than SE and LS, respectively. Finally, we show that how our new method can be applied to optimize an HW ECC module.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제14권2호
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• pp.738-756
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• 2020

The modular multiplication is the key module of public-key cryptosystems such as RSA (Rivest-Shamir-Adleman) and ECC (Elliptic Curve Cryptography). However, the efficiency of the modular multiplication, especially the modular square, is very low. In order to reduce their operation cycles and power consumption, and improve the efficiency of the public-key cryptosystems, a dual-field efficient FIPS (Finely Integrated Product Scanning) modular multiplication algorithm is proposed. The algorithm makes a full use of the correlation of the data in the case of equal operands so as to avoid some redundant operations. The experimental results show that the operation speed of the modular square is increased by 23.8% compared to the traditional algorithm after the multiplication and addition operations are reduced about (s² - s) / 2, and the read operations are reduced about s² - s, where s = n / 32 for n-bit operands. In addition, since the algorithm supports the length scalable and dual-field modular multiplication, distinct applications focused on performance or cost could be satisfied by adjusting the relevant parameters.

• 대한수학회지
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• 제36권4호
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• pp.707-723
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• 1999

Since the modular curve $X(4)=\Gamma(4)/{\mathfrak{}}^*$ has genus 0, we have a field isomorphism K(X(4)){\approx}\mathcal{C}(j_{4})$ where $j_{4}(z)={\theta}_{3}(\frac{z}{2})/{\theta}_{4}(\frac{z}{2})$ is a quotient of Jacobi theta series ([9]). We derive recursion formulas for the Fourier coefficients of $j_4$ and $N(j_{4})$ (=the normalized generator), respectively. And we apply these modular functions to Thompson series and the construction of class fields.

• 한국정보통신학회논문지
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• 제26권4호
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• pp.548-555
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• 2022

본 논문은 타원곡선 디지털 서명 알고리듬 (Elliptic Curve Digital Signature Algorithm; ECDSA)의 핵심 연산으로 사용되는 타원곡선 암호 (Elliptic Curve Cryptography; ECC)의 하드웨어 구현에 대해 기술한다. 설계된 ECC 프로세서는 NIST P-521 곡선 상의 8가지 연산 모드 (점 연산 4가지, 모듈러 연산 4가지)를 지원한다. 점 스칼라 곱셈 (PSM)에 필요한 연산량을 최소화하기 위해 5가지 PSM 알고리듬과 4가지 좌표계에 따른 연산 복잡도 분석을 토대로 radix-4 Booth 인코딩과 수정된 자코비안 좌표계를 적용하여 설계하였다. 모듈러 곱셈은 수정형 3-Way Toom-Cook 정수 곱셈과 수정형 고속 축약 알고리듬을 적용하여 구현되었다. 설계된 ECC 프로세서는 xczu7ev FPGA 디바이스에 구현하여 하드웨어 동작을 검증하였다. 101,921개의 LUT와 18,357개의 플립플롭 그리고 101개의 DSP 블록이 사용되었고, 최대 동작주파수 45 MHz에서 초당 약 370번의 PSM 연산이 가능한 것으로 평가되었다.

• 한국전산유체공학회:학술대회논문집
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• 한국전산유체공학회 2010년 춘계학술대회논문집
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• pp.181-186
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• 2010

A numerical study was conducted to predict the performance curve of a canned motor pump for SMART(System Integrated Modular Advanced ReacTor). The study used a computational domain which included not only the pump but also a suction pipe and a volute casing with a discharging pipe in order to simulate an experimental setup. The ANSYS CFX program was utilized to obtain flow characteristics inside the pump as well as the overall pressure rise across the pump operating on- and off-design points. Computed results showed that the performance of the pump at off-design points was much lower than expected. Special attention was made to find the cause of the low performance of the pump operating at low flow rate.

• 충청수학회지
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• 제23권2호
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• pp.267-270
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• 2010

In this work, we determine all the curves $X^+_1(N)$ which are sub-hyperelliptic.