• Communications of the Korean Mathematical Society
• /
• v.12 no.3
• /
• pp.553-560
• /
• 1997

A class number formula over global function field, extending the formula obtained by Shu, is proved.

• Journal of KIISE:Computer Systems and Theory
• /
• v.31 no.8
• /
• pp.453-464
• /
• 2004

In order to solve the well-known drawback of reduced flexibility that is associate with ASIC implementations, this paper proposes a novel arithmetic unit over GF(2$^{m}$ ) for field programmable gate arrays (FPGAs) implementations of elliptic curve cryptographic processor. The proposed arithmetic unit is based on the binary extended GCD algorithm and the MSB-first multiplication scheme, and designed as systolic architecture to remove global signals broadcasting. The proposed architecture can perform both division and multiplication in GF(2$^{m}$ ). In other word, when input data come in continuously, it produces division results at a rate of one per m clock cycles after an initial delay of 5m-2 in division mode and multiplication results at a rate of one per m clock cycles after an initial delay of 3m in multiplication mode respectively. Analysis shows that while previously proposed dividers have area complexity of Ο(m$^2$) or Ο(mㆍ(log$_2$$^{m}$ )), the Proposed architecture has area complexity of Ο(m), In addition, the proposed architecture has significantly less computational delay time compared with the divider which has area complexity of Ο(mㆍ(log$_2$$^{m}$ )). FPGA implementation results of the proposed arithmetic unit, in which Altera's EP2A70F1508C-7 was used as the target device, show that it ran at maximum 121MHz and utilized 52% of the chip area in GF(2$^{571}$ ). Therefore, when elliptic curve cryptographic processor is implemented on FPGAs, the proposed arithmetic unit is well suited for both division and multiplication circuit.

• Journal of the Korean Institute of Telematics and Electronics
• /
• v.23 no.4
• /
• pp.571-577
• /
• 1986

In this paper, we have investigated pole-zero shifting due to variable stopband frequency Ws and passband ripple Ap of elliptic function filters. Also we have studied the phase, group-delay, unit step response and impulse response of elliptic filters. We show that in the passband the phase linearity improves as Ws increases, and eventually it approaches that of a chebyshev function filter.

• The Journal of Korean Institute of Communications and Information Sciences
• /
• v.28 no.7A
• /
• pp.547-556
• /
• 2003

This paper proposes a novel arithmetic unit over GF(2$^{m}$ ) for low-area elliptic curve cryptographic processor. The proposed arithmetic unit, which is linear feed back shift register (LFSR) architecture, is designed by using hardware sharing between the binary GCD algorithm and the most significant bit (MSB)-first multiplication scheme, and it can perform both division and multiplication in GF(2$^{m}$ ). In other word, the proposed architecture produce division results at a rate of one per 2m-1 clock cycles in division mode and multiplication results at a rate of one per m clock cycles in multiplication mode. Analysis shows that the computational delay time of the proposed architecture, for division, is less than previously proposed dividers with reduced transistor counts. In addition, since the proposed arithmetic unit does not restrict the choice of irreducible polynomials and has regularity and modularity, it provides a high flexibility and scalability with respect to the field size m. Therefore, the proposed novel architecture can be used for both division and multiplication circuit of elliptic curve cryptographic processor. Specially, it is well suited to low-area applications such as smart cards and hand held devices.

• Journal of the Korean Institute of Telematics and Electronics
• /
• v.20 no.5
• /
• pp.37-42
• /
• 1983

The elliptic functions have transmission zeros on the imaginary axis and exhibit equal ripples in the stopband as well as in the passband. As a consequence they can be made optimal in the sense that the transition band is minimal. However the time domain behaviors turned out to be inferior to those of Chebyshev and Butterworth responses. This paper investigates the unit step responses and impulse responses in order to analyze the effects of various parameters such as passband attenuation, stopband frequencies M. etc., The following are the prominent features. Step responses of elliptic filters rise faster and produce larger overshoots and undershoots with higher natural frequencies. In the case of even functions, the initial values are non-zero which decreases as $\omega$s increases. Unlike Butter-worth or Chebyshev cases the impulse responses start with nonzero valses which also decrease as $\omega$s or order of the function increases. Eight figures are included to illustrate above analysis.

• East Asian mathematical journal
• /
• v.39 no.1
• /
• pp.11-21
• /
• 2023

Non-trivial automorphisms of the unit disc in the complex plane can be classified by three classes; elliptic, parabolic and hyperbolic automorphisms. This classification is due to a representation in the projective special linear group of the real field, or in terms of fixed points on the closure of the unit disc. In this paper, we will characterize this classification by the distance function of the Poincaré metric on the interior of the unit disc.

• Journal of applied mathematics & informatics
• /
• v.8 no.2
• /
• pp.371-386
• /
• 2001

In this work we investigate the finite element preconditioning method for the $C^1$-cubic spline collocation discretizations for an elliptic operator A defined by $Au := -{\Delta}u + a_1u_x+a_2u_y+a_0u$ in the unit square with some boundary conditions. We compute the condition number and the numerical solution of the preconditioning system for the several example problems. Finally, we compare the this preconditioning system with the another preconditioning system.

• Journal of the Institute of Electronics Engineers of Korea SD
• /
• v.42 no.1
• /
• pp.57-67
• /
• 2005

Fast scalar multiplication of points on elliptic curve is important for elliptic curve cryptography applications. In order to vary field sizes depending on security situations, the cryptography coprocessors should support variable length finite field arithmetic units. To determine the effective variable length finite field arithmetic architecture, two well-known curve scalar multiplication algorithms were implemented on FPGA. The affine coordinates algorithm must use a hardware division unit, but the projective coordinates algorithm only uses a fast multiplication unit. The former algorithm needs the division hardware. The latter only requires a multiplication hardware, but it need more space to store intermediate results. To make the division unit versatile, we need to add a feedback signal line at every bit position. We proposed a method to mitigate this problem. For multiplication in projective coordinates implementation, we use a widely used digit serial multiplication hardware, which is simpler to be made versatile. We experimented with our implemented ECC coprocessors using variable length finite field arithmetic unit which has the maximum field size 256. On the clock speed 40 MHz, the scalar multiplication time is 6.0 msec for affine implementation while it is 1.15 msec for projective implementation. As a result of the study, we found that the projective coordinates algorithm which does not use the division hardware was faster than the affine coordinate algorithm. In addition, the memory implementation effectiveness relative to logic implementation will have a large influence on the implementation space requirements of the two algorithms.

• Journal of the Korean Mathematical Society
• /
• v.37 no.5
• /
• pp.763-777
• /
• 2000

We investigate the existence of group invariant solutions of the Emden-Fowler equation, - u=$\mid$x$\mid$$\sigma$$\mid$u$\mid$p-1u in B, u=0 on B and u(gx)=u(x) in B for g G. Here B is the unit ball in n 2, 1$\sigma$ 0 and G is a closed subgrop of the orthogonal group. A soultion of the problem is called a G in variant solution. We prove that there exists a G invariant non-radial solution if and only if G is not transitive on the unit sphere.

• ETRI Journal
• /
• v.25 no.5
• /
• pp.315-320
• /
• 2003

Elliptic curve cryptography (ECC) offers the highest security per bit among the known public key cryptosystems. The operation of ECC is based on the arithmetic of the finite field. This paper presents the design of a 193-bit finite field multiplier and an inversion unit based on a normal basis representation in which the inversion and the square operation units are easy to implement. This scalable multiplier can be constructed in a variable structure depending on the performance area trade-off. We implement it using Verilog HDL and a 0.35 ${\mu}m$ CMOS cell library and verify the operation by simulation.