• 충청수학회지
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• 제15권2호
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• pp.1-12
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• 2003

L. H. Encinas, A. J. Menezes, and J. M. Masque in [2] proposed a classification of isomorphism classes of hyperelliptic curve of genus 2 over finite fields with characteristic different from 2 and 5. Y. Choie and D. Yun in [1] obtained for the number of isomorphic classes of hyperelliptic curves of genus 2 over $F_q$ using direct counting method. In this paper we will classify the isomorphism classes of hyperelliptic curves of genus 2 over $F_{2^n}$ for odd n, represented by an equation of the form $y^2+a_5y=x^5+a_8x+a_{10}(a_5{\neq}0)$.

• Journal of applied mathematics & informatics
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• 제13권1_2호
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• pp.413-424
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• 2003

L. H Encinas, A. J. Menezes and J. M. Masque in [3] proposed a classification of isomorphism classes of hyperelliptic curve of genus 2 over finite fields with characteristic different from 2 and 5. Y. Choie and D. Yun in [2] obtained the number of isomorphic classes of hyperelliptic curves of genus 2 over $F_{2-}$ using direct counting method. We have obtained isomorphism classes of hyperelliptic curves of genus 2 over $F_{2n}$ for odd n, represented by an equation of the form $y^2$ + $a_{5}$ y = $x^{5}$ + $a_{8}$ x + $a_{10}$ ( $a_{5}$ $\neq$0) [1]. In this paper we characterize hyperelliptic curves of genus 2 over $F_{2n}$ for even n, represented by an equation of the form $y^2$ + $a_{5}$ y = $x^{5}$ + $a_{5}$ x + $a_{10}$ ( $a_{5}$ $\neq$0).>0).

• Journal of applied mathematics & informatics
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• 제32권1_2호
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• pp.17-26
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• 2014

In this paper, we present an algorithm for computing the number of points on the Jacobian varieties of genus 3 hyperelliptic curves of type $y^2=x^7+ax$ over finite prime fields. The problem of determining the group order of the Jacobian varieties of algebraic curves defined over finite fields is important not only arithmetic geometry but also curve-based cryptosystems in order to find a secure curve. Based on this, we provide the explicit formula of the characteristic polynomial of the Frobenius endomorphism of the Jacobian variety of hyperelliptic curve $y^2=x^7+ax$ over a finite field $\mathbb{F}_p$ with $p{\equiv}1$ modulo 12. Moreover, we also introduce some implementation results by using our algorithm.

• ETRI Journal
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• 제37권1호
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• pp.107-117
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• 2015

This paper presents an energy-efficient (low power) prime-field hyperelliptic curve cryptography (HECC) processor with uniform power draw. The HECC processor performs divisor scalar multiplication on the Jacobian of genus 2 hyperelliptic curves defined over prime fields for arbitrary field and curve parameters. It supports the most frequent case of divisor doubling and addition. The optimized implementation, which is synthesized in a $0.13{\mu}m$ standard CMOS technology, performs an 81-bit divisor multiplication in 503 ms consuming only $6.55{\mu}J$ of energy (average power consumption is $12.76{\mu}W$). In addition, we present a technique to make the power consumption of the HECC processor more uniform and lower the peaks of its power consumption.

• 대한수학회보
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• 제56권2호
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• pp.333-349
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• 2019

We give explicitly an average value formula under the multiplication-by-2 map for the x-coordinates of the 2-division points D on the Jacobian variety J(C) of a hyperelliptic curve C with genus g if $2D{\equiv}2P-2{\infty}$ (mod Pic(C)) for $P=(x_P,y_P){\in}C$ with $y_P{\neq}0$. Moreover, if g = 2, we give a more explicit formula for D such that $2D{\equiv}P-{\infty}$ (mod Pic(C)).

• Journal of applied mathematics & informatics
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• 제33권1_2호
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• pp.145-155
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• 2015

In this paper, we study the bounds of the coefficients of the characteristic polynomial of the Frobenius endomorphism of the Jacobian of dimension three over a finite field. We provide explicitly computable bounds for the coefficients of the characteristic polynomial. In addition, we present the counting points algorithm for computing a group of the Jacobian of genus 3 hyperelliptic curves over a finite field with large characteristic. Based on these bounds, we found an efficient search space that was used in the counting points algorithm on genus 3 curves. The algorithm was explained and verified through simple examples.

• 대한수학회지
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• 제45권4호
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• pp.1057-1073
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• 2008

We present an explicit Eta pairing approach for computing the Tate pairing on general divisors of hyperelliptic curves $H_d$ of genus 2, where $H_d\;:\;y^2+y=x^5+x^3+d$ is defined over ${\mathbb{F}}_{2^n}$ with d=0 or 1. We use the resultant for computing the Eta pairing on general divisors. Our method is very general in the sense that it can be used for general divisors, not only for degenerate divisors. In the pairing-based cryptography, the efficient pairing implementation on general divisors is significantly important because the decryption process definitely requires computing a pairing of general divisors.

• 대한수학회보
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• 제46권4호
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• pp.789-802
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• 2009

We show that computation of a sequence of Richelot isogenies from specified supersingular Jacobians of genus-2 curves over $\mathbb{F}_p$ can be executed in $\mathbb{F}_{p2}$ or $\mathbb{F}_{p4}$ . Based on this, we describe a practical algorithm for computing a Richelot isogeny sequence.

• ETRI Journal
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• 제27권5호
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• pp.617-627
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• 2005

This paper proposes an efficient scalar multiplication algorithm for hyperelliptic curves, which is based on the idea that efficient endomorphisms can be used to speed up scalar multiplication. We first present a new Frobenius expansion method for special hyperelliptic curves that have Gallant-Lambert-Vanstone (GLV) endomorphisms. To compute kD for an integer k and a divisor D, we expand the integer k by the Frobenius endomorphism and the GLV endomorphism. We also present improved scalar multiplication algorithms that use the new expansion method. By our new expansion method, the number of divisor doublings in a scalar multiplication is reduced to a quarter, while the number of divisor additions is almost the same. Our experiments show that the overall throughputs of scalar multiplications are increased by 15.6 to 28.3 % over the previous algorithms when the algorithms are implemented over finite fields of odd characteristics.

• 대한수학회논문집
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• 제9권4호
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• pp.905-915
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• 1994

Let $M_g$ be the moduli space of isomorphism classes of genus g smooth curves. It is a quasi-projective variety of dimension 3g - 3, when $g > 2$. It is known that a complete subvariety of $M_g$ has dimension $< g-1 [D]$. In general it is not known whether this bound is rigid. For example, it is not known whether $M_4$ has a complete surface in it. But one knows that there is a complete curve through any given finite points [H]. Recently, an explicit example of a complete curve in moduli space is given in [G-H]. In [G-H] they constructed a complete curve of $M_3$ as an intersection of five hypersurfaces of the Satake compactification of $M_3$. One way to get a complete curve of $M_3$ is to find a complete one dimensional family $p : X \to B$ of plane quartics which gives a nontrivial morphism from the base space B to the moduli space $M_3$. This is because every non-hyperelliptic smooth curve of genus three can be realized as a nonsingular plane quartic and vice versa. This paper has come out from the effort to find such a complete family of plane quartics. Since nonsingular quartics form an affine space some fibers of p must be singular ones. In this paper, due to the semistable reduction theorem [M], we search singular plane quartics which can occur as singular fibers of the family above. We first list all distinct plane quartics in terms of singularities.