• Journal of Information Processing Systems
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• v.8 no.1
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• pp.145-158
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• 2012

Hyperelliptic Curve Cryptosystem (HECC) is well suited for all kinds of embedded processor architectures, where resources such as storage, time, or power are constrained due to short operand sizes. We can construct genus 3 HECC on 54-bit finite fields in order to achieve the same security level as 160-bit ECC or 1024-bit RSA due to the algebraic structure of Hyperelliptic Curve. This paper explores various possible attacks to the discrete logarithm in the Jacobian of a Hyperelliptic Curve (HEC) and addition and doubling of the divisor using explicit formula to speed up the scalar multiplication. Our aim is to develop a cryptosystem that can sign and authenticate documents and encrypt / decrypt messages efficiently for constrained devices in wireless networks. The performance of our proposed cryptosystem is comparable with that of ECC and the security analysis shows that it can resist the major attacks in wireless networks.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.379-384
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• 2015

In this paper, we find all Weierstrass points on the hyperelliptic modular curves $X_1(N)$.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.2
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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.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.869-893
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• 2016

We construct the extension of a hyperelliptic K3 surface to a Fano 6-fold with extraordinary properties in moduli. This leads us to a family of surfaces of general type with $p_g=1$, q = 0, $K^2=2$ and hyperelliptic canonical curve, each of which is a weighted complete inter-section inside a Fano 6-fold. Finally, we use these hyperelliptic surfaces to determine an 8-parameter family of Godeaux surfaces with ${\pi}_1={\mathbb{Z}}/2$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.4 no.1
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• pp.23-28
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• 2000

We compute the order of jacobian groups of hyperelliptic curves on a finite field of characteristic 3 and we determine which curves are secure against known attacks.

• East Asian mathematical journal
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• v.37 no.5
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• pp.585-590
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• 2021

This paper consider the Jacobian variety of a hyperelliptic curve over a finite field with the formal structure of the mixed type. We present the Newton polygon of the characteristic polynomial of the Frobenius endomorphism of the Jacobian variety. It gives an useful tool for finding the local decomposition of the Jacobian variety into isotypic components.

• Journal of the Chungcheong Mathematical Society
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• v.15 no.2
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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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• v.13 no.1_2
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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).

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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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)).

• ETRI Journal
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• v.27 no.5
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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.