• East Asian mathematical journal
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• v.38 no.5
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• pp.611-616
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• 2022

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

• ETRI Journal
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• v.30 no.3
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• pp.365-376
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• 2008

This paper presents the design and implementation of a hyperelliptic curve cryptography (HECC) coprocessor over affine and projective coordinates, along with measurements of its performance, hardware complexity, and power consumption. We applied several design techniques, including parallelism, pipelining, and loop unrolling, in designing field arithmetic units, group operation units, and scalar multiplication units to improve the performance and power consumption. Our affine and projective coordinate-based HECC processors execute in 0.436 ms and 0.531 ms, respectively, based on the underlying field GF($2^{89}$). These results are about five times faster than those for previous hardware implementations and at least 13 times better in terms of area-time products. Further results suggest that neither case is superior to the other when considering the hardware complexity and performance. The characteristics of our proposed HECC coprocessor show that it is applicable to high-speed network applications as well as resource-constrained environments, such as PDAs, smart cards, and so on.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.1
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• pp.97-102
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• 2015

In this work, we study some arithmetic properties of an intermediate modular curve $X_{\Delta}(21)$.

• 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.

• The Pure and Applied Mathematics
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• v.22 no.4
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• pp.403-412
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• 2015

Kato[6] and Torres[9] characterized the Weierstrass semigroup of ramification points on h-hyperelliptic curves. Also they showed the converse results that if the Weierstrass semigroup of a point P on a curve C satisfies certain numerical condition then C can be a double cover of some curve and P is a ramification point of that double covering map. In this paper we expand their results on the Weierstrass semigroup of a ramification point of a double covering map to the Weierstrass semigroup of a pair (P, Q). We characterized the Weierstrass semigroup of a pair (P, Q) which lie on the same fiber of a double covering map to a curve with relatively small genus. Also we proved the converse: if the Weierstrass semigroup of a pair (P, Q) satisfies certain numerical condition then C can be a double cover of some curve and P, Q map to the same point under that double covering map.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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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.

• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.419-424
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• 2009

Let J be the Jacobian variety of a hyperelliptic curve over $\mathbb{Q}$. Let M be the field generated by all square roots of rational integers over a finite number field K. Then we prove that the Mordell-Weil group J(M) is the direct sum of a finite torsion group and a free $\mathbb{Z}$-module of infinite rank. In particular, J(M) is not a divisible group. On the other hand, if $\widetilde{M}$ is an extension of M which contains all the torsion points of J over $\widetilde{\mathbb{Q}}$, then $J(\widetilde{M}^{sol})/J(\widetilde{M}^{sol})_{tors}$ is a divisible group of infinite rank, where $\widetilde{M}^{sol}$ is the maximal solvable extension of $\widetilde{M}$.

• Journal of applied mathematics & informatics
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• v.33 no.1_2
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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.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.12 no.1
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• pp.43-54
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• 2002

McEliece introduced a public-key cryptosystem based on Algebraic codes, specially binary classical Goppa which have a good decoding algorithm and vast number of inequivalent codes with given parameters. And the advantage of this system low cost of their encryption and decryption procedures compared with other public-key systems specially RSA, ECC based on DLP(discrete logarithm problem). But in [1], they resent new attack based on probabilistic algorithm to find minimum weight codeword, so for a sufficient security level, much larger parameter size [2048, 1608,81]is required. Then the big size of public key make McEliece PKC more inefficient. So in this paper, we will propose New Type PKC using q-ary Hyperelliptic code so that with smaller parameter(1 over 3) but still work factor as hi인 as McEliece PKC and faster encryption, decryption can be maintained.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.3
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• pp.443-449
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• 2015

In this paper we propose a method of computing the number of points on the reduction of non-hyperelliptic modular curves of genus greater than or equal to 3 over finite fields.