• 대한수학회보
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• 제46권5호
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• pp.917-930
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• 2009

We find all distinct representatives of isomorphism classes of genus-3 pointed trigonal curves and compute the number of isomorphism classes of a special class of genus-3 pointed trigonal curves including that of Picard curves over a finite field F of characteristic 2.

• Korean Journal of Mathematics
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• 제29권3호
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• pp.483-492
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• 2021

This article concerns the property that for any element a in a ring, if a²ⁿ = aⁿ for some n ≥ 2 then a² = a. The class of rings with this property is large, but there also exist many kinds of rings without that, for example, rings of characteristic ≠2 and finite fields of characteristic ≥ 3. Rings with such a property is called reduced-over-idempotent. The study of reduced-over-idempotent rings is based on the fact that the characteristic is 2 and every nonzero non-identity element generates an infinite multiplicative semigroup without identity. It is proved that the reduced-over-idempotent property pass to polynomial rings, and we provide power series rings with a partial affirmative argument. It is also proved that every finitely generated subring of a locally finite reduced-over-idempotent ring is isomorphic to a finite direct product of copies of the prime field {0, 1}. A method to construct reduced-over-idempotent fields is also provided.

• 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.

• Journal of applied mathematics & informatics
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• 제30권1_2호
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• pp.101-109
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• 2012

In this paper, we present an algorithm for computing the number of points on the Jacobian varieties of one-dimensional family of genus 3 nonhyperelliptic curves over finite fields. We also provide the explicit formula of the characteristic polynomial of the Frobenius endomorphism of the Jacobian of $C:y^3=x^4+{\alpha}$ over a finite field $\mathbb{F}_p$ with $p{\equiv}1$ (mod 3) and $p{\neq}1$ (mod 4). Moreover, we give some implementation results using Gaudry-Schost method. A 162-bit order is computed in 97 s on a Pentium IV 2.13 GHz computer using our algorithm.

• Smart Structures and Systems
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• 제18권2호
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• pp.233-247
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• 2016

The vibration characteristic analysis of sandwich cylindrical shells subjected with magnetorheological (MR) elastomer and constraining layer are considered in this study. And, the discrete finite element method is adopted to calculate the vibration and damping characteristics of the sandwich cylindrical shell system. The effects of thickness of the MR elastomer, constraining layer, applied magnetic fields on the vibration characteristics of the sandwich shell system are also studied in this paper. Additionally, the rheological properties of the MR elastomer can be changed by applying various magnetic fields and the properties of the MR elastomer are described by complex quantities. The natural frequencies and modal loss factor of the sandwich cylindrical shells are calculated for many designed parameters. The core layer of MR elastomer is found to have significant effects on the damping behavior of the sandwich cylindrical shells.

• 정보보호학회논문지
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• 제12권5호
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• pp.45-51
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• 2002

Koblitz 타원곡선과 같이 표수(characteristic)가 2인 작은 유한체 위에서 정의된 non-supersingular 타원곡선은 스칼라 곱을 효율적으로 구현하기 위하여 프로베니우스 자기준동형 (Frobenius endomorphism)이 유용하게 사용된다. 본 논문은 확장된 프로베니우스 함수를 사용하여 스칼라 곱의 고속연산을 가능하게 하는 방법을 소개한다. 이 방법은 Muller［5］가 제안한 블록방법(block method) 보다 선행계산을 위해 사용되는 덧셈량을 줄이는 반면에 확장길이는 거의 같게 하므로 M(equation omitted )ller의 방법보다 효율적이다.

• 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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• 제54권2호
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• pp.417-426
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• 2017

We explicitly determine fundamental units and regulators of an infinite family of cyclic quartic function fields $L_h$ of unit rank 3 with a parameter h in a polynomial ring $\mathbb{F}_q[t]$, where $\mathbb{F}_q$ is the finite field of order q with characteristic not equal to 2. This result resolves the second part of Lehmer's project for the function field case.

• East Asian mathematical journal
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• 제35권5호
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• pp.589-595
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• 2019

In this paper, we prove that a special supersingular K3 surface of Artin invariant ${\sigma}$ over a field of odd characteristic p has a model over a finite field of $p^{2{\sigma}}$ elements.

• 대한수학회보
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• 제46권5호
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• pp.941-947
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• 2009

Let q be a power of 16. Every polynomial $P\in\mathbb{F}_q$[t] is a strict sum $P=A^2+A+B^3+C^3+D^3+E^3$. The values of A,B,C,D,E are effectively obtained from the coefficients of P. The proof uses the new result that every polynomial $Q\in\mathbb{F}_q$[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: $Q=F^2+F+tG^2$. This improves for such q's and such Q's a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials F,G,H for the strict representation $Q=F^2$+F+GH. Observe that the latter representation may be considered as an analogue in characteristic 2 of the strict representation of a polynomial Q by three squares in odd characteristic.