• East Asian mathematical journal
• /
• 제39권5호
• /
• pp.529-536
• /
• 2023

This paper introduces the Frobenius endomophisms on the binary Hessian curves. It provides an efficient and computable homomorphism for computing point multiplication on binary Hessian curves. As an application, it is possible to construct the GLV method combined with the Frobenius endomorphism to accelerate scalar multiplication over the curve.

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

• 정보보호학회논문지
• /
• 제12권2호
• /
• pp.3-10
• /
• 2002

Koblitz 타원곡선에서 스칼라 곱을 효율적으로 구현하기 위하여 프로베니우스 자기준동형 (Frobenius endomorphism)이 유용하게 사용된다. 스칼라 곱 연산시 스칼라를 이진 전개하는 대신에 프로베니우스 확장을 사용하여 고속연산을 가능하게 할 수 있으며 따라서 연산의 속도는 확장길이와 밀접한 관계가 있다. 본 논문은 스칼라의 프로베니우스 확장길이를 줄임으로써 스칼라 곱의 고속연산을 가능하게 하는 새로운 방법을 제안한다. 타원곡선의 위수를 노름(Norm)으로 갖는 원소대신 큰 소수 위수를 노름으로 갖는 원소를 사용하여 프로베니우스 확장길이를 최적화시키는 이 방법은 Solinas, Smart가 제안한 방법보다 프로베니우스 확장길이를 더 감소시킬 수 있다.

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

• 정보보호학회논문지
• /
• 제12권1호
• /
• pp.81-88
• /
• 2002

작은 홀수 표수를 갖는 유한체 위에 정의된 타원곡선에서 스칼라 곱을 효율적으로 구현하기 위해 프로베니우스 자기준동형(Frobenius endomorphism)이 유용하게 사용된다. 본 논문은 이러한 타원곡선에서 스칼라 곱 연산속도를 향상 시키는 새로운 방법을 소개한다. 이 방법은 스칼라의 프로베니우스 자기준동형 확장길이를 기존의 것보다 줄이므로 속도개선을 얻는다.

• East Asian mathematical journal
• /
• 제37권5호
• /
• pp.585-590
• /
• 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.

• East Asian mathematical journal
• /
• 제38권5호
• /
• pp.611-616
• /
• 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.

• Journal of applied mathematics & informatics
• /
• 제32권1_2호
• /
• pp.17-26
• /
• 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
• /
• 제33권1_2호
• /
• pp.145-155
• /
• 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 applied mathematics & informatics
• /
• 제30권1_2호
• /
• pp.101-109
• /
• 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.