• ETRI Journal
• /
• 제30권6호
• /
• pp.818-825
• /
• 2008

This paper proposes an exponentiation method with Frobenius mappings. The main target is an exponentiation in an extension field. This idea can be applied for scalar multiplication of a rational point of an elliptic curve defined over an extension field. The proposed method is closely related to so-called interleaving exponentiation. Unlike interleaving exponentiation methods, it can carry out several exponentiations of the same base at once. This happens in some pairing-based applications. The efficiency of using Frobenius mappings for exponentiation in an extension field was well demonstrated by Avanzi and Mihailescu. Their exponentiation method efficiently decreases the number of multiplications by inversely using many Frobenius mappings. Compared to their method, although the number of multiplications needed for the proposed method increases about 20%, the number of Frobenius mappings becomes small. The proposed method is efficient for cases in which Frobenius mapping cannot be carried out quickly.

• 호남수학학술지
• /
• 제42권2호
• /
• pp.213-218
• /
• 2020

Assume that K is a number field and K_ur is the maximal unramified extension of it. When Gal(K_ur/K) is an infinite group. It is known that Gal(K_ur/K) is generated by finitely many Frobenius classes of Gal(K_ur/K) by Y. Ihara. In this paper, we will give the explicit number of Frobenius classes which generate whole group Gal(K_ur/K).

• Korean Journal of Mathematics
• /
• 제10권2호
• /
• pp.191-194
• /
• 2002

Let $S_i$ be the numerical semigroup generated by the set $\{a,a+d,{\cdots},a+(i-1)d,a+(i+1)d, {\cdots},a+rd\}$. In this paper, we will formulate the largest nonmember, the Frobenius number, of each set $S_i$.

• 대한수학회보
• /
• 제57권3호
• /
• pp.623-647
• /
• 2020

The greatest integer that does not belong to a numerical semigroup S is called the Frobenius number of S. The Frobenius problem, which is also called the coin problem or the money changing problem, is a mathematical problem of finding the Frobenius number. In this paper, we introduce the Frobenius problem for two kinds of numerical semigroups generated by the Thabit numbers of the first kind, and the second kind base b, and by the Cunningham numbers. We provide detailed proofs for the Thabit numbers of the second kind base b and omit the proofs for the Thabit numbers of the first kind base b and Cunningham numbers.

• Korean Journal of Mathematics
• /
• 제10권1호
• /
• pp.67-74
• /
• 2002

Suppose linear forms whose coefficients are arithmetic progressions are given. In this paper, we will find the upper and lower bounds of the minimal ranks of subforms of these forms which left the Frobenius numbers invariant. This is an improvement of Ritter's bounds.

• 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.

• 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.

• 한국통신학회논문지
• /
• 제40권7호
• /
• pp.1257-1265
• /
• 2015

본 논문은 투영 행렬을 이용한 높은 채널 용량을 획득할 수 있는 사용자 선택 기법을 제안한다. 기존의 Frobenius norm기반의 사용자 선택 기법이 다수의 사용자를 선택할수록 정확도가 감소하는 점을 투영 행렬의 근사화 형태를 이용하여 개선하였다. Flop count를 이용한 제안 기준의 계산복잡도를 분석하였고, 모의 실험을 통하여 안테나 구성에 따른 획득 가능한 채널 용량에 대해서 비교하였다. 모의 실험 결과를 통해 제안한 사용자 선택 기법이 더 높은 채널 용량을 획득하는 것을 확인하였다.

• Structural Engineering and Mechanics
• /
• 제66권1호
• /
• pp.125-138
• /
• 2018

This paper proposes a transfer matrix method for the bending vibration of two types of tapered beams subjected to axial force, and it is applied to analyze tapered beams with an edge or multiple edge open cracks. One beam type is assumed to be reduced linearly in the cross-section height along the beam length. The other type is a tapered beam in which the cross-section height and width with the same taper ratio is linearly reduced simultaneously. Each crack is modeled as two sub-elements connected by a rotational spring, and the method can evaluate the effect of cracking on the desired number of eigenfrequencies using a minimum number of subdivisions. Among the power series available for the solutions, the roots of the differential equation are computed using the Frobenius method. The computed results confirm the accuracy of the method and are compared with previously reported results. The effectiveness of the proposed methods is demonstrated by examining specific examples, and the effects of cracking and axial loading are carefully examined by a comparison of the single and double tapered beam results.

• Kyungpook Mathematical Journal
• /
• 제54권4호
• /
• pp.607-617
• /
• 2014

Let S = C(r,m), the finite cyclic semigroup with index r and period m. Each subsemigroup of S is cyclic if and only if either r = 1; r = 2; or r = 3 with m odd. For $r{\neq}1$, the maximum value of the minimum number of elements in a (minimal) generating set of a subsemigroup of S is 1 if r = 3 and m is odd; 2 if r = 3 and m is even; (r-1)/2 if r is odd and unequal to 3; and r/2 if r is even. The number of cyclic subsemigroups of S is $r-1+{\tau}(m)$. Formulas are also given for the number of 2-generated subsemigroups of S and the total number of subsemigroups of S. The minimal generating sets of subsemigroups of S are characterized, and the problem of counting them is analyzed.