• 한국전자통신학회논문지
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• 제11권7호
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• pp.693-700
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• 2016

현재 사용되고 있는 유한체 GF(q)위의 non-supersingular 타원곡선 이산대수문제에 기반한 공개키 암호법의 안전성을 보장하기 위해서는 타원곡선의 위수의 크기와 소인수의 크기를 계산하는 일이 매우 중요하다. 그런데 타원곡선의 위수를 구하는 전통적인 방법인 Schoof 알고리즘은 매우 복잡하여 지금도 개선작업이 진행중이다. 본 논문에서는 복잡한 Schoof 알고리즘을 피하기 위하여, 표수가 2인 유한체의 합성체$GF(2^m)=GF(2^{rs})=GF((2^r)^s)$ 위에서 Weil 정리를 이용하여 타원곡선의 위수를 계산하는 방법을 제안한다. 또한, 그에 따른 알고리즘과 그 알고리즘을 적용한 프로그램을 실행하여 타원곡선 암호법에 사용될 수 있는 효율적인 곡선으로 ${\sharp}E(GF(2^5))=36$일 때의 합성체 $GF(2^5)^{31})$ 위에서 위수에 $10^{40}$ 이상인 소인수를 포함하는 non-supersingular 타원곡선을 찾을 수 있었다.

• 대한수학회보
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• 제60권4호
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• pp.1035-1059
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• 2023

Let p be an odd prime. Let E be an elliptic curve defined over a quadratic imaginary field, where p splits completely. Suppose E has supersingular reduction at primes above p. Under appropriate hypotheses, we extend the results of [17] to ℤ²_p-extensions. We define and study the fine double-signed residual Selmer groups in these settings. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed 𝜇-invariants of one elliptic curve implies the vanishing of the signed 𝜇-invariants of the other. Finally, we show that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions.

• 충청수학회지
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• 제22권3호
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• pp.299-307
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• 2009

We count the isomorphism classes of elliptic curves over finite fields $\mathbb{F}_{3^{n}}$ and list a representative of each isomorphism class. Also we give the number of rational points for each supersingular elliptic curve over $\mathbb{F}_{3^{n}}$.

• 대한수학회보
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• 제50권2호
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• pp.407-416
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• 2013

Let E be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime of good supersingular reduction for E. Although the Iwasawa theory of E over the cyclotomic ${\mathbb{Z}}_p$-extension of $\mathbb{Q}$ is well known to be fundamentally different from the case of good ordinary reduction at p, we are able to combine the method of our earlier paper with the theory of Kobayashi [5] and Pollack [8], to give an explicit upper bound for the number of copies of ${\mathbb{Q}}_p/{\mathbb{Z}}_p$ occurring in the $p$-primary part of the Tate-Shafarevich group of E over $\mathbb{Q}$.

• ETRI Journal
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• 제21권1호
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• pp.28-39
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• 1999

Koblitz has suggested to use "anomalous" elliptic curves defined over ${\mathbb{F}}_2$, which are non-supersingular and allow or efficient multiplication of a point by and integer, For these curves, Meier and Staffelbach gave a method to find a polynomial of the Frobenius map corresponding to a given multiplier. Muller generalized their method to arbitrary non-supersingular elliptic curves defined over a small field of characteristic 2. in this paper, we propose an algorithm to speed up scalar multiplication on an elliptic curve defined over a small field. The proposed algorithm uses the same field. The proposed algorithm uses the same technique as Muller's to get an expansion by the Frobenius map, but its expansion length is half of Muller's due to the reduction step (Algorithm 1). Also, it uses a more efficient algorithm (Algorithm 3) to perform multiplication using the Frobenius expansion. Consequently, the proposed algorithm is two times faster than Muller's. Moreover, it can be applied to an elliptic curve defined over a finite field with odd characteristic and does not require any precomputation or additional memory.

• 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.

• ETRI Journal
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• 제31권2호
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• pp.129-139
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• 2009

This paper provides an efficient algorithm for computing the ${\eta}_T$ pairing on supersingular elliptic curves over fields of characteristic two. In the proposed algorithm, we deploy a modified multiplication in $F_{2^{4n}}$ using the Vandermonde matrix. For F, G ${\in}$ $F_{2^{4n}}$ the proposed multiplication method computes ${\beta}{\cdot}F{\cdot}G$ instead of $F{\cdot}G$ with some ${\beta}$ ${\in}$ $F^*_{2n}$ because ${\beta}$ is eliminated by the final exponentiation of the ${\eta}_T$ pairing computation. The proposed multiplication method asymptotically requires only 7 multiplications in $F_{2^n}$ as n ${\rightarrow}$ ${\infty}$, while the cost of the previously fastest Karatsuba method is 9 multiplications in $F_{2^n}$. Consequently, the cost of the ${\eta}_T$ pairing computation is reduced by 14.3%.