• The Journal of the Korea institute of electronic communication sciences
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• v.11 no.7
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• pp.693-700
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• 2016

It is important that we can calculate the order of non-supersingular elliptic curves with large prime factors over the finite field GF(q) to guarantee the security of public key cryptosystems based on discrete logarithm problem(DLP). Schoof algorithm, however, which is used to calculate the order of the non-supersingular elliptic curves currently is so complicated that many papers are appeared recently to update the algorithm. To avoid Schoof algorithm, in this paper, we propose an algorithm to calculate orders of elliptic curves over finite composite fields of the forms $GF(2^m)=GF(2^{rs})=GF((2^r)^s)$ using Weil's theorem. Implementing the program based on the proposed algorithm, we find a efficient non-supersingular elliptic curve over the finite composite field $GF(2^5)^{31})$ of the order larger than $10^{40}$ with prime factor larger than $10^{40}$ using the elliptic curve $E(GF(2^5))$ of the order 36.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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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.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.3
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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 $\mathbb{F}_{3^{n}}$.

• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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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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• v.21 no.1
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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.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.12 no.5
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• pp.45-51
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• 2002

As Koblitz curve, the Frobenius endomorphism is know to be useful in efficient implementation of multiplication on non-supersingular elliptic cures defined on small finite fields of characteristic two. In this paper a method using the extended Frobenius endomorphism to speed up scalar multiplication is introduced. It will be shown that the proposed method is more efficient than Muller's block method in ［5］ because the number of point addition for precomputation is small but on the other hand the expansion length is almost same.

• ETRI Journal
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• v.31 no.2
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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%.