• Journal of the Korean Mathematical Society
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• v.31 no.1
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• pp.89-95
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• 1994

• Journal of the Chungcheong Mathematical Society
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• v.17 no.2
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• pp.191-196
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• 2004

• ETRI Journal
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• v.35 no.5
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• pp.880-888
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• 2013

The preexisting pairings ate, $ate_i$, R-ate, and optimal-ate use q-expansion, where q is the size of the defining field for the elliptic curves. Elliptic curves with small embedding degrees only allow a few of these pairings. In such cases, efficiently computable endomorphisms can be used, as in [11] and [12]. They used the endomorphisms that have characteristic polynomials with very small coefficients, which led to some restrictions in finding various pairing-friendly curves. To construct more pairing-friendly curves, we consider ${\mu}$-expansion using the Gallant-Lambert-Vanstone (GLV) decomposition method, where ${\mu}$ is an arbitrary integer. We illustrate some pairing-friendly curves that provide more efficient pairing from the ${\mu}$-expansion than from the ate pairing. The proposed method can achieve timing results at least 20% faster than the ate pairing.

• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.869-889
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• 2015

Let ${\Gamma}^+$ be the positive cone in a totally ordered abelian group ${\Gamma}$, and ${\alpha}$ an action of ${\Gamma}^+$ by extendible endomorphisms of a $C^*$-algebra A. Suppose I is an extendible ${\alpha}$-invariant ideal of A. We prove that the partial-isometric crossed product $\mathcal{I}:=I{\times}^{piso}_{\alpha}{\Gamma}^+$ embeds naturally as an ideal of $A{\times}^{piso}_{\alpha}{\Gamma}^+$, such that the quotient is the partial-isometric crossed product of the quotient algebra. We claim that this ideal $\mathcal{I}$ together with the kernel of a natural homomorphism $\phi:A{\times}^{piso}_{\alpha}{\Gamma}^+{\rightarrow}A{\times}^{iso}_{\alpha}{\Gamma}^+$ gives a composition series of ideals of $A{\times}^{piso}_{\alpha}{\Gamma}^+$ studied by Lindiarni and Raeburn.

• Journal of the Korean Mathematical Society
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• v.59 no.1
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• pp.105-127
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• 2022

It is known that the maximal invariant set of a continuous iterative dynamical system in a compact Hausdorff space is equal to the intersection of its forward image sets, which we will call the first minimal image set. In this article, we investigate the corresponding relation for a class of discontinuous self maps that are on the verge of continuity, or topologically almost continuous endomorphisms. We prove that the iterative dynamics of a topologically almost continuous endomorphisms yields a chain of minimal image sets that attains a unique transfinite length, which we call the maximal invariance order, as it stabilizes itself at the maximal invariant set. We prove the converse, too. Given ordinal number ξ, there exists a topologically almost continuous endomorphism f on a compact Hausdorff space X with the maximal invariance order ξ. We also discuss some further results regarding the maximal invariance order as more layers of topological restrictions are added.

• East Asian mathematical journal
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• v.6 no.2
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• pp.167-182
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• 1990

The relationships between submodules of a module and ideals of the endomorphism ring of a module had been studied in [1]. For a submodule L of a moudle M, the set $I^L$ of all endomorphisms whose images are contained in L is a left ideal of the endomorphism ring End (M) and for a submodule N of M, the set $I_N$ of all endomorphisms whose kernels contain N is a right ideal of End (M). In this paper, author defines an H-invariant module and proves that every submodule of an H-invariant module is the image and kernel of unique endomorphisms. Every ideal $I^L(I_N)$ of the endomorphism ring End(M) when M is H-invariant is a left (respectively, right) principal ideal of End(M). From the above results, if a module M is H-invariant then each left, right, or both sided ideal I of End(M) is an intersection of a left, right, or both sided principal ideal and I itself appropriately. If M is an H-invariant module then the ACC on the set of all left ideals of type $I^L$ implies the ACC on M. Also if the set of all right ideals of type $I^L$ has DCC, then H-invariant module M satisfies ACC. If the set of all left ideals of type $I^L$ satisfies DCC, then H-invariant module M satisfies DCC. If the set of all right ideals of type $I_N$ satisfies ACC then H-invariant module M satisfies DCC. Therefore for an H-invariant module M, if the endomorphism ring End(M) is left Noetherian, then M satisfies ACC. And if End(M) is right Noetherian then M satisfies DCC. For an H-invariant module M, if End(M) is left Artinian then M satisfies DCC. Also if End(M) is right Artinian then M satisfies ACC.

• Communications of the Korean Mathematical Society
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• v.12 no.2
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• pp.257-267
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• 1997

It is shown that if S is a positively expansive endomorphism on a one-sided mixing SFT (X,T), then (X,S) is conjugate to a one-sided mixing SFT, and the Parry measures of (X,T) and (X,S) are identical.

• Honam Mathematical Journal
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• v.31 no.4
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• pp.529-540
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• 2009

The authors have investigated several properties of endomorphism rings of modules. In particular, a closedly simple module and a closedly semi-simple module were defined and then the regularities of its rings of endomorphisms were discussed.

• East Asian mathematical journal
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• v.39 no.5
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• pp.529-536
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• 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.

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.45-56
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• 2015

In this paper, we study the relation between two dynamical systems (V, f) and (V, g) with $f{\circ}g=g{\circ}f$. As an application, we show that an endomorphism (respectively a polynomial map with Zariski dense, of bounded Preper(f)) has only finitely many endomorphisms (respectively polynomial maps) of bounded degree which are commutable with f.