• 호남수학학술지
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• 제39권2호
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• pp.137-141
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• 2017

Let A be an abelian variety defined over a number field K and let L be a finite Galois extension of K. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Let $Res_{L/K}(A)$ be the restriction of scalars of A from L to K and let B be an abelian subvariety of $Res_{L/K}(A)$ defined over K. Assuming that III(A/L) is finite, we compute [III(B/K)][III(C/K)]/[III(A/L)], where [X] is the order of a finite abelian group X and the abelian variety C is defined by the exact sequence defined over K $0{\longrightarrow}B{\longrightarrow}Res_{L/K}(A){\longrightarrow}C{\longrightarrow}0$.

• International Journal of Computer Science & Network Security
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• 제21권4호
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• pp.289-300
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• 2021

Non-abelian group based Cryptography is a field which has become a latest trend in research due to increasing vulnerabilities associated with the abelian group based cryptosystems which are in use at present and the interesting algebraic properties associated that can be thought to provide higher security. When developing cryptographic primitives based on non-abelian groups, the researchers have tried to extend the similar layouts associated with the traditional underlying mathematical problems and assumptions by almost mimicking their operations which is fascinating even to observe. This survey contributes in highlighting the different analogous extensions of traditional assumptions presented by various authors and a set of open problems. Further, suggestions to apply the Hamiltonian Cycle/Path Problem in a similar direction is presented.

• International Journal of Computer Science & Network Security
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• 제21권8호
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• pp.17-27
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• 2021

Non-abelian group based cryptosystems are a latest research inspiration, since they offer better security due to their non-abelian properties. In this paper, we propose a novel approach to non-abelian group based public-key cryptographic protocols using semidirect products of finite groups. An intractable problem of determining automorphisms and generating elements of a group is introduced as the underlying mathematical problem for the suggested protocols. Then, we show that the difficult problem of determining paths and cycles of Cayley graphs including Hamiltonian paths and cycles could be reduced to this intractable problem. The applicability of Hamiltonian paths, and in fact any random path in Cayley graphs in the above cryptographic schemes and an application of the same concept to two previous cryptographic protocols based on a Generalized Discrete Logarithm Problem is discussed. Moreover, an alternative method of improving the security is also presented.

• 호남수학학술지
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• 제38권1호
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• pp.85-93
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• 2016

Let A be an abelian variety defined over a number field K and let L be a degree 3 non-Galois extension of K. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Assuming that III(A/L) is finite, we compute [III(A/K)][III($A_{\varphi}/K$)]/[III(A/L)], where [X] is the order of a finite abelian group X.

• 대한수학회보
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• 제38권1호
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• pp.7-16
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• 2001

We study the Fourier transformation on the Gelfand-Bruhat space of type S and characterize this space by means of Fourier transform on a locally compact abelian group G. Also, we extend Bochner-Schwartz theorem to the dual space of the Gelfand-Bruhat space and the space of Fourier hyperfunctions on G. respectively.

• 대한수학회지
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• 제58권1호
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• pp.149-171
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• 2021

Let Cl(A) denote the class group of an arbitrary integral domain A introduced by Bouvier in 1982. Then Cl(A) is the ideal class (resp., divisor class) group of A if A is a Dedekind or a Prüfer (resp., Krull) domain. Let G be an abelian group. In this paper, we show that there is a ring of Krull type D such that Cl(D) = G but D is not a Krull domain. We then use this ring to construct a Prüfer ring of Krull type E such that Cl(E) = G but E is not a Dedekind domain. This is a generalization of Claborn's result that every abelian group is the ideal class group of a Dedekind domain.

• Korean Journal of Mathematics
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• 제29권3호
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• pp.603-612
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• 2021

We generalize 3-manifolds supporting non-positively curved metric to construct manifolds which have the following properties : (1) They are not locally CAT(0). (2) Their fundamental groups are residually finite. (3) They have subgroup separability for some abelian subgroups.

• 호남수학학술지
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• 제45권3호
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• pp.410-417
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• 2023

Suppose that there are 8 abelian varieties defined over a number field K which satisfy a commutative diagram. We show that if we know that three out of four short exact sequences satisfy the rate formula of Tate-Shafarevich groups, then the unknown short exact sequence satisfies the rate formula of Tate-Shafarevich groups, too.

• 대한수학회보
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• 제42권4호
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• pp.807-815
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• 2005

Let R be a ring with identity, X the set of all nonzero, nonunits of Rand G the group of all units of R. We will consider two group actions on X by G, the regular action and the conjugate action. In this paper, by investigating two group actions we can have some results as follows: First, if G is a finitely generated abelian group, then the orbit O(x) under the regular action on X by G is finite for all nilpotents x $\in$ X. Secondly, if F is a field in which 2 is a unit and F $\backslash\;\{0\}$ is a finitley generated abelian group, then F is finite. Finally, if G in a unit-regular ring R is a torsion group and 2 is a unit in R, then the conjugate action on X by G is trivial if and only if G is abelian if and only if R is commutative.

• 대한수학회지
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• 제46권2호
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• pp.237-248
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• 2009

We give an explicit formula for the Mordell-Weil rank of an abelian fibered variety and some of its applications for an abelian fibered $hyperk{\ddot{a}}hler$ manifold. As a byproduct, we also give an explicit example of an abelian fibered variety in which the Picard number of the generic fiber in the sense of scheme is different from the Picard number of generic closed fibers.