• 대한수학회보
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• 제51권6호
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• pp.1689-1696
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• 2014

Recently, it was proved that every p-Schur ring over an abelian group of order $p^3$ is Schurian. In this paper, we prove that every commutative p-Schur ring over a non-abelian group of order $p^3$ is Schurian.

• 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.

• 대한수학회보
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• 제52권1호
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• pp.239-246
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• 2015

If for every elements x and y of an associative algebraic structure (S, ${\cdot}$) there exists a positive integer r such that $ab=b^ra$, then S is called quasi-commutative. Evidently, every abelian group or commutative semigroup is quasi-commutative. Also every finite Hamiltonian group that may be considered as a semigroup, is quasi-commutative however, there are quasi-commutative semigroups which are non-group and non commutative. In this paper, we provide three finitely presented non-commutative semigroups which are quasi-commutative. These are the first given concrete examples of finite semigroups of this type.

• 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.

• 대한수학회지
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• 제57권1호
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• pp.21-59
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• 2020

Let L_K denote the hypothetical automorphic Langlands group of a number field K. In our recent study, we briefly introduced a certain unconditional non-commutative topological group ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$, called the Weil-Arthur idèle group of K, which, assuming the existence of L_K, comes equipped with a natural topological group homomorphism $NR{\frac{\varphi}{K}^{Langlands}}$ : ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$ → L_K that we called the "Langlands form" of the global nonabelian norm-residue symbol of K. In this work, we present a detailed construction of ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$ and $NR{\frac{\varphi}{K}^{Langlands}}$ : ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$ → L_K, and discuss their basic properties.

• 충청수학회지
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• 제1권1호
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• pp.85-90
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• 1988

We develope a signature invariant for odd higher dimensional links. This signature has an advantage that it is defined as a G-signature for a non-abelian group G so that it can distinguish two links whose different were not detected by other invariants defined on commutative set-ups.