• Journal of the Korean Mathematical Society
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• v.44 no.3
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• pp.499-509
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• 2007

Let m, n be positive integers or monic polynomials in $\mathbb{F}_q[T]$ with n|m. Let $K_m\;and\;K^+_m$ be the m-th cyclotomic field and its maximal real subfield, respectively. In this paper we define two matrices $D^+_{m,n}\;and\;D^-_{m,n}$ whose determinants give us the ratios $\frac{h(\mathcal{O}_{K^+_m})}{h(\mathcal{O}_{K^+_n})}$ and $\frac{h-(\mathcal{O}_K_m)}{h-(\mathcal{O}_K_n)}$ with some factors, respectively.

• Honam Mathematical Journal
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• v.40 no.2
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• pp.281-291
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• 2018

We associate a positive integer n and a subgroup H of the group $({\mathbb{Z}}/n{\mathbb{Z}})^{\times}$ with a polynomial $J_n,H(x)$, which is called the Galois polynomial. It turns out that $J_n,H(x)$ is a polynomial with integer coefficients for any n and H. In this paper, we provide an equivalent condition for a subgroup H to provide the Galois polynomial which is irreducible over ${\mathbb{Q}}$ in the case of $n=p^{e_1}_1{\cdots}p^{e_r}_r$ (prime decomposition) with all $e_i{\geq}2$.

• The Pure and Applied Mathematics
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• v.8 no.2
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• pp.127-135
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• 2001

The main object of this paper is to present a transformation formula for a finite series involving $_3F_2$ and some identities associated with the binomial coefficients by making use of the theory of Legendre polynomials $P_{n}$(x) and some summation theorems for hypergeometric functions $_pF_q$. Some integral formulas are also considered.

• Honam Mathematical Journal
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• v.46 no.2
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• pp.267-278
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• 2024

In this paper, we investigate the uniqueness of a meromorphic function f(z) and its difference polynomial of difference operator with two sharing values counting multiplicities. Our two results improve and generalize the recent results of Barki Mahesh, Dyavanal Renukadevi S and Bhoosnurmath Subhas S [4] and for the case q ≥ 2, this allows for a highly unique generalization. To further demonstrate the validity of our main result, we provide an example.

• Korean Journal of Mathematics
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• v.32 no.3
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• pp.485-496
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• 2024

In this study, we examine ℓ-quasi-cyclic self-dual codes of length ℓm over 𝔽₂, provided that the polynomial X^m - 1 has exactly four distinct irreducible factors in 𝔽₂[X]. We find the standard form of generator matrices of codes over the ring R ≅ 𝔽_q[X]/(X^m - 1) and the conditions for the codes to be self-dual. We explicitly determine the forms of generator matrices of self-dual codes of lengths 2 and 4 over R.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.24 no.8
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• pp.1044-1051
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• 2020

The age of quantum computers is coming soon. In order to prepare for the upcoming future, the National Institute of Standards and Technology has recruited candidates to set standards for post quantum cryptography to establish a future cryptography standard. The submitted ciphers are expected to be safe from quantum algorithm attacks, but it is necessary to verify that the submitted algorithm is safe from quantum attacks using quantum algorithm even when it is actually operated on a quantum computer. Therefore, in this paper, we investigate an efficient quantum gate implementation for binary field multiplication of code based post quantum cryptography to work on quantum computers. We implemented the binary field multiplication for two field polynomials presented by Classic McEliece and three field polynomials presented by ROLLO in generic algorithm and Karatsuba algorithm.