• 한국통신학회논문지
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• 제30권6C호
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• pp.577-584
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• 2005

본 연구는 협대역 통신시스템을 위한 전처리기-등화기 구조의 여파기에서, 곱셈기를 사용하지 않는 최소 복잡도의 디지털 FIR 여파기를 설계하는 방법을 제안한다. 제안하는 여파기는 순환 다항식(cyclotomic polynomial, CP) 여파기와 2차 내삽 다항식(interpolated second order polynomial, ISOP) 등화기로 구성되며, 이 두 여파기가 동시에 혼합 정수 선형 계획법(mixed integer linear programming (MILP))으로 최적 설계되어 최소의 복잡도를 갖는 특성을 갖게 된다. 제안된 방식으로 설계된 여파기들은, 설계 규격을 만족하면서도 기존의 여파기에 비하여 복잡도면에서 월등히 간단함을 확인하였다.

• 대한전자공학회논문지SP
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• 제42권4호
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• pp.143-152
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• 2005

본 연구는 협대역 응용 시스템을 위한 전처리기-등화기 구조의 여파기에서, 최소의 복잡도를 갖는 곱셈기 없는 디지털 IIR 여파기의 설계 방식을 제안한다. 제안하는 여파기는 순환 다항식 (cyclotomic polynomial (CP)) 여파기와 1차 내삽 다항식(interpolated second order polynomial (EOP))을 근간으로 하는 al1-pole 등화기로 구성 되며, 이 두 여파기가 동시에 혼합 정수 선형계획법(miked integer linear programming (MILP))으로 최적 설계된다. 설계된 여파기는 최소의 복잡도를 갖는 특성을 가지고 있다. 뿐만 아니라, 이 MILP 방식은 계산 복잡도와 위상 응답의 비선형 특성을 모두 최소화하도록 설계한다. 설계 예제를 통하여 제안된 설계 방식으로 설계된 여파기는 구현 요구사항을 만족하면서 기존의 설계 방식에 비하여 복잡도면에서 월등히 우수한 특성을 보임을 확인하였다.

• 호남수학학술지
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• 제40권2호
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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 $e_i{\geq}2$.

• 대한수학회지
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• 제59권3호
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• pp.621-634
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• 2022

The scaled inverse of a nonzero element a(x) ∈ ℤ[x]/f(x), where f(x) is an irreducible polynomial over ℤ, is the element b(x) ∈ ℤ[x]/f(x) such that a(x)b(x) = c (mod f(x)) for the smallest possible positive integer scale c. In this paper, we investigate the scaled inverse of (xⁱ - x^j) modulo cyclotomic polynomial of the form Φ_p^s (x) or Φ_p^sq^t (x), where p, q are primes with p < q and s, t are positive integers. Our main results are that the coefficient size of the scaled inverse of (xⁱ - x^j) is bounded by p - 1 with the scale p modulo Φ_p^s (x), and is bounded by q - 1 with the scale not greater than q modulo Φ_p^sq^t (x). Previously, the analogous result on cyclotomic polynomials of the form Φ_2ⁿ (x) gave rise to many lattice-based cryptosystems, especially, zero-knowledge proofs. Our result provides more flexible choice of cyclotomic polynomials in such cryptosystems. Along the way of proving the theorems, we also prove several properties of {x^k}_k∈ℤ in ℤ[x]/Φ_pq(x) which might be of independent interest.

• 대한수학회보
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• 제59권6호
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• pp.1511-1522
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• 2022

Let H be a subgroup of $\mathbb{Z}^*_n$ (the multiplicative group of integers modulo n) and h₁, h₂, …, h_l distinct representatives of the cosets of H in $\mathbb{Z}^*_n$. We now define a polynomial J_n,H(x) to be $$J_{n,H}(x)=\prod^l_{j=1} \left( x-\sum_{h{\in}H} {\zeta}^{h_jh}_n\right)$$, where ${\zeta}_n=e^{\frac{2{\pi}i}{n}}$ is the nth primitive root of unity. Polynomials of such form generalize the nth cyclotomic polynomial $\Phi_n(x)={\prod}_{k{\in}\mathbb{Z}^*_n}(x-{\zeta}^k_n)$ as J_n,{1}(x) = Φ_n(x). While the nth cyclotomic polynomial Φ_n(x) is irreducible over ℚ, J_n,H(x) is not necessarily irreducible. In this paper, we determine the subgroups H for which J_n,H(x) is irreducible over ℚ.

• 대한수학회지
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• 제56권2호
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• pp.559-578
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• 2019

We list all subfields of cyclotomic function fields over rational function fields with class number three. We also determine all the imaginary abelian extensions with relative class number three, explicitly.

• ETRI Journal
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• 제33권4호
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• pp.656-659
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• 2011

In ICISC 2007, Comuta and others showed that among the methods for constructing pairing-friendly curves, those using cyclotomic polynomials, that is, the Brezing-Weng method and the Freeman-Scott-Teske method, are affected by Cheon's algorithm. This paper proposes a method for searching parameters of pairing-friendly elliptic curves that induces minimal security loss by Cheon's algorithm. We also provide a sample set of parameters of BN-curves, FST-curves, and KSS-curves for pairing-based cryptography.

• 대한수학회논문집
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• 제31권4호
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• pp.667-682
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• 2016

In this article we show a recursive method to compute the coefficients of the minimal polynomial of cos($2{\pi}/n$) explicitly for $n{\geq}3$. The recursion is not on n but on the coefficient index. Namely, for a given n, we show how to compute ei of the minimal polynomial ${\sum_{i=0}^{d}}(-1)^ie_ix^{d-i}$ for $i{\geq}2$ with initial data $e_0=1$, $e_1={\mu}(n)/2$, where ${\mu}(n)$ is the $M{\ddot{o}}bius$ function.

• 충청수학회지
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• 제20권4호
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• pp.433-438
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• 2007

Let $\mathbb{A}=\mathbb{F}_q[T]$ and $k=\mathbb{F}_q(T)$. Assume q is odd, and fix a prime divisor ${\ell}$ of q - 1. Let P be a monic irreducible polynomial in A whose degree d is divisible by ${\ell}$. In this paper we define a subgroup $\tilde{C}_F$ of $\mathcal{O}^*_F$ which is generated by $\mathbb{F}^*_q$ and $\{{\eta}^{{\tau}^i}:0{\leq}i{\leq}{\ell}-1\}$ in $F=k(\sqrt[{\ell}]{P})$ and calculate the unit-index $[\mathcal{O}^*_F:\tilde{C}_F]={\ell}^{\ell-2}h(\mathcal{O}_F)$. This is a generalization of [3, Theorem 16.15].