• 정보보호학회논문지
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• 제19권6호
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• pp.3-15
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• 2009

최근 Wu는 효율적인 비트-병렬 곱셈기를 위한 세 가지 종류의 이진체 제안하였다. 제안된 곱셈기는 오항 기약다항식을 사용하는 기존의 결과보다 효율적이다. 본 논문에서는 비트-병렬 곱셈에서 효율적인 이진체 위의 새로운 반복다항식(Repeated Polynomial:RP)을 제안한다. 제안하는 RP를 case 1, case 2와 case 3 3가지로 구분할 때, 제안하는 RP를 위한 비트-병렬 곱셈기는 기존의 오항 기약다항식의 결과보다 효율적이다. 유한체의 차수가 1,000이하에서 EPS 또는 삼항 기약다항식이 없는 차수를 고려할 때, Wu의 단지 11개의 유한체만 존재한다. 그러나 제안하는 결과는 case 1에서 181, case 2에서 232 그리고 case 3에서 443개의 유한체가 존재한다.

• 대한수학회지
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• 제48권5호
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• pp.1065-1081
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• 2011

For any field F, a polynomial f $\in$ F[$x_1,x_2,{\ldots},x_k$] can be associated with a polytope, called its Newton polytope. If the polynomial f has integrally indecomposable Newton polytope, in the sense of Minkowski sum, then it is absolutely irreducible over F, i.e., irreducible over every algebraic extension of F. We present some results giving new integrally indecomposable classes of polygons. Consequently, we have some criteria giving many types of absolutely irreducible bivariate polynomials over arbitrary fields.

• Journal of information and communication convergence engineering
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• 제16권3호
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• pp.166-172
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• 2018

A pseudo-random sequence generated by using a primitive polynomial, trace function, and Legendre symbol has been researched in our previous work. Our previous sequence has some interesting features such as period, autocorrelation, and linear complexity. A pseudo-random sequence widely used in cryptography. However, from the aspect of the practical use in cryptographic systems sequence needs to generate swiftly. Our previous sequence generated by utilizing a primitive polynomial, however, finding a primitive polynomial requires high calculating cost when the degree or the characteristic is large. It’s a shortcoming of our previous work. The main contribution of this work is to find some relation between the generated sequence and irreducible polynomials. The purpose of this relationship is to generate the same sequence without utilizing a primitive polynomial. From the experimental observation, it is found that there are (p - 1)/2 kinds of polynomial, which generates the same sequence. In addition, some of these polynomials are non-primitive polynomial. In this paper, these relationships between the sequence and the polynomials are shown by some examples. Furthermore, these relationships are proven theoretically also.

• Korean Journal of Mathematics
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• 제19권3호
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• pp.263-272
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• 2011

In the paper [1], an explicit correspondence between certain cubic irreducible polynomials over $\mathbb{F}_q$ and cubic irreducible polynomials of special type over $\mathbb{F}_{q^2}$ was established. In this paper, we show that we can mimic such a correspondence for quintic polynomials. Our transformations are rather constructive so that it can be used to generate irreducible polynomials in one of the finite fields, by using certain irreducible polynomials given in the other field.

• 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 $e_i{\geq}2$.

• Korean Journal of Mathematics
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• 제27권2호
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• pp.465-474
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• 2019

Let ${\mathbb{Z}}$ be the ring of integers and ${\mathbb{Z}}[X]$ (resp., $h({\mathbb{Z}})$) be the ring of polynomials (resp., Hurwitz polynomials) over ${\mathbb{Z}}$. In this paper, we study the irreducibility of Hurwitz polynomials in $h({\mathbb{Z}})$. We give a sufficient condition for Hurwitz polynomials in $h({\mathbb{Z}})$ to be irreducible, and we then show that $h({\mathbb{Z}})$ is not isomorphic to ${\mathbb{Z}}[X]$. By using a relation between usual polynomials in ${\mathbb{Z}}[X]$ and Hurwitz polynomials in $h({\mathbb{Z}})$, we give a necessary and sufficient condition for Hurwitz polynomials over ${\mathbb{Z}}$ to be irreducible under additional conditions on the coefficients of Hurwitz polynomials.

• 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.

• 정보보호학회논문지
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• 제20권5호
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• pp.11-22
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• 2010

최근 Fan과 Dai는 이진체 곱셈기의 효율성을 개선하기 위하여 Shifted Polynomial Basis(SPB)를 제안하고 이를 이용한 non-pipeline 비트-병렬 곱셈기를 제안하였다. SPB는 PB에 {1, ${\alpha}$, $\cdots$, ${\alpha}^{n-l}$}에 ${\alpha}^{-\upsilon}$를 곱한 것으로, 이 둘 사이는 매우 적은 비용으로 쉽게 기저 변환이 된다. 이후 삼항 기약다항식 $f(x)=x^n+x^k+1$을 사용하여 Modified Shifted Polynomial Basis(MSPB) 기반의 SPB 비트-병렬 Mastrovito type I과 type II 곱셈기가 제안되었다. 본 논문에서는 SPB를 이용한 비트-병렬 곱셈기를 제안한다. n ${\neq}$ 2k 일 때 제안하는 곱셈기 구조는 기존의 모든 SPB 곱셈기와 비교하여 효율적인 공간 복잡도를 가진다. 또한, 기존의 가장 작은 공간 복잡도를 가지는 곱셈기와 비교하여 1 ${\leq}$ k ${\leq}$ (n+1)/3인 경우 항상 효율적이다. 또한, (n+2)/3 $\leq$ k < n/2인 경우에도 일분 경우를 제외하고 기존 결과보다 항상 작은 공간 복잡도를 가진다.

• Korean Journal of Mathematics
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• 제30권3호
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• pp.425-431
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• 2022

Let ℤ and ℚ be the ring of integers and the field of rational numbers, respectively. Let h(ℤ, ℚ) be the ring of composite Hurwitz polynomials. In this paper, we study the factorization of composite Hurwitz polynomials in h(ℤ, ℚ). We show that every nonzero nonunit element of h(ℤ, ℚ) is a finite *-product of quasi-primary elements and irreducible elements of h(ℤ, ℚ). By using a relation between usual polynomials in ℚ[x] and composite Hurwitz polynomials in h(ℤ, ℚ), we also give a necessary and sufficient condition for composite Hurwitz polynomials of degree ≤ 3 in h(ℤ, ℚ) to be irreducible.

• 충청수학회지
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• 제25권4호
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• pp.771-777
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• 2012

We give a result on the density of certain additive arithmetic functions which count the number of restricted irreducible polynomials on the polynomial ring.