• The Transactions of the Korea Information Processing Society
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• v.2 no.1
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• pp.75-80
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• 1995

In this paper, we show that it is more efficient to use a new algorithm than to use a method of trace definition and property when we use trace calculation method on trinomial irreducible polynomial of reed-solomon code. This implementation has been done in SUN SPARC2 workstation using C-language.

• Bulletin of the Korean Mathematical Society
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• v.51 no.6
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• pp.1615-1623
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• 2014

Let $G{\leq}S_n$ and ${\chi}$ be any nonzero complex valued function on G. We first study the irreducibility of the generalized matrix polynomial $d^G_{\chi}(X)$, where $X=(x_{ij})$ is an n-by-n matrix whose entries are $n^2$ commuting independent indeterminates over $\mathbb{C}$. In particular, we show that if $\mathcal{X}$ is an irreducible character of G, then $d^G_{\chi}(X)$ is an irreducible polynomial, where either $G=S_n$ or $G=A_n$ and $n{\neq}2$. We then give a necessary and sufficient condition for the equality of two generalized matrix functions on the set of the so-called ${\chi}$-singular (${\chi}$-nonsingular) matrices.

• Proceedings of the Korea Institutes of Information Security and Cryptology Conference
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• 1991.11a
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• pp.144-151
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• 1991

• Honam Mathematical Journal
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• v.41 no.3
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• pp.651-659
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• 2019

A certain class of polynomials with integer coefficients are considered. If one of the coefficients is large enough in modulus with additional assumptions, then the irreducibility over the field of rationals is proved.

• ETRI Journal
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• v.39 no.4
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• pp.570-581
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• 2017

Multiplication in finite fields is used in many applications, especially in cryptography. It is a basic and the most computationally intensive operation from among all such operations. Several systolic multipliers are proposed in the literature that offer low hardware complexity or high speed. In this paper, a bit-parallel polynomial basis systolic multiplier for generic irreducible polynomials is proposed based on a modified interleaved multiplication method. The hardware complexity and delay of the proposed multiplier are estimated, and a comparison with the corresponding multipliers available in the literature is presented. Of the corresponding multipliers, the proposed multiplier achieves a reduction in the hardware complexity of up to 20% when compared to the best multiplier for m = 163. The synthesis results of application-specific integrated circuit and field-programmable gate array implementations of the proposed multiplier are also presented. From the synthesis results, it is inferred that the proposed multiplier achieves low power consumption and low area complexitywhen compared to the best of the corresponding multipliers.

• Journal of KIISE:Computer Systems and Theory
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• v.31 no.1_2
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• pp.112-117
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• 2004

In this paper, we propose a suitable multiplication architecture for cellular automata in a finite field $GF(2^m)$. Proposed least significant bit first multiplier is based on irreducible all one Polynomial, and has a latency of (m+1) and a critical path of $ 1-D_{AND}＋1-D{XOR}$.Specially it is efficient for implementing VLSI architecture and has potential for use as a basic architecture for division, exponentiation and inverses since it is a parallel structure with regularity and modularity. Moreover our architecture can be used as a basic architecture for well-known public-key information service in $GF(2^m)$ such as Diffie-Hellman key exchange protocol, Digital Signature Algorithm and ElGamal cryptosystem.

• The Journal of the Korea institute of electronic communication sciences
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• v.13 no.2
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• pp.383-390
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• 2018

Since cellular automata(CA) is superior to LFSR in randomness, it is applied as an alternative of LFSR in various fields. However, constructing CA corresponding to a given polynomial is more difficult than LFSR. Cattell et al. and Cho et al. showed that irreducible polynomials are CA-polynomials. And Cho et al. and Sabater et al. gave a synthesis method of 90/150 CA corresponding to the power of an irreducible polynomial, which is applicable as a shrinking generator. Swan characterizes the parity of the number of irreducible factors of a trinomial over the finite field GF(2). These polynomials are of practical importance when implementing finite field extensions. In this paper, we show that the trinomial $x^{2^n-1}+X+1$ ($n{\geq}2$) are CA-polynomials. Also the trinomial $x^{2^a(2^n-1)}+x^{2^a}+1$ ($n{\geq}2$, $a{\geq}0$) are CA-polynomials.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.21 no.2
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• pp.3-11
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• 2011

Evaluation of cube roots in characteristic three finite fields is required for Tate (or modified Tate) pairing computation. The Hamming weights (the number of nonzero coefficients) in the polynomial representations of $x^{1/3}$ and $x^{2/3}$ determine the efficiency of cube roots computation, where $F_{3^m}$is represented as $F_3[x]/(f)$ and $f(x)=x^m+ax^k+b{\in}F_3[x]$ (a, $b{\in}F_3$) is an irreducible trinomial. O. Ahmadi et al. determined the Hamming weights of $x^{1/3}$ and $x^{2/3}$ for all irreducible trinomials. In this paper, we present formulas for cube roots in $F_{3^m}$ using the shifted polynomial basis(SPB). Moreover, we provide the suitable shifted polynomial basis bring no further modular reduction process.

• Korean Journal of Mathematics
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• v.21 no.3
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• pp.319-323
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• 2013

Let D be a principal ideal domain, X be an indeterminate over D, D[X] be the polynomial ring over D, and $R_n=D[X]/(X^n)$ for an integer $n{\geq}1$. Clearly, $R_n$ is a commutative Noetherian ring with identity, and hence each nonzero nonunit of $R_n$ can be written as a finite product of irreducible elements. In this paper, we show that every irreducible element of $R_n$ is a primary element, and thus every nonunit element of $R_n$ can be written as a finite product of primary elements.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.833-842
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• 2016

For a certain kind of reciprocal polynomials $P(x),Q(x){\in}{\mathbb{Z}}[x]$, their sums are considered. We demonstrate that the Mahler measure of polynomials plays a role to prove the irreducibility of the sums over the field of rationals.