• Aerospace Engineering and Technology
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• v.6 no.2
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• pp.157-163
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• 2007

The purpose of this study is to design and simulate Reed Solomon encoder(255,223) in PCM/FM communication system in order to transmit the KSLV-I onboard video data. Especially in the compressed video data transmission applications, the communication system is required to have a very low BER performance because of interframe or interframe compression techniques. We have used the primitive polynomial of CCSDS standard and calculated the various coefficients and then the encoder have been simulated as a part of RF interface FPGA hardware in a video compression unit.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.14 no.6
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• pp.25-30
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• 2004

Linear cellular automata(CA) which generate maximum-length cycles, have wide applications in generation of pseudo-random patterns, signature analysis, cryptography and error correcting codes etc. Linear CA whose characteristic polynomial is primitive has been studied. In this paper Ive propose a effective method for generation of a variety of maximum-length CA(MLCA). And we show that the complemented CA's derived from a linear MLCA are all MLCA. Also we analyze the Properties of complemented MLCA. And we prove that the number of n-cell MLCA is ${\phi}(2^{n}-1)2^{n+1}$/n.

• Journal of the Institute of Electronics Engineers of Korea SC
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• v.38 no.1
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• pp.27-34
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• 2001

The multiplication algorithm using the primitive irreducible trinomial $x^m+x+1$ over $GF(2^m)$ was proposed by Mastrovito. The multiplier proposed in this paper consisted of the multiplicative operation unit, the primitive irreducible operation unit and mod operation unit. Among three units mentioned above, the Primitive irreducible operation was modified to primitive irreducible trinomial $x^m+x+1$ that satisfies the range of 1$x^m,{\cdots},x^{2m-2}\;to\;x^{m-1},{\cdots},x^0$ is reduced. In this paper, the primitive irreducible polynomial was reduced to the primitive irreducible trinomial proposed. As a result of this reduction, the primitive irreducible trinomial reduced the size of circuit. In addition, the proposed design of multiplier was suitable for VLSI implementation because the circuit became regular and modular in structure, and required simple control signal.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.14 no.3
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• pp.41-48
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• 2004

Finite field multiplication and division are important arithmetic operation in error-correcting codes and cryptosystems. The elements of the finite field GF($2^m$) are represented by bases with a primitive polynomial of degree m over GF(2). We can be easily realized for multiplication or computing multiplicative inverse in GF($2^m$) based on a normal basis representation. The number of product terms of logic function determines a complexity of the Messay-Omura multiplier. A normal basis exists for every finite field. It is not easy to find the optimal normal element for a given primitive polynomial. In this paper, the generating method of normal basis is investigated. The normal bases whose product terms are less than other bases for multiplication in GF($2^m$) are found. For each primitive polynomial, a list of normal elements and number of product terms are presented.

• Journal of the Korean Institute of Telematics and Electronics A
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• v.31A no.8
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• pp.1-6
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• 1994

In this paper, we introduce the nonlinear combiner structure which improves linear complexity and randomness properties on maximum length sequences generated by LFSR. Choosing the primitive polynomial over GF(2S04T) as feedback tap polynomial, we devise nonlinear combiner structure and analyze the random output sequences generated by LFSR with nonlinear function.

• The Journal of the Korea institute of electronic communication sciences
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• v.13 no.1
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• pp.133-140
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• 2018

In this paper we discover the generation principle of a sequence when the characteristic polynomial of the sequence is a power of a primitive polynomial. With the generation principle, we can rearrange a sequence. Also we get the linear complexity and the required term of the sequence efficiently.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.2
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• pp.137-145
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• 1999

In this paper, We consider two-dimensional Maximum Length Cellular Automata (2-D MLCA) as an extension of the 1-D MLCA. 2-D MLCA can display much better random patterns than those generated by 1-D CA and LFSR. To generate random pattern, a CA should have a maximum length cycle. So, it is necessary to find MLCA that the characteristic polynomial of the transition matrix is primitive. New boundary conditions of 3 types are proposed and some rules having primitive polynomials of 2-D MLCA are found.

• Korean Journal of Mathematics
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• v.27 no.2
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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.

• The Mathematical Education
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• v.48 no.1
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• pp.81-91
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• 2009

This paper focuses on two problems in the 10th grade mathematics, the rational zero theorem and the content(the integer divisor) of a polynomial Among 138 students participated in the problem solving, 58 of them (42 %) has used the rational zero theorem for the factorization of polynomials. However, 30 of 58 students (52 %) consider the rational zero theorem is a mathematical fake(false statement) and they only use it to get a correct answer. There are three different types in the textbooks in dealing with the content of a polynomial with integer coefficients. Computing the greatest common divisor of polynomials, some textbooks consider the content of polynomials, some do not and others suggest both methods. This also makes students confused. We suggests that a separate section of the rational zero theorem must be included in the text. As for the content of a polynomial, we consider the polynomials are contained in the polynomial ring over the rational numbers. So computing the gcd of polynomials, guide the students to give a monic(or primitive) polynomial as ail answer.

• Journal of the Korea Society of Computer and Information
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• v.18 no.2
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• pp.9-17
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• 2013

In this paper, we propose a new multiplication algorithm for two polynomials using primitive polynomial with all 1 of coefficient on finite fields GF($2^m$), and design the multiplier with high-speed parallel input-output module structure using the presented multiplication algorithm. The proposed multiplier is designed $m^2$ same basic cells that have a 2-input XOR gate and a 2-input AND gate. Since the basic cell have no a latch circuit, the multiplicative circuit is very simple and is short the delay time $D_A+D_X$ per cell unit. The proposed multiplier is easy to extend the circuit with large m having regularity and modularity by cell array, and is suitable to the implementation of VLSI circuit.