• Communications of the Korean Mathematical Society
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• v.28 no.1
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• pp.71-77
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• 2013

Consider a multiplicative group of integers modulo $n$, denoted by $\mathbb{Z}_n^*$. Any element $a{\in}\mathbb{Z}_n^*$ is said to be a semi-primitive root if the order of $a$ modulo $n$ is ${\phi}(n)/2$, where ${\phi}(n)$ is the Euler phi-function. In this paper, we discuss some interesting properties of the multiplicative groups of integers possessing semi-primitive roots and give its applications to solving certain congruences.

• Honam Mathematical Journal
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• v.33 no.2
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• pp.181-186
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• 2011

Consider a multiplicative group of integers modulo n, denoted by $\mathbb{Z}_n^*$. Any element $a{\in}\mathbb{Z}_n^*$ n is said to be a semi-primitive root if the order of a modulo n is $\phi$(n)/2, where $\phi$(n) is the Euler phi-function. In this paper, we classify the multiplicative groups of integers having semi-primitive roots and give interesting properties of such groups.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.403-410
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• 2006

For a finite group G, #Cent(G) denotes the number of centralizers of its elements. A group G is called n-centralizer if #Cent(G) = n, and primitive n-centralizer if #Cent(G) = #Cent($\frac{G}{Z(G)}$) = n. The first author in [1], characterized the primitive 6-centralizer finite groups. In this paper we continue this problem and characterize the primitive 7-centralizer finite groups. We prove that a finite group G is primitive 7-centralizer if and only if $\frac{G}{Z(G)}{\simeq}D_{10}$ or R, where R is the semidirect product of a cyclic group of order 5 by a cyclic group of order 4 acting faithfully. Also, we compute #Cent(G) for some finite groups, using the structure of G modulu its center.

• The Journal of Korea Institute of Information, Electronics, and Communication Technology
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• v.4 no.3
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• pp.197-201
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• 2011

The use of gestures provides an attractive alternate to cumbersome interfaces for human-computer devices interaction. This has motivated a very active research area concerned with computer vision-based recognition of hand gestures. The most important issues in hand gesture recognition is to recognize the directivity of finger. The primitive elements extracted to a hand gesture include in very important information on the directivity of finger. In this paper, we propose the recognition algorithm of finger directivity by using the cross points of circle and sub-primitive element. The radius of circle is increased from minimum radius including main-primitive element to it including sub-primitive elements. Through the experiment, we demonstrated the efficiency of proposed algorithm.

• Convergence Security Journal
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• v.18 no.4
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• pp.27-33
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• 2018

Stream cipher is an one-time-pad type encryption algorithm that encrypt plaintext using simple operation such as XOR with random stream of bits (or characters) as symmetric key and its security depends on the randomness of used stream. Therefore we can design more secure stream cipher algorithm by using mathematical analysis of the stream such as period, linear complexity, non-linearity, correlation-immunity, etc. The key stream in stream cipher is generated in linear feedback shift register(LFSR) having characteristic polynomial. The primitive polynomial is the characteristic polynomial which has the best security property. It is used widely not only in stream cipher but also in SEED, a block cipher using 8-degree primitive polynomial, and in Chor-Rivest(CR) cipher, a public-key cryptosystem using 24-degree primitive polynomial. In this paper we present the concept and various properties of primitive polynomials in Galois field and prove the theorem finding the number of irreducible polynomials and primitive polynomials over $F_p$ when p is larger than 2. This kind of research can be the foundation of finding primitive polynomials of higher security and developing new cipher algorithms using them.

• The Transactions of the Korea Information Processing Society
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• v.7 no.9
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• pp.2913-2919
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• 2000

It is essential to get large prime numbers in the design of asymmetric encryption algorithm. However, the pseudoprime numbers with high possibility to be primes have been generally used in the asymmetric encryption algorithms, because it is very difficult to find large deterministic prime numbers. In this paper, we propose a new method of deterministic prime number generation. The prime numbers generated by the proposed method have a 100% precise prime characteristic. They are also guaranteed reliability, security strength, and an ability of primitive element generation.

• Proceedings of the IEEK Conference
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• 2001.06d
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• pp.37-40
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• 2001

The most important issues in gesture recognition are the simplification of algorithm and the reduction of processing time. The mathematical morphology based on geometrical set theory is best used to perform the real-time processing. A key idea of the algorithm proposed in this paper is to apply morphological shape decomposition. The primitive elements extracted from a hand gesture have very important information including the directivity of the hand gestures. Based on this algorithm, we proposed the morphological hand-gesture recognition algorithm using feature vectors extracted from lines connecting the center points of a main-primitive element and sub-primitive elements. Through the experiments, we applied to the video contents browsing system with natural interactions and demonstrated the efficiency of this algorithm.

• Proceedings of the Korea Institutes of Information Security and Cryptology Conference
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• 1992.11a
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• pp.127-130
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• 1992

GF(2m) 이론은 switching 이론과 컴퓨터 연산, 오류 정정 부호(error correcting codes), 암호학(cryptography) 등에 대한 폭넓은 응용 때문에 주목을 받아 왔다. 특히 유한체에서의 이산 대수(discrete logarithm)는 one-way 함수의 대표적인 예로서 Massey-Omura Scheme을 비롯한 여러 암호에서 사용하고 있다. 이러한 암호 system에서는 암호화 시간을 동일하게 두면 고속 연산은 유한체의 크기를 크게 할 수 있어 비도(crypto-degree)를 향상시킨다. 따라서 고속 연산의 필요성이 요구된다. 1981년 Massey와 Omura가 정규기저(normal basis)를 이용한 고속 연산 방법을 제시한 이래 Wang, Troung 둥 여러 사람이 이 방법의 구현(implementation) 및 곱셈기(Multiplier)의 설계에 힘써왔다. 1988년 Itoh와 Tsujii는 국제 정보 학회에서 유한체의 역원을 구하는 획기적인 방법을 제시했다. 1987년에 H, W. Lenstra와 Schoof는 유한체의 임의의 확대체는 원시정규기저(primitive normal basis)를 갖는다는 것을 증명하였다. 1991년 Stepanov와 Shparlinskiy는 유한체에서의 원시원소(primitive element), 정규기저를 찾는 고속 연산 알고리즘을 개발하였다. 이 논문에서는 원시 정규기저를 찾는 Algorithm을 구현(Implementation)하고 이것이 응용되는 문제들에 관해서 연구했다.

• 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.

• Transactions of Materials Processing
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• v.4 no.1
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• pp.28-38
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• 1995

An integrated process planning system can determine desirable operation sequences even if they have little experience in the design of multistage cold forging process. This system is composed of seven major modules such as input module, pre-design module, formability check module, forming sequence design module, forming analysis module, FEM verification module, and output module which are used independently or in all. The forming sequence for the part can be determined by means of primitive geometries such as cylinder, cone, convex, and concave. By utilizing this geometrical characteristics(diameter, height, and radius), the part geometry is expressed by a list of the primitive geometries. Accordingly, the forming sequence design is formulated as the search problem which starts with a billet geometry and finishes with a given product one. Using the developed system, the sequence drawing with all dimensions, which includes the dimensional tolerances and the proper sequence of operations for parts, is generated under the environment of AutoCAD. Several forming sequences generated by the planning system can be checked by the forming analysis module. The acceptable forming sequences can be verified further, using FE simulation.