• Journal of Korea Society of Industrial Information Systems
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• v.8 no.1
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• pp.92-96
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• 2003

본 논문에서는 비대칭 터보부호를 사용 무선 ATM환경에서 ATM셀을 전송할 때 비트 오류확률, 셀 손실확률을 대칭 터보부호와 비교하였다. 일반적으로 터보부호기는 동일한 재귀 컨벌루션코드(Resursive Systematic Convolution Code)로 구성된다. 비대칭 터보부호와 대칭적 터보부호를 비교하기 위한 생성다항식을 Primitive Polynomial과 Non-Primitive Polynomial 의 4가지 조합을 고려하여 시뮬레이션을 수행하였다. Feedback 다항식이 Primitive 다항식인 사용된 터보부호기가 Non-Primitive다항식이 사용된 터보부호기와 비교하여 1.2dB이상 성능 향상을 보였고, Primitive Polynomial과 Non-Primitive Polynomial이 같이 사용된 비대칭 터보부호기는 Non-Primitive Polynomial이 사용된 대칭 터보부호기와 비교하여 0.7dB 성능 향상을 보였다.

• 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.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.11 no.1
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• pp.35-42
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• 2001

The primitive polynomial on GF(q) is used in the area of the scrambler, the error correcting code and decode, the random generator and the cipher, etc. The algorithm that generates efficiently the primitive polynomial on GF(q) was proposed by A.D. Porto. The algorithm is a method that generates the sequence of the primitive polynomial by repeating to find another primitive polynomial with a known primitive polynomial. In this paper, we propose the algorithm that is improved in the A.D. Porto algorithm. The running rime of the A.D. Porto a1gorithm is O($\textrm{km}^2$), the running time of the improved algorithm is 0(m(m+k)). Here, k is gcd(k, $q^m$-1). When we find the primitive polynomial with m odor, it is efficient that we use the improved algorithm in the condition k, m>>1.

• Journal of information and communication convergence engineering
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• v.16 no.3
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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.

• Honam Mathematical Journal
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• v.46 no.1
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• pp.136-145
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• 2024

In this paper we obtained several properties that the characteristic polynomial of the unit-primitive matrix satisfies. In addition, using these properties we have shown that the recurrence relation given as in the formula (1) is true. In fact, Xin and Zhong ([4]) showed it earlier. However, we provide a simpler method here.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.449-454
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• 2012

In this paper, we find the minimal polynomial of a primitive root of unity over cyclotomic fields. From this, we factorize cyclotomic polynomials over cyclotomic fields and investigate the coefficients of ${\Phi}_{3n}(x)$ when 3∤$n$.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.341-360
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• 1998

Some interesting properties of Carlitz cyclotomic polynomials analogous to those of classical cyclotomic polynomials are given.

• Communications of the Korean Mathematical Society
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• v.32 no.1
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• pp.1-6
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• 2017

In this paper, the fundamental theorem of Galois Theory is used to generalize cyclotomic polynomials and construct irreducible polynomials associated with the n-th primitive roots of unity.

• East Asian mathematical journal
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• v.16 no.1
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• pp.71-79
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• 2000

We show that the intersection of two standard torus knots of type (${\lambda}_1$, ${\lambda}_2$) and (${\beta}_1$, ${\beta}_2$) induces an automorphism of the cyclic group ${\mathbb{Z}}_d$, where d is the intersection number of the two torus knots and give an elementary proof of the fact that all non-trivial torus knots are strongly invertiable knots. We also show that the intersection of two standard knots on the 3-torus $S^1{\times}S^1{\times}S^1$ induces an isomorphism of cyclic groups.

• The Journal of the Korea institute of electronic communication sciences
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• v.6 no.1
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• pp.13-19
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• 2011

Games gave the calculation method for the crosscorrelation function of a Kasami sequence and a No sequence that have been generated by the same primitive polynomial. In this paper, we calculate the crosscorrelation function of a Kasami sequence and a No sequence that have been generated by the same primitive polynomial with the periodic crosscorrelation function of two base sequences. Our method is different from the Games's method.