• Journal of applied mathematics & informatics
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• v.32 no.3_4
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• pp.529-537
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• 2014

The paper introduces a new product of polynomials defined over a field. It is a generalization of the ordinary product with inner polynomial getting non-overlapping segments obtained by multiplying with coefficients and variable with expanding powers. It has been called 'Ordered Power Product' (OPP). Considering two rings of polynomials $R_m[x]=F[x]modulox^m-1$ and $R_n[x]=F[x]modulox^n-1$, over a field F, the paper then considers the newly introduced product of the two polynomial rings. Properties and algebraic structure of the product of two rings of polynomials are studied and it is shown to be a ring. Using the new type of product of polynomials, we define a new product of two cyclic codes and devise a method of getting a cyclic code from the 'ordered power product' of two cyclic codes. Conditions for the OPP of the generators polynomials of component codes, giving a cyclic code are examined. It is shown that OPP cyclic code so obtained is more efficient than the one that can be obtained by Kronecker type of product of the same component codes.

• Proceedings of the KSRS Conference
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• 2007.10a
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• pp.537-539
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• 2007

Telemetry processing system requires minimum bit transition level in data streams to maintain a bit synchronization while receiving telemetry signal. PN code has a capability of providing the bit transition and is widely used in the packet communication of CCSDS. CCSDS PN code that generator polynomial is $h(x)=x^{8}+x^{7}+x^{5}+x^{3}+1$, and the random bit sequence that is generated from this polynomial is repeated with the cycle of 255 bits. As the resolution of satellite image increases, the size and transmission rate of data increases. To process of huge and bulky size of satellite image, the speed of CCSDS PN Processing is very important. This paper introduces the way of improving the CCSDS PN Processing speed through processing 128 bits at one time using the feature of cyclic structure that repeats after first 255 bytes by grouping the random bit sequence with 1 byte and Intel Streaming SIMD Extensions 2. And this paper includes the comparison data of processing speed between SSE2-applied implementation and not-applied implementation, in addition, the measured value of speed improvement.

• 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.

• The Journal of the Korea institute of electronic communication sciences
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• v.13 no.4
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• pp.819-826
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• 2018

The generalized Hamming weight is one of the important parameters of the linear code. It determines the performance of the code when the linear codes are applied to a cryptographic system. In addition, when the block code is decoded by soft decision using the lattice diagram, it becomes a measure for evaluating the state complexity required for the implementation. In particular, a bit-parallel multiplier on finite fields based on trinomials have been studied. Cellular automata(CA) has superior randomness over LFSR due to its ability to update its state simultaneously by local interaction. In this paper, we deal with the efficient synthesis of the pseudo random number generator, which is one of the important factors in the design of effective cryptosystem. We analyze the property of the characteristic polynomial of the simple 90/150 transition rule block, and propose a synthesis algorithm of the reversible 90/150 CA corresponding to the trinomials $x^2^n+x^{2^n-1}+1$($n{\geq}2$) and the 90/150 reversible CA(RCA) corresponding to the maximum weight polynomial with $2^n$ degree by using this rule block.

• Journal of IKEEE
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• v.22 no.4
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• pp.896-900
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• 2018

Traditionally, Linear Shift Feedback Register (LFSR) has been widely employed to implement Cyclic Redundant Check (CRC) codes for a serial input. Since many applications including network and storage systems demand as high throughput as ever, various efforts have been made to implement CRC hardware to support parallel inputs. Among various parallel schemes, the look-ahead scheme is one of the most widely used schemes due to its short critical path. However, it is very cumbersome to design HDL codes for parallel CRC codes since the look-ahead scheme is inevitable to consider how register and input values move in the next cycles. Thus, this paper proposes a novel CRC hardware generator, which automatically produces HDL codes given a CRC polynomial and parallel factor. The experimental results verify the applicability to use the proposed generator by analyzing the synthesis results from the generated HDL code.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.24 no.1 s.173
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• pp.225-231
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• 2000

Dimensional synthesis of planar and spatial mechanisms mostly requires solution-finding, procedure for a system of polynomial equations. In case the system is nonlinear, numerical techniques like Newton-Raphson are often used. But there are no logical ways for finding all possible solutions in such iterative methods. In this paper, algebraic elimination is used to get all solutions for the synthesis of 5-SS spatial mechanism with seven prescribed positions. The proposed algorithm is more suitable for computer implementation and takes less time than existing one. Two numerical examples are given to demonstrate the implemented algorithm.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.487-497
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• 2006

The purpose of this paper is to describe the structure of the rings $\mathbb{Z}_{p^2}[X]/({\alpha}(X))$ with ${\alpha}(X)$ a monic polynomial and $\={X}^{\kappa}=0$ for some nonnegative integer ${\kappa}$. Especially we will see that any ideal of such rings can be generated by at most two elements of the special form and we will find the 'minimal' set of generators of the ideals. We indicate how to identify the isomorphism types of the ideals as $\mathbb{Z}_{p^2}-modules$ by finding the isomorphism types of the ideals of some particular ring. Also we will find the annihilators of the ideals by finding the most 'economical' way of annihilating the generators of the ideal.

• Journal of IKEEE
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• v.21 no.2
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• pp.130-135
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• 2017

Normally, in data transmission, extra parity bits are added to the input message which were derived from its input and a pre-defined algorithm. The same algorithm is used by the receiver to check the consistency of the delivered information, to determine if it is corrupted or not. It recovers and compares the received information, to provide matching and correcting the corrupted transmitted bits if there is any. This paper aims the following objectives: to use an alternative error detection-correction method, to lessens both the fixed number of the required redundancy bits 'r' in cyclic redundancy checking (CRC) because of the required polynomial generator and the overhead of interspersing the r in Hamming code. The experimental results were synthesized using Xilinx Virtex-5 FPGA and showed a significant increase in both the transmission rate and detection of random errors. Moreover, this proposal can be a better option for detecting and correcting errors.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.25 no.10A
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• pp.1576-1581
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• 2000

We investigated into the minimum distance of a primitive binary cyclic code C with a generator polynomial g(x)=$m_1(x)m_{d}(x)$. It is known that the necessary and sufficient condition for C to have minimum distance five is the fact that \ulcorner is an APN power function. In this paper we derived the new proof of minimum distance for the primitive binary cyclic codes with minimum distance five.

• Journal of Korea Multimedia Society
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• v.24 no.3
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• pp.406-415
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• 2021

The worldwide spread of the Internet and the digital information revolution have resulted in a rapid increase in the use and transmission of multimedia information due to the rapid development of communication technologies. It is important to protect images in order to prevent problems such as piracy and illegal distribution. To solve this problem, I propose a new digital color image encryption algorithm in this paper. I design a new pseudo-random number generator based on 1D five-neighborhood maximum length cellular automata (FN-MLCA) to change the pixel values of the plain image into unpredictable values. And then I use a 3D chaotic cat map to effectively shuffle the positions of the image pixel. In this paper, I propose a method to construct a new MLCA by modeling 1D FN-MLCA. This result is an extension of 1D 3-neighborhood CA and shows that more 1D MLCAs can be synthesized. The safety of the proposed algorithm is verified through various statistical analyses.