• Proceedings of the Korea Contents Association Conference
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• 2005.11a
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• pp.389-392
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• 2005

Boolean matrices are applied to a variety of areas and used successfully in many applications. There are many researches on the application and multiplication of boolean matrices. Most researches deal with the multiplication of boolean matrices, but all of them focus on the multiplication of just two boolean matrices and very few researches deal with the multiplication of many pairs of two boolean matrices. The paper discusses it is not suitable to use for the multiplication of many pairs of two boolean matrices the algorithm for the multiplication of two boolean matrices that is considered optimal up to now, and suggests a method that can improve the multiplication of a $n{\times}m$ boolean matrix and all $m{\times}k$ boolean matrices.

• Bulletin of the Korean Mathematical Society
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• v.46 no.2
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• pp.373-385
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• 2009

An m${\times}$n Boolean matrix A is called regular if there exists an n${\times}$m Boolean matrix X such that AXA = A. We have characterizations of Boolean regular matrices. We also determine the linear operators that strongly preserve Boolean regular matrices.

• Journal of the Korea Computer Industry Society
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• v.7 no.5
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• pp.473-480
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• 2006

Extraction is tile most important step in global minimization. Its approache is to identify and extract subexpressions, which are multiple-cubes or single-cubes, common to two or more expressions which can be used to reduce the total number of literals in a Boolean network. Extraction is described as either algebraic or Boolean according to the trade-off between run-time and optimization. Boolean extraction is capable of providing better results, but difficulty in finding common Boolean divisors arises. In this paper, we present a new method for Boolean extraction to remove the difficulty. The key idea is to identify and extract two-cube Boolean subexpression pairs from each expression in a Boolean network. Experimental results show the improvements in the literal counts over the extraction in SIS for some benchmark circuits.

• Korean Journal of Optics and Photonics
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• v.13 no.1
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• pp.84-87
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• 2002

By using SOA (Semiconductor Optical Amplifier), all-optical XOR gate has been demonstrated at 5 Gb/s in RZ format. Firstly, Boolean AB-and Boolean AB have been obtained. Then, Boolean AB and Boolean AB have been combined to achieve the all-optical XOR gate, which has Boolean logic of AB+AB.

• Journal of Korea Society of Industrial Information Systems
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• v.12 no.2
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• pp.68-78
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• 2007

D-class is defined as a set of equivalent $n{\times}n$ boolean matrices according to a given equivalence relation. The D-class computation requires the multiplication of three boolean matrices for each of all possible triples of $n{\times}n$ boolean matrices. However, almost all the researches on boolean matrices focused on the efficient multiplication of only two boolean matrices and a few researches have recently been shown to deal with the multiplication of all boolean matrices. The paper suggests a mathematical theory that enables the efficient multiplication for all possible boolean matrix triples and the efficient computation of all D-classes, and discusses algorithms designed with the theory and their execution results.

• Journal of Information Technology Services
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• v.9 no.4
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• pp.219-229
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• 2010

The Green's equivalence relations have played a fundamental role in the development of semigroup theory. They are concerned with mutual divisibility of various kinds, and all of them reduce to the universal equivalence in a group. Boolean matrices have been successfully used in various areas, and many researches have been performed on them. Studying Green's relations on a monoid of boolean matrices will reveal important characteristics about boolean matrices, which may be useful in diverse applications. Although there are known algorithms that can compute Green relations, most of them are concerned with finding one equivalence class in a specific Green's relation and only a few algorithms have been appeared quite recently to deal with the problem of finding the whole D or J equivalence relations on the monoid of all $n{\times}n$ Boolean matrices. However, their results are far from satisfaction since their computational complexity is exponential-their computation requires multiplication of three Boolean matrices for each of all possible triples of $n{\times}n$ Boolean matrices and the size of the monoid of all $n{\times}n$ Boolean matrices grows exponentially as n increases. As an effort to reduce the execution time, this paper shows an isomorphism between the R relation and L relation on the monoid of all $n{\times}n$ Boolean matrices in terms of transposition. introduces theorems based on it discusses an improved algorithm for the J relation computation whose design reflects those theorems and gives its execution results.

• Journal of the Chungcheong Mathematical Society
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• v.17 no.1
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• pp.47-56
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• 2004

The nonlinearity of a Boolean function f on $GF(2)^n$ is the minimum hamming distance between f and all affine functions on $GF(2)^n$ and it measures the ability of a cryptographic system using the functions to resist against being expressed as a set of linear equations. Finding out the exact value of the nonlinearity of given Boolean functions is not an easy problem therefore one wants to estimate the nonlinearity using extra information on given functions, or wants to find a lower bound or an upper bound on the nonlinearity. In this paper we extend the notion of auto-correlations of Boolean functions to vector Boolean functions and obtain upper bounds and a lower bound on the nonlinearity of vector Boolean functions in the context of their auto-correlations. Also we can describe avalanche characteristics of vector Boolean functions by examining the extended notion of auto-correlations.

• Proceedings of the Korea Contents Association Conference
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• 2006.05a
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• pp.459-461
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• 2006

The successive multiplication of all $n{\times}n$ boolean matrices is necessary for applications such as D-class computation. But, no research has been performed on it despite many researches dealing with boolean matrices. The paper suggests a theory with which successively multiplying a $n{\times}n$ boolean matrix by all $n{\times}n$ boolean matrices can be done efficiently, applies it to the successive multiplication of all $n{\times}n$ boolean matrices and shows its execution results.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.10 no.11
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• pp.3227-3233
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• 2009

A method for performing Boolean resubstitution is proposed. This method is efficiently implemented using division matrix. It begins by creating an algebraic division matrix from given two logic expressions. By introducing Boolean properties and adding literals into the algebraic division matrix, we make the Boolean division matrix. Using this extended division matrix, Boolean substituted expressions are found. Experimental results show the improvements in the literal counts over well-known logic synthesis tools for some benchmark circuits.

• Journal of Information Technology Services
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• v.6 no.1
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• pp.151-158
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• 2007

D-class computation requires multiplication of three Boolean matrices for each of all possible triples of $n{\times}n$ Boolean matrices and search for equivalent $n{\times}n$ Boolean matrices according to a specific equivalence relation. It is easy to see that even multiplying all $n{\times}n$ Boolean matrices with themselves shows exponential time complexity and D-Class computation was left an unsolved problem due to its computational complexity. The vector-based multiplication theory shows that the multiplication of three Boolean matrices for each of all possible triples of $n{\times}n$ Boolean matrices can be done much more efficiently. However, D-Class computation requires computation of equivalent classes in addition to the efficient multiplication. The paper discusses a theory and an algorithm for efficient D-class computation, and shows execution results of the algorithm.