• The Mathematical Education
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• v.49 no.4
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• pp.489-500
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• 2010

This study makes clear justification of contents of set in secondary school through the scientific consideration and contents consideration of curriculum about two points - lattice and ring - of set deal with 'number and operation'. In this process, we make clear the greatest common divisor, the least common multiple and operation of set, especially the meaning of symmetric difference, we suggest direction about constitution of contents of set in secondary school. This study helps to raise the specificity on the elements of textbook and presents the first step about the range of teaching in a construct of curriculum.

• 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.

• Communications of Mathematical Education
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• v.34 no.4
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• pp.587-603
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• 2020

This study is a basic study to effectively develop a mathematics experience object, an important tool and educational tool in mathematics education. Recently, as mathematics education based on action theory is emphasized, various mathematics experience objects are being developed. It is also used through various after-school activities in the school. However, there are insufficient cases in which a mathematics experience teaching tools is developed and used as a tool for explaining mathematics concepts in mathematics classrooms. Also, the mathematical background of the mathematics experience teaching tools used by students is unclear. For this reason, the mathematical understanding of the toolst for mathematics experience is also very insufficient. Therefore, in this study, a development model is proposed as a systematic method for developing a mathematics experience teaching tools. Also, in this study, we developed 'the Great Common Divisor' mathematics experience teaching tool according to the development model. Through the model proposed through this study and the actual mathematics experience teaching tool, the development of various tools for mathematical experience will be practically implemented. In addition, it is expected that various tools for experiencing mathematics based on mathematical foundations will be developed.

• Kyungpook Mathematical Journal
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• v.54 no.4
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• pp.607-617
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• 2014

Let S = C(r,m), the finite cyclic semigroup with index r and period m. Each subsemigroup of S is cyclic if and only if either r = 1; r = 2; or r = 3 with m odd. For $r{\neq}1$, the maximum value of the minimum number of elements in a (minimal) generating set of a subsemigroup of S is 1 if r = 3 and m is odd; 2 if r = 3 and m is even; (r-1)/2 if r is odd and unequal to 3; and r/2 if r is even. The number of cyclic subsemigroups of S is $r-1+{\tau}(m)$. Formulas are also given for the number of 2-generated subsemigroups of S and the total number of subsemigroups of S. The minimal generating sets of subsemigroups of S are characterized, and the problem of counting them is analyzed.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.7 no.10
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• pp.2497-2513
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• 2013

In 2010, Dijk et al. demonstrated a simple somewhat homomorphic encryption (HE) scheme over the integers of which this simplicity came at the cost of a public key size in $\tilde{O}({\lambda}^{10})$. Although in 2011 Coron et al. reduced the public key size to $\tilde{O}({\lambda}^7)$, it is still too large for practical applications, especially for the cloud computing. In this paper, we propose a new form of somewhat HE scheme to reduce further the public key size and a variation of the scheme to optimize the ciphertext size. First of all, we propose a new somewhat HE scheme which is built on the hardness of the approximate greatest common divisor (GCD) problem of two integers, where the public key size in the scheme is reduced to $\tilde{O}({\lambda}^3)$. Furthermore, we can reduce the length of the ciphertext of the new somewhat HE scheme by applying the modular reduction technique. Additionally, we give simulation results for evaluating ability of the proposed scheme.

• The KIPS Transactions:PartC
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• v.9C no.2
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• pp.163-172
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• 2002

Java Card API porvide benifit for development program based on smart card using limmited resource. This APIs does not support arithmetic operations such as modular arithmetic, greatest common divisor calculation, and generation and certification of prime number, which is necessary arithmetic in PKI algorithm implementation. In this paper, we implement class BigInteger acted in the Java Card platform because that Java Card APIs does not support class BigInteger necessary in implementation of PKI algorithm.

• Journal of Educational Research in Mathematics
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• v.23 no.1
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• pp.17-35
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• 2013

The purpose of this study was to analyze the extendibility of division algorithm and Euclid algorithm for integers to algorithms for rational numbers based on word problems of fraction division. This study serviced to upgrade professional development of elementary and secondary mathematics teachers. In this paper, fractions were used as expressions of rational numbers, and they also represent rational numbers. According to discrete context and continuous context, and measurement division and partition division etc, divisibility was classified into two types; one is an abstract algebraic point of view and the other is a generalizing view which preserves division algorithms for integers. In the second view, we raised some contextual problems that can be used in school mathematics and then we discussed division algorithm, the greatest common divisor and the least common multiple, and Euclid algorithm for fractions.

• The Mathematical Education
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• v.58 no.2
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• pp.319-335
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• 2019

In this paper, we examine the effectiveness of calculation according to automation, which is one of Computational Thinking, by coding the conceptual process into Python language, focusing on the concept of divisibility in elementary school textbooks. The educational implications of these considerations are as follows. First, it is possible to make a field of learning that can revise the new mathematical concept through the opportunity to reinterpret the Conceptual Thinking learned in school mathematics from the perspective of Computational Thinking. Second, from the analysis of college students, it can be seen that many students do not have mathematical concepts in terms of efficiency of computation related to the divisibility. This phenomenon is a characteristic of the mathematics curriculum that emphasizes concepts. Therefore, it is necessary to study new mathematical concepts when considering the aspect of utilization. Third, all algorithms related to the concept of divisibility covered in elementary mathematics textbooks can be found to contain the notion of iteration in terms of automation, but little recursive activity can be found. Considering that recursive thinking is frequently used with repetitive thinking in terms of automation (in Computational Thinking), it is necessary to consider low level recursive activities at elementary school. Finally, it is necessary to think about mathematical Conceptual Thinking from the point of view of Computational Thinking, and conversely, to extract mathematical concepts from computer science's Computational Thinking.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.24 no.1
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• pp.5-14
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• 2014

Searchable encryption systems provide search on encrypted data while preserving the privacy of the data and the search keywords used in queries. Recently, interest on data outsourcing has increased due to proliferation of cloud computing services. Many researches are on going to minimize the trust put on external servers and searchable encryption is one of them. However, most of previous searchable encryption schemes provide only a single keyword boolean search. Although, there have been proposals to provide conjunctive keyword search, most of these works use a fixed field which limit their application. In this paper, we propose a field-free conjunctive keyword searchable encryption that also provides rank information of search results. Our system uses prime tables and greatest common divisor operation, making our system very efficient. Moreover, our system is practical and can be implemented very easily since it does not require sophisticated cryptographic module.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.11 no.2
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• pp.107-112
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• 2011

It is very difficult to factorize composite number, $n=pq$ to integer factorization, p and q that is almost similar length of digits. Integer factorization algorithms, for the most part, find ($a,b$) that is congruence of squares ($a^2{\equiv}b^2$ (mod $n$)) with using factoring(factor base, B) and get the result, $p=GCD(a-b,n)$, $q=GCD(a+b,n)$ with taking the greatest common divisor of Euclid based on the formula $a^2-b^2=(a-b)(a+b)$. The efficiency of these algorithms hangs on finding ($a,b$) and deciding factor base, B. This paper proposes a efficient algorithm. The proposed algorithm extracts B from integer factorization with 3 digits prime numbers of $n+1$ and decides f, the combination of B. And then it obtains $x$(this is, $a=fxy$, $\sqrt{n}$ < $a$ < $\sqrt{2n}$) from integer factorization of $n-2$ and gets $y=\frac{a}{fx}$, $y_1$={1,3,7,9}. Our algorithm is much more effective in comparison with the conventional Fermat algorithm that sequentially finds $\sqrt{n}$ < $a$.