• Journal of Applied and Pure Mathematics
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• v.6 no.3_4
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• pp.167-176
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• 2024

This study is about division and right multiplication in matrices. The discussion of the properties of multiplication and division is examined. Some results between multiplication based on the row-column relationship and division based on the same relationship are discussed. The commonalities of these results between the processes are emphasized. Examples of unrealized properties are given. The algebraic properties of the newly defined right product and division are clarified in matrices. The properties of the known multiplication operation and new situations between right multiplication and division are investigated. Some results are declared between the transpositions of matrices and the obtained rules of operations. New results are discussed belong the equations ${\underleftarrow{XA}}=B$ and ${\underleftarrow{AX}}=B$. New ideas are proposed for solving these equations. The contribution The contribution is explained the equation $AB={\underleftarrow{BA}}$ to division operation. Many new properties, lemmas and theorems are presented on this subject.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.13 no.11
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• pp.5616-5630
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• 2019

Fully homomorphic encryption allows a third-party to perform arbitrary computation over encrypted data and is especially suitable for secure outsourced computation. This paper investigates secure outsourced computation of multiple matrix multiplication based on fully homomorphic encryption. Our work significantly improves the latest Mishra et al.'s work. We improve Mishra et al.'s matrix encoding method by introducing a column-order matrix encoding method which requires smaller parameter. This enables us to develop a binary multiplication method for multiple matrix multiplication, which multiplies pairwise two adjacent matrices in the tree structure instead of Mishra et al.'s sequential matrix multiplication from left to right. The binary multiplication method results in a logarithmic-depth circuit, thus is much more efficient than the sequential matrix multiplication method with linear-depth circuit. Experimental results show that for the product of ten 32×32 (64×64) square matrices our method takes only several thousand seconds while Mishra et al.'s method will take about tens of thousands of years which is astonishingly impractical. In addition, we further generalize our result from square matrix to non-square matrix. Experimental results show that the binary multiplication method and the classical dynamic programming method have a similar performance for ten non-square matrices multiplication.

• JSTS:Journal of Semiconductor Technology and Science
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• v.16 no.1
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• pp.118-125
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• 2016

This paper presents a new high-efficient algorithm and architecture for an elliptic curve cryptographic processor. To reduce the computational complexity, novel modified Lopez-Dahab scalar point multiplication and left-to-right algorithms are proposed for point multiplication operation. Moreover, bit-serial Galois-field multiplication is used in order to decrease hardware complexity. The field multiplication operations are performed in parallel to improve system latency. As a result, our approach can reduce hardware costs, while the total time required for point multiplication is kept to a reasonable amount. The results on a Xilinx Virtex-5, Virtex-7 FPGAs and VLSI implementation show that the proposed architecture has less hardware complexity, number of clock cycles and higher efficiency than the previous works.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.2
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• pp.9-17
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• 2001

A ring R is called right mininjective if every isomorphsim between simple right ideals is given by left multiplication by an element of R. In this paper we consider that the necessary and sufficient condition for that Trivial extension of R by V, i.e. T(R; V ) is mininjective. We also study the split null extension R and S by V.

• Proceedings of the IEEK Conference
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• 2005.11a
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• pp.813-816
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• 2005

In last several years, the need for the right of privacy and mobile banking has increased. The RSA system is one of the most widely used public key cryptography systems, and its core arithmetic operation IS modular multiplication. P. L. Montgomery proposed a very efficient modular multiplication technique that is well suited to hardware implementation. In this paper, the montgomery modular multiplication algorithms(CIOS, SOS, FIOS) , developed by Cetin Kaya Koc, is presented and implemented using radix-16 and Altera FPGA. Also, we undertake comparisons of power dissipation using Quatrus II PowerPlay Power Analyzer.

• Honam Mathematical Journal
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• v.29 no.3
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• pp.415-425
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• 2007

The normal basis has the advantage that the result of squaring an element is simply the right cyclic shift of its coordinates in hardware implementation over finite fields. In particular, the optimal normal basis is the most efficient to hardware implementation over finite fields. In this paper, we propose an efficient parallel architecture which transforms the Gaussian normal basis multiplication in GF($2^m$) into the type-I optimal normal basis multiplication in GF($2^{mk}$), which is based on the palindromic representation of polynomials.

• East Asian mathematical journal
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• v.18 no.2
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• pp.245-259
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• 2002

In this paper, all rings or (left)near-rings R are associative, and for near-ring R, all R-groups are right R action and all modules are right R-modules. First, we begin with the study of rings in which all the additive endomorphisms or only the left multiplication endomorphisms are generated by ring endomorphisms and their properties. This study was motivated by the work on the Sullivan's Problem [14]. Next, for any right R-module M, we will introduce AM modules and investigate their basic properties. Finally, for any nearring R, we will also introduce MR-groups and study some of their properties.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.1
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• pp.85-103
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• 2001

Suppose that Hand K are paired Hopf algebras and that A is an H - K - bicomodule algebra with multiplication which is a left H-comodule map and is a right K-comodule map. We define a new twisted algebra, $A^{\tau}$ and define $M^{\tau}$ for $M{\in}M_A^K$. We find an equivalent condition for $M^{\tau}{\in}M_{A^{\tau}}^K$. We show that the above defined twisted multiplication is the special case of Beattie's twist multiplication. We show that if K is commutative, then A is an H-module algebra and show that if $H^*$ is cocommutative then the construction of smash product appears as a special case of the new twist product.

• School Mathematics
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• v.6 no.3
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• pp.267-281
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• 2004

The purpose of this study is to understand Preservice elementary teachers' knowledge about multiplication of fractions by focusing on their computation abilities, understanding of meanings, generating appropriate problem contexts and representations. A total of 115 preservice elementary teachers participated in the present study. The results of this study indicated that most of preservice elementary teachers have little difficulty in computing multiplication of fractions for right answers, but they have big difficulty in understanding meanings and generating appropriate problem contexts for multiplication of fractions when the multiplier is not an integer, called 'multiplier effect.' Likewise, the rate of appropriate representations surprisingly decreased for multiplication of fractions when the multiplier is not an integer. The findings also point out that an ability to make problem contexts is highly correlated with representations and meanings. This study implies that teacher education programs need to improve preservice elementary teachers' profound understanding of elementary mathematics in order to fundamentally improve the quality of teaching practices in classrooms.

• Journal of KIISE:Computer Systems and Theory
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• v.26 no.7
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• pp.743-751
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• 1999

공개키 암호 시스템에서 모듈러 지수 연산은 주된 연산으로, 이 연산은 내부적으로 모듈러 곱셈을 반복적으로 수행함으로써 계산된다. 본 논문에서는 GF(2m)상에서 수행할 수 있는 Montgomery 알고리즘을 분석하여 right-to-left 방식의 모듈러 지수 연산에서 공통으로 계산 가능한 부분을 이용하여 모듈러 제곱과 모듈러 곱셈을 동시에 수행하는 선형 시스톨릭 어레이를 설계한다. 본 논문에서 설계한 시스톨릭 어레이는 기존의 곱셈기보다 모듈러 지수 연산시 약 0.67배 처리속도 향상을 가진다. 그리고, VLSI 칩과 같은 하드웨어로 구현함으로써 IC 카드에 이용될 수 있다.Abstract One of the main operations for the public key cryptographic system is the modular exponentiation, it is computed by performing the repetitive modular multiplications. In this paper, we analyze Montgomery's algorithm and design a linear systolic array to perform modular multiplication and modular squaring simultaneously. It is done by using common-multiplicand modular multiplication in the right-to-left modular exponentiation over GF(2m). The systolic array presented in this paper improves about 0.67 times than existing multipliers for performing the modular exponentiation. It could be designed on VLSI hardware and used in IC cards.