• The Journal of Korean Institute of Communications and Information Sciences
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• v.33 no.1C
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• pp.131-139
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• 2008

This paper presents a scalable dual-field Montgomery multiplier based on a new multi-precision carry save adder(MP-CSA), which operates in both types of finite fields GF(p) and GF($2^m$). The new MP-CSA consists of two carry save adders(CSA). Each CSA is composed of n = [w/b] carry propagation adders(CPA) for a modular multiplication with w-bit words, where b is the number of dual field adders(DFA) in a CPA. The proposed Montgomery multiplier has roughly the same timing complexity compared with the previous result, however, it has the advantage of reduced chip area requirements. In addition, the proposed circuit produces the exact modular multiplication result at the end of operation unlike the previous architecture. Furthermore, the proposed Montgomery multiplier has a high scalability in terms of w and m. Therefore, it can be used to multiplier over GF(p) and GF($2^m$) for cryptographic applications.

• Proceedings of the KIEE Conference
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• 2006.10c
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• pp.142-144
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• 2006

In this paper, the Ternary adder and multiplier are implemented by current-mode CMOS. First, we implement the ternary T-gate using current-mode CMOS which have an effective availability of integrated circuit design. Second, we implement the circuits to be realized 2-variable ternary addition table and multiplication table over finite fields GF(3) with the ternary T-gates. Finally, these operation circuits are simulated by Spice under $1.5{\mu}m$ CMOS standard technology, $1.5{\mu}m$ unit current, and 3.3V VDD voltage. The simulation results have shown the satisfying current characteristics. The ternary adder and multiplier implemented by current-mode CMOS are simple and regular for wire routing and possess the property of modularity with cell array.

• The KIPS Transactions:PartA
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• v.11A no.2
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• pp.109-114
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• 2004

This paper presents a high-speed bit-parallel systolic divider for computing modular division A($\chi$)/B($\chi$) mod G($\chi$) in finite fields GF$(2^m)$. The presented divider is based on the binary GCD algorithm and verified through FPGA implementation. The proposed architecture produces division results at a rate of one every 1 clock cycles after an initial delay of 5m-2. Analysis shows that the proposed divider provides a significant reduction in both chip area and computational delay time compared to previously proposed systolic dividers with the same I/O format. In addition, since the proposed architecture does not restrict the choice of irreducible polynomials and has regularity and modularity, it provides a high flexibility and Scalability with respect to the field size m. Therefore, the proposed divider is well suited to VLSI implementation.

• Journal of Korea Society of Industrial Information Systems
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• v.7 no.5
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• pp.167-174
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• 2002

This paper analyzes the characteristics of three multipliers and squarers in finite fields GF(2/sup m/) from the point of view of processing time and area complexity. First, we analyze structures of three multipliers and squarers: 1) Systolic array structure, 2), LFSR structure, and 3) CA structure. To make performance analysis, each multiplier and squarer was modeled in VHDL and was synthesized for FPGA implementation. The simulation results show that CA structure is the best from the point view of processing time, and LFSR structure is the best from the point of view of area complexity.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.29 no.3A
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• pp.331-336
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• 2004

This study focuses on the hardware implementation of fast and low-system-complexity multiplier over GF(2$^{m}$ ). From the properties of an irreducible AOP of degree m. the modular reduction in GF(2$^{m}$ ) multiplicative operation can be simplified using cyclic shift operation. And then, GF(2$^{m}$ ) multiplicative operation can be established using the away structure of AND and XOR gates. The proposed multiplier is composed of m(m+1) 2-input AND gates and (m+1)$^2$ 2-input XOR gates. And the minimum critical path delay is Τ$_{A+}$〔lo $g_2$$^{m}$ 〕Τ$_{x}$ proposed multiplier obtained have low circuit complexity and delay time, and the interconnections of the circuit are regular, well-suited for VLSI realization.n.

• The Journal of the Korea institute of electronic communication sciences
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• v.8 no.11
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• pp.1749-1754
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• 2013

Binary sequences of period $2^n-1$ are widely used in many areas of engineering and sciences. Some well-known applications include coding theory, code-division multiple-access (CDMA) communications, and stream cipher systems. In this paper we analyze different solutions to $x^5+bx^3+b^{2^m}x^2+1=0$ over $GF(2^n)$. The number of different solutions determines frequencies of cross-correlations of nonlinear binary sequences generated by $d=3{\cdot}2^m-2$, n=2m, m=4k($k{\geq}2$). Also we give an algorithm for determination of number of different solutions to the equation.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.8 no.2
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• pp.15-21
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• 2008

This paper discuss the sequential digital logic systems and arithmetic operation algorithms which is the important material in computer architecture using analysis and synthesis which is based on extension logic for binary logic over galois fields. In sequential digital logic systems, we construct the moore model without feedback sequential logic systems after we obtain the next state function and output function using building block T-gate. Also, we obtain each algorithms of the addition, subtraction, multiplication, division based on the finite fields mathematical properties. Especially, in case of P=2 over GF($P^m$), the proposed algorithm have a advantage which will be able to apply traditional binary logic directly.The proposed method can construct more efficiency digital logic systems because it can be extended traditional binary logic to extension logic.

• Journal of KIISE:Computer Systems and Theory
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• v.31 no.1_2
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• pp.1-9
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• 2004

The important arithmetic operations over finite fields include exponentiation, division, and inversion. An exponentiation operation can be implemented using a series of squaring and multiplication operations using a binary method, while division and inversion can be performed by the iterative application of an AB$^2$ operation. Hence, it is important to develop a fast algorithm and efficient hardware for this operations. In this paper presents new bit-serial architectures for the simultaneous computation of multiplication and squaring operations, and the computation of an $AB^2$ operation over $GF(2^m)$ generated by an irreducible AOP of degree m. The proposed architectures offer a significant improvement in reducing the hardware complexity compared with previous architectures, and can also be used as a kernel circuit for exponentiation, division, and inversion architectures. Furthermore, since the Proposed architectures include regularity and modularity, they can be easily designed on VLSI hardware and used in IC cards.

• Journal of Korea Society of Industrial Information Systems
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• v.8 no.3
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• pp.85-90
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• 2003

This paper proposes a modular multiplier based on LFSR (Linear Feedback Shift Register) architecture with efficient area complexity over GF(2/sup m/). At first, we examine the modular exponentiation algorithm and propose it's architecture, which is basic module for public-key cryptosystems. Furthermore, this paper proposes on efficient modular multiplier as a basic architecture for the modular exponentiation. The multiplier uses AOP (All One Polynomial) as an irreducible polynomial, which has the properties of all coefficients with '1 ' and has a more efficient hardware complexity compared to existing architectures.

• Journal of Korea Society of Industrial Information Systems
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• v.9 no.1
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• pp.43-48
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• 2004

This paper presents new architectures based on the linear feedback shia resister architecture over GF(2m). First we design a modular multiplier and a modular squarer, then propose an architecture by combing the multiplier and the squarer. All architectures use an irreducible AOP (All One Polynomial) as a modulus, which has the properties of all coefficients with '1'. The proposed architectures have lower hardware complexity than previous architectures. They could be. Therefore it is useful for implementing the exponentiation architecture, which is the con operation in public-key cryptosystems.