• The Journal of Korean Institute of Communications and Information Sciences
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• v.35 no.4C
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• pp.337-342
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• 2010

In this paper, a new architecture for digit-parallel/bit-serial GF($2^m$) multiplier with low complexity is proposed. The proposed multiplier operates in polynomial basis of GF($2^m$) and produces multiplication results at a rate of one per D clock cycles, where D is the selected digit size. The digit-parallel/bit-serial multiplier is faster than bit-serial ones but with lower area complexity than bit-parallel ones. The most significant feature of the digit-parallel/bit-serial architecture is that a trade-off between hardware complexity and delay time can be achieved. But the traditional digit-parallel/bit-serial multiplier needs extra hardware for high speed. In this paper a new low complexity efficient digit-parallel/bit-serial multiplier is presented.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.12 no.7
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• pp.1209-1217
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• 2008

In this paper, we present a new bit-parallel multiplier for performing the bit-parallel multiplication of two polynomials in the finite fields $GF(2^m)$. Prior to construct the multiplier circuits, we consist of the vector code generator(VCG) to generate the result of bit-parallel multiplication with one coefficient of a multiplicative polynomial after performing the parallel multiplication of a multiplicand polynomial with a irreducible polynomial. The basic cells of VCG have two AND gates and two XOR gates. Using these VCG, we can obtain the multiplication results performing the bit-parallel multiplication of two polynomials. Extending this process, we show the design of the generalized circuits for degree m and a simple example of constructing the multiplier circuit over finite fields $GF(2^4)$. Also, the presented multiplier is simulated by PSpice. The multiplier presented in this paper use the VCGs with the basic cells repeatedly, and is easy to extend the multiplication of two polynomials in the finite fields with very large degree m, and is suitable to VLSI.

• Journal of the Korean Institute of Telematics and Electronics B
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• v.31B no.10
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• pp.67-72
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• 1994

In this paper we suggest a 32 bit high speed parallel multiplier which plays an important role in digital signal processing. We employ a bit-pair recoding Booth algoritham that gurantees n/2 partial product terms, which uniformly handles the signed-operand case. While partial product terms are generated, a special method is suggested to reduce time delay by employing 1's complement instead of 2's complement. Later when partial products are added, the additional 1 bit's are packed in a single partial product term and added to in the parallel counter. Then 16 partial product terms are reduced to two summands by using successive parallel counters. Final multiplication value is obtained by a BLC adder. When this multiplier is simulated under 0.8$\mu$CMOS standard cell we obtain 30ns multiplier speed.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.33 no.11C
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• pp.892-897
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• 2008

In this paper, a new architecture for digit-parallel/bit-serial GF$(2^m)$ multiplier with low latency is proposed. The proposed multiplier operates in polynomial basis of GF$(2^m)$ and produces multiplication results at a rate of one per D clock cycles, where D is the selected digit size. The digit-parallel/bit-serial multiplier is faster than bit-serial ones but with lower area complexity than bit-parallel ones. The most significant feature of the proposed architecture is that a trade-off between hardware complexity and delay time can be achieved.

• Proceedings of the KIEE Conference
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• 2005.05a
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• pp.144-147
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• 2005

This paper is proposed an 8$\times$8 bit parallel multiplier for low power consumption. The 8$\times$8 bit parallel multiplier is used for the comparison between the proposed Low-Swing CVSL full adder with conventional CVSL full adder. Comparing tile previous works, this circuit is reduced the power consumption rate of 8.2% and the power-delay-product of 11.1%. The validity and effectiveness of the proposed circuits are verified through the HSPICE under Hynix 0.35$\{\mu}m$ standard CMOS process.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.20 no.5
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• pp.11-22
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• 2010

Recently, Fan and Dai introduced a Shifted Polynomial Basis and construct a non-pipeline bit-parallel multiplier for $F_{2^n}$. As the name implies, the SPB is obtained by multiplying the polynomial basis 1, ${\alpha}$, ${\cdots}$, ${\alpha}^{n-1}$ by ${\alpha}^{-\upsilon}$. Therefore, it is easy to transform the elements PB and SPB representations. After, based on the Modified Shifted Polynomial Basis(MSPB), SPB bit-parallel Mastrovito type I and type II multipliers for all irreducible trinomials are presented. In this paper, we present a bit-parallel architecture to multiply in SPB. This multiplier have a space complexity efficient than all previously presented architecture when n ${\neq}$ 2k. The proposed multiplier has more efficient space complexity than the best-result when 1 ${\leq}$ k ${\leq}$ (n+1)/3. Also, when (n+2)/3 ${\leq}$ k < n/2 the proposed multiplier has more efficient space complexity than the best-result except for some cases.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.19 no.2
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• pp.49-61
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• 2009

Finite Field multiplication operation is one of the most important operations in the finite field arithmetic. Recently, Fan and Dai introduced a Shifted Polynomial Basis(SPB) and construct a non-pipeline bit-parallel multiplier for $F_{2^n}$. In this paper, we propose a new bit-parallel shifted polynomial basis type I and type II multipliers for $F_{2^n}$ defined by an irreducible trinomial $x^{n}+x^{k}+1$. The proposed type I multiplier has more efficient the space and time complexity than the previous ones. And, proposed type II multiplier have a smaller space complexity than all previously SPB multiplier(include our type I multiplier). However, the time complexity of proposed type II is increased by 1 XOR time-delay in the worst case.

• Journal of the Institute of Electronics Engineers of Korea TC
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• v.43 no.7 s.349
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• pp.115-121
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• 2006

The efficiency of the multiplier largely depends on the representation of finite filed elements such as normal basis, polynomial basis, dual basis, and redundant representation, and so on. In particular, the redundant representation is attractive since it can simply implement squaring and modular reduction. In this paper, we propose an efficient bit-parallel multiplier for GF(2m) defined by an irreducible all-one polynomial using a redundant representation. We modify the well-known multiplication method which was proposed by Karatsuba to improve the efficiency of the proposed bit-parallel multiplier. As a result, the proposed multiplier has a lower space complexity compared to the previously known multipliers using all-one polynomials. On the other hand, its time complexity is similar to the previously proposed ones.

• Journal of IKEEE
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• v.18 no.2
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• pp.228-233
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• 2014

In this paper, we propose a CMOS-memristor hybrid circuit that can perform 4-bit multiplication for future energy-efficient computing in nano-scale digital systems. The proposed CMOS-memristor hybrid circuit is based on the parallel architecture with AND and OR planes. This parallel architecture can be very useful in improving the power-delay product of the proposed circuit compared to the conventional CMOS array multiplier. Particularly, from the SPECTRE simulation of the proposed hybrid circuit with 0.13-mm CMOS devices and memristors, this proposed multiplier is estimated to have better power-delay product by 48% compared to the conventional CMOS array multiplier. In addition to this improvement in energy efficiency, this 4-bit multiplier circuit can occupy smaller area than the conventional array multiplier, because each cross-point memristor can be made only as small as $4F^2$.

• The Journal of the Korea institute of electronic communication sciences
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• v.9 no.4
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• pp.409-416
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• 2014

In H/W implementation for the finite field, the use of normal basis has several advantages, especially the optimal normal basis is the most efficient to H/W implementation in $GF(2^m)$. The finite field $GF(2^m)$ with type I optimal normal basis(ONB) has the disadvantage not applicable to some cryptography since m is even. The finite field $GF(2^m)$ with type II ONB, however, such as $GF(2^{233})$ are applicable to ECDSA recommended by NIST. In this paper, we propose a bit-parallel multiplier over $GF(2^m)$ having a type II ONB, which performs multiplication over $GF(2^m)$ in the extension field $GF(2^{2m})$. The time and area complexity of the proposed multiplier is the same as or partially better than the best known type II ONB bit-parallel multiplier.