• 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.

• ETRI Journal
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• v.35 no.3
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• pp.523-529
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• 2013

Subquadratic space complexity multipliers for optimal normal bases (ONBs) have been proposed for practical applications. However, for the Gaussian normal basis (GNB) of type t > 2 as well as the normal basis (NB), there is no known subquadratic space complexity multiplier. In this paper, we propose the first subquadratic space complexity multipliers for the type 4 GNB. The idea is based on the fact that the finite field GF($2^n$) with the type 4 GNB can be embedded into fields with an ONB.

• Proceedings of the IEEK Conference
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• 2002.07a
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• pp.26-29
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• 2002

Radial Basis Function (RBF) networks is known as efficient method in classification problems and function approximation. The basis function of RBF networks is usual adopted normal distribution like the Gaussian function. The output of the Gaussian function has the maximum at the center and decrease as increase the distance from the center. For learning of neural network, the method treating the limited area of input space is sometimes more useful than the method treating the whole of input space. The q-normal distribution is the set of probability density function include the Gaussian function. In this paper, we introduce the RBF networks with the basis function of q-normal distribution and actually approximate a function using the RBF networks.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.31 no.11C
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• pp.1120-1127
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• 2006

[ $GF(2^m)$ ] for elliptic curve cryptosystem. The proposed multiplier uses Gaussian normal basis representation and produces multiplication results at a rate of one per [m/w] clock cycles, where w is the selected we.4 size. We implement the p.oposed design using Xilinx XC2V1000 FPGA device. Our design has significantly less critical path delay compared with previously proposed hard ware implementations.

• The Journal of the Korea institute of electronic communication sciences
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• v.4 no.3
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• pp.197-203
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• 2009

Finite field arithmetic is very important in the area of cryptographic applications and coding theory, and it is efficient to use normal bases in hardware implementation. Using the fact that $GF(2^{mk})$ having a type-I optimal normal basis becomes the extension field of $GF(2^m)$, we, in this paper, propose a new serial multiplier which reduce the critical XOR path delay of the best known Reyhani-Masoleh and Hasan's serial multiplier by 25% and the number of XOR gates of Kwon et al.'s multiplier by 2 based on the Reyhani-Masoleh and Hasan's serial multiplier for type-I optimal normal basis.

• Journal of KIISE:Computer Systems and Theory
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• v.34 no.3
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• pp.113-123
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• 2007

This paper presents a high-performance elliptic curve cryptographic processor over $GF(2^m)$. The proposed design adopts Lopez-Dahab Montgomery algorithm for elliptic curve point multiplication and uses Gaussian normal basis for $GF(2^m)$ field arithmetic operations. We select m=163 which is the smallest value among five recommended $GF(2^m)$ field sizes by NIST and it is Gaussian normal basis of type 4. The proposed elliptic curve cryptographic processor consists of host interface, data memory, instruction memory, and control. We implement the proposed design using Xilinx XCV2000E FPGA device. Based on the FPGA implementation results, we can see that our design is 2.6 times faster and requires significantly less hardware resources compared with the previously proposed best hardware implementation.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.34 no.1C
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• pp.21-27
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• 2009

We present an efficient exponentiation algorithm for a finite field $GF(2^n)$ determined by an optimal normal basis of type II using signed digit representation of the exponents. Our signed digit representation uses a non-adjacent form (NAF) for $GF(2^n)$. It is generally believed that a signed digit representation is hard to use when a normal basis is given because the inversion of a normal element requires quite a computational delay. However our result shows that a special normal basis, called an optimal normal basis (ONB) of type II, has a nice property which admits an effective exponentiation using signed digit representations of the exponents.

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.43 no.2 s.344
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• pp.84-95
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• 2006

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 $GF(2^m)$. In this paper, we propose a new, simpler, parallel multiplier over $GF(2^m)$ having a Gaussian normal basis of type k, which performs multiplication over $GF(2^m)$ in the extension field $GF(2^{mk})$ containing a type-I optimal normal basis. For k=2,4,6 the time and area complexity of the proposed multiplier is the same as tha of the best known Reyhani-Masoleh and Hasan multiplier.

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.43 no.1 s.343
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• pp.45-58
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• 2006

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 $GF(2^m)$. In this paper, we propose a new, simpler, parallel multiplier over $GF(2^m)$ having a Gaussian normal basis of type k, which performs multiplication over $GF(2^m)$ in the extension field $GF(2^{mk})$ containing a type-I optimal normal basis. For k=2,4,6 the time and area complexity of the proposed multiplier is the same as tha of the best known Reyhani-Masoleh and Hasan multiplier

• The KIPS Transactions:PartC
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• v.15C no.2
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• pp.133-140
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• 2008

In this paper, we propose a high performance elliptic curve cryptographic processor over $GF(2^{163})$. The proposed architecture is based on a modified Lopez-Dahab elliptic curve point multiplication algorithm and uses Gaussian normal basis for $GF(2^{163})$ field arithmetic. To achieve a high throughput rates, we design two new word-level arithmetic units over $GF(2^{163})$ and derive a parallelized elliptic curve point doubling and point addition algorithm with uniform addressing based on the Lopez-Dahab method. We implement our design using Xilinx XC4VLX80 FPGA device which uses 24,263 slices and has a maximum frequency of 143MHz. Our design is roughly 4.8 times faster with 2 times increased hardware complexity compared with the previous hardware implementation proposed by Shu. et. al. Therefore, the proposed elliptic curve cryptographic processor is well suited to elliptic curve cryptosystems requiring high throughput rates such as network processors and web servers.