• Title/Summary/Keyword: normal basis

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An Efficient Algorithm for Computing Multiplicative Inverses in GF($2^m$) Using Optimal Normal Bases (최적 정규기저를 이용한 효율적인 역수연산 알고리즘에 관한 연구)

  • 윤석웅;유형선
    • The Journal of Society for e-Business Studies
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    • v.8 no.1
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    • pp.113-119
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    • 2003
  • This paper proposes a new multiplicative inverse algorithm for the Galois field GF (2/sup m/) whose elements are represented by optimal normal basis type Ⅱ. One advantage of the normal basis is that the squaring of an element is computed by a cyclic shift of the binary representation. A normal basis element is always possible to rewrite canonical basis form. The proposed algorithm combines normal basis and canonical basis. The new algorithm is more suitable for implementation than conventional algorithm.

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AFFINE TRANSFORMATION OF A NORMAL ELEMENT AND ITS APPLICATION

  • Kim, Kitae;Namgoong, Jeongil;Yie, Ikkwon
    • Korean Journal of Mathematics
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    • v.22 no.3
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    • pp.517-527
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    • 2014
  • In this paper, we study affine transformations of normal bases and give an explicit formulation of the multiplication table of an affine transformation of a normal basis. We then discuss constructions of self-dual normal bases using affine transformations of traces of a type I optimal normal basis and of a Gauss period normal basis.

EFFICIENT PARALLEL GAUSSIAN NORMAL BASES MULTIPLIERS OVER FINITE FIELDS

  • Kim, Young-Tae
    • 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.

NAP and Optimal Normal Basis of Type II and Efficient Exponentiation in $GF(2^n)$ (NAF와 타입 II 최적정규기저를 이용한 $GF(2^n)$ 상의 효율적인 지수승 연산)

  • Kwon, Soon-Hak;Go, Byeong-Hwan;Koo, Nam-Hun;Kim, Chang-Hoon
    • 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.

Subquadratic Space Complexity Multiplier for GF($2^n$) Using Type 4 Gaussian Normal Bases

  • Park, Sun-Mi;Hong, Dowon;Seo, Changho
    • 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.

Arithmetic of finite fields with shifted polynomial basis (변형된 다항식 기저를 이용한 유한체의 연산)

  • 이성재
    • Journal of the Korea Institute of Information Security & Cryptology
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    • v.9 no.4
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    • pp.3-10
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    • 1999
  • More concerns are concentrated in finite fields arithmetic as finite fields being applied for Elliptic curve cryptosystem coding theory and etc. Finite fields arithmetic is affected in represen -tation of those. Optimal normal basis is effective in hardware implementation and polynomial field which is effective in the basis conversion with optimal normal basis and show that the arithmetic of finite field with the basis is effective in software implementation.

Basis Translation Matrix between Two Isomorphic Extension Fields via Optimal Normal Basis

  • Nogami, Yasuyuki;Namba, Ryo;Morikawa, Yoshitaka
    • ETRI Journal
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    • v.30 no.2
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    • pp.326-334
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    • 2008
  • This paper proposes a method for generating a basis translation matrix between isomorphic extension fields. To generate a basis translation matrix, we need the equality correspondence of a basis between the isomorphic extension fields. Consider an extension field $F_{p^m}$ where p is characteristic. As a brute force method, when $p^m$ is small, we can check the equality correspondence by using the minimal polynomial of a basis element; however, when $p^m$ is large, it becomes too difficult. The proposed methods are based on the fact that Type I and Type II optimal normal bases (ONBs) can be easily identified in each isomorphic extension field. The proposed methods efficiently use Type I and Type II ONBs and can generate a pair of basis translation matrices within 15 ms on Pentium 4 (3.6 GHz) when $mlog_2p$ = 160.

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On the Radial Basis Function Networks with the Basis Function of q-Normal Distribution

  • Eccyuya, Kotaro;Tanaka, Masaru
    • 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.

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Efficient Serial Gaussian Normal Basis Multipliers over Binary Extension Fields

  • Kim, Yong-Tae
    • 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.

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GRӦBNER-SHIRSHOV BASIS AND ITS APPLICATION

  • Oh, Sei-Qwon;Park, Mi-Yeon
    • Journal of the Chungcheong Mathematical Society
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    • v.15 no.2
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    • pp.97-107
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    • 2003
  • An efficient algorithm for the multiplication in a binary finite filed using a normal basis representation of $F_{2^m}$ is discussed and proposed for software implementation of elliptic curve cryptography. The algorithm is developed by using the storage scheme of sparse matrices.

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