A method for constructing a self-dual normal basis in odd characteristic extension fields

Nogami, Yasuyuki; Nasu, Hiroaki; Morikawa, Yoshitaka; Uehara, Satoshi

doi:10.1016/j.ffa.2008.04.001

Cited by 13 publications

(3 citation statements)

References 5 publications

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“…Constructions of normal bases and self-dual bases have been extensively studied in the past two decades. A non exhaustive list of references is [10,11,12,13,14]. The latest results can be found for instance in [13] and [14], where explicit constructions of self-dual (integral) normal bases in abelian extensions of finite and local fields were given.…”

Section: Introductionmentioning

confidence: 99%

Finding normal bases over finite fields with prescribed trace self-orthogonal relations

Zhang¹,

Feng²,

Liao³

et al. 2014

Finite Fields and Their Applications

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Normal bases and self-dual normal bases over finite fields have been found to be very useful in many fast arithmetic computations. It is well-known that there exists a self-dual normal basis of F2n over F2 if and only if 4 ∤ n. In this paper, we prove there exists a normal element α of F2n over F2 corresponding to a prescribed vector a = (a0, a1, · · · , an−1) ∈ F n 2 such that ai = Tr 2 n |2 (α 1+2 i ) for 0 ≤ i ≤ n − 1, where n is a 2-power or odd, if and only if the given vector a is symmetric (ai = an−i for all i, 1 ≤ i ≤ n − 1), and one of the following is true. 1) n = 2 s ≥ 4, a0 = 1, a n/2 = 0,Furthermore we give an algorithm to obtain normal elements corresponding to prescribed vectors in the above two cases. For a general positive integer n with 4|n, some necessary conditions for a vector to be the corresponding vector of a normal element of F2n over F2 are given. And for all n with 4|n, we prove that there exists a normal element of F2n over F2 such that the Hamming weight of its corresponding vector is 3, which is the lowest possible Hamming weight.

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Section: Introductionmentioning

confidence: 99%

Finding normal bases over finite fields with prescribed trace self-orthogonal relations

Zhang¹,

Feng²,

Liao³

et al. 2014

Finite Fields and Their Applications

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“…The theoretical techniques used to construct normal bases with low complexity sometimes yield self-dual normal bases, see e.g. [7, §5.4] or [3, §5], [9,Corollary 3.5], [5,Theorem 5], [20].…”

Section: Introductionmentioning

confidence: 99%

Construction of self-dual normal bases and their complexity

Arnault

Pickett

Vinatier

2012

Finite Fields and Their Applications

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Recent work of Pickett has given a construction of self-dual normal bases for extensions of finite fields, whenever they exist. In this article we present these results in an explicit and constructive manner and apply them, through computer search, to identify the lowest complexity of self-dual normal bases for extensions of low degree. Comparisons to similar searches amongst normal bases show that the lowest complexity is often achieved from a self-dual normal basis

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“…Aside from their intrinsic interest, self-dual normal bases for extensions of finite fields are of use in encryption and have been used by Wang for constructing the Massey-Omura finitefield multiplier (see [27]). There are results in the literature constructing such bases, see [9], [15] and [27] but all put some restrictions on the degree of the extension or the characteristic of the base field. We present here constructions for self-dual normal bases for any extension of finite fields for which they exist.…”

Section: Introductionmentioning

confidence: 99%

Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields

Pickett

2010

Int. J. Number Theory

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Let F/E be a finite Galois extension of fields with abelian Galois group Γ. A self-dual normal basis for F/E is a normal basis with the additional property that T r F/E (g(x), h(x)) = δ g,h for g, h ∈ Γ. Bayer-Fluckiger and Lenstra have shown that when char(E) = 2, then F admits a self-dual normal basis if and only if [F : E] is odd. If F/E is an extension of finite fields and char(E) = 2, then F admits a self-dual normal basis if and only if the exponent of Γ is not divisible by 4. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let K be a finite extension of Q p , let L/K be a finite abelian Galois extension of odd degree and let O L be the valuation ring of L. We define A L/K to be the unique fractional O L -ideal with square equal to the inverse different of L/K. It is known that a self-dual integral normal basis exists for A L/K if and only if L/K is weakly ramified. Assuming p = 2, we construct such bases whenever they exist.

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A method for constructing a self-dual normal basis in odd characteristic extension fields

Cited by 13 publications

References 5 publications

Finding normal bases over finite fields with prescribed trace self-orthogonal relations

Finding normal bases over finite fields with prescribed trace self-orthogonal relations

Construction of self-dual normal bases and their complexity

Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields

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