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Trace self-orthogonal relations of normal bases. (English) Zbl 1331.11113

Summary: Normal bases with specific trace self-orthogonal relations over finite fields have been found to be very useful for many fast arithmetic computations, especially when the extensions of finite fields have no self-dual normal basis. Recent work in [Finite Fields Appl. 28, 1–21 (2014; Zbl 1331.11112)] has given the necessary and sufficient conditions for the existence of normal bases of \(\mathrm{GF}(2^n)\) with a prescribed trace vector when \(n\) is odd or \(n\) is a power of two. However, the methods in the paper cited above cannot work in general cases. In this paper, using methods different from that paper, we give a complete characterization of the trace self-orthogonal relations of arbitrary normal bases. Furthermore, we provide a combination method to construct normal elements with the prescribed trace vectors. These generalize the results in [loc. cit.] to general cases. The main result of this paper is shown as follows.
Let \(\underline{a} = (a_0, a_1, \cdots, a_{n - 1})\) be a prescribed \(n\)-vector over \(\mathrm{GF}(q)\), with corresponding polynomial \(f_a(x) = \sum_{i = 0}^{n - 1} a_i x^i\). We present that there exists a normal element \(\alpha\) of \(\mathrm{GF}(q^n)\) over \(\mathrm{GF}(q)\), with trace vector \(\underline{a}\), such that \(a_i = \mathrm{Tr}_{q^n\mid q}(\alpha^{1 + q^i})\) for all \(0 \leq i \leq n - 1\), if and only if \(a_i = a_{n - i}\) for all \(1 \leq i \leq n - 1\) and
1) when \(q\) is odd, \(f_a(x)\) is prime to \(x^n - 1\); \(f_a(1)\) is a quadratic residue in \(\mathrm{GF}(q)\); for even \(n\), if \(f_a(-1) \neq 0\), then \(f_a(-1)\) is not a quadratic residue in \(\mathrm{GF}(q)\);
2) when \(q\) is even, \(f_a(x)\) is prime to \(x^n - 1\); for even \(n\), \(a_{n / 2} = 0\) and if \(4\mid n\), then \[ \mathrm{Tr}_{q | 2}\left(\sum_{i = 0}^{n / 4 - 1} a_0^{- 1} a_{2 i + 1}\right) = 1. \]

MSC:

11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)
12E20 Finite fields (field-theoretic aspects)

Citations:

Zbl 1331.11112
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References:

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