Abstract
A finite extension φ = GF( q n ) of a finite field F = GF( q) has a self-complementary normal basis over F iff either n is odd or n ≡ 2 (mod 4) and q is even. This paper presents a complete characterization of square matrices A of order n over F such that the set { A q k } n−1 k=0 is a self-complementary normal basis of a matrix representation of φ over F. Based on this characterization, we derive a method of synthesizing irreducible polynomials of degree n over F, whose roots form such a basis whenever one exists.
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