Normal basis

English

Noun

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1. Template:Lb For a given Galois field 𝔽_q^m and a suitable element β, a basis that has the form {β, β^q, β^q2, ... , β^qm-1}.

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   • 1989, Willi Geiselmann, Dieter Gollmann, Symmetry and Duality in Normal Basis Multiplication, T. Mora (editor), Applied Algebra, Algebraic Algorithms, and Error-correcting Codes: 6th International Conference, Proceedings, Springer, Template:W 357, page 230,

     We also combine dual basis and normal basis techniques. The duality of normal bases is shown to be equivalent to the symmetry of the logic array of the serial input / parallel output architectures proposed in this paper.

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   • 2015, Sergey Abrahamyan, Melsik Kyureghyan, New recursive construction of normal polynomials over finite fields, Gohar Kyureghyan, Gary L. Mullen, Alexander Pott (editors, Topics in Finite Fields, Template:W, page 1,

     The set of conjugates of normal element is called normal basis. A monic irreducible polynomial $F\in {𝔽}_q[x]$ is called normal or N-polynomial if its roots form a normal basis or, equivalently, if they are linearly independent over ${𝔽}_q$. The minimal polynomial of an element in a normal basis $\{\alpha ,{\alpha }^q,\dots ,{\alpha }^{q^{n-1}}\}$ is $m(x)={\prod }_{i=0}^{n-1}(x-{\alpha }^{q^i})\in {𝔽}_q[x]$ which is irreducible over ${𝔽}_q$. The elements of a normal basis are exactly the roots of some N-polynomial. Hence an N-polynomial is just another way of describing a normal basis.

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