normal basis

Noun
$$$$


 * 1)  For a given Galois field 𝔽qm and a suitable element β, a basis that has the form { β, βq, βq 2, ... , βq m-1 }.
 * 2) * 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, 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.
 * 1) * 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,, page 1,
 * The set of conjugates of normal element is called normal basis. A monic irreducible polynomial $$F\in\mathbb F_q[x]$$ is called normal or N-polynomial if its roots form a normal basis or, equivalently, if they are linearly independent over $$\mathbb F_q$$. The minimal polynomial of an element in a normal basis $$\{\alpha,\alpha^q,\dots,\alpha^{q^{n-1}}\}$$ is $$\textstyle m(x)=\prod_{i=0}^{n-1}(x-\alpha^{q^i})\in\mathbb F_q[x]$$ which is irreducible over $$\mathbb F_q$$. 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.
 * 1) * 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,, page 1,
 * The set of conjugates of normal element is called normal basis. A monic irreducible polynomial $$F\in\mathbb F_q[x]$$ is called normal or N-polynomial if its roots form a normal basis or, equivalently, if they are linearly independent over $$\mathbb F_q$$. The minimal polynomial of an element in a normal basis $$\{\alpha,\alpha^q,\dots,\alpha^{q^{n-1}}\}$$ is $$\textstyle m(x)=\prod_{i=0}^{n-1}(x-\alpha^{q^i})\in\mathbb F_q[x]$$ which is irreducible over $$\mathbb F_q$$. 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.
 * The set of conjugates of normal element is called normal basis. A monic irreducible polynomial $$F\in\mathbb F_q[x]$$ is called normal or N-polynomial if its roots form a normal basis or, equivalently, if they are linearly independent over $$\mathbb F_q$$. The minimal polynomial of an element in a normal basis $$\{\alpha,\alpha^q,\dots,\alpha^{q^{n-1}}\}$$ is $$\textstyle m(x)=\prod_{i=0}^{n-1}(x-\alpha^{q^i})\in\mathbb F_q[x]$$ which is irreducible over $$\mathbb F_q$$. 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.

Translations

 * French: base normale