normal basis

Definitions

For a given Galois field 𝔽_(qᵐ) and a suitable element β, a basis that has the form {β, β^q, β^(q2), ... , β^(qm-1)}.

Equivalents

Examples

“A normal basis is generated by the repeated action of the Frobenius endomorphism on a suitable element β; it is the orbit of β for that endomorphism.”

“It is a characterising property of normal bases that #92;beta#123;qᵐ#125;#61;#92;beta.”

“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, LNCS 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.”

“12.3 In the finite field E#61;#92;mathbbF#95;#123;3³#125;, find⟳: (a) a primitive root of E whose conjugates do not form⟳ a normal basis of E#47;#92;mathbbF#95;3; (b) a normal basis that does not consist⟳ of primitive roots of E. For an arbitrary field E with prime field #92;mathbbF#95;p, the extension E#47;#92;mathbbF#95;p does always have⟳ at least one normal basis consisting of primitive roots.”

“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, American Mathematical Society, page 1, The set⟳ of conjugates of normal element is called normal basis. A monic irreducible polynomial F∈ 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 α,α^q,…,α is m(x)=∏ᵢ₌₀ⁿ⁻¹(x-α)∈ 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.”

CEFR level