Introduction to finite fields and their applications
Introduction to finite fields and their applications
Optimal normal bases in GF(pn)
Discrete Applied Mathematics
Discrete Applied Mathematics
Designs, Codes and Cryptography
Normal and Self-dual Normal Bases from Factorization of$c x^{q+1} + d x^{q} - ax - b$
SIAM Journal on Discrete Mathematics
Handbook of Applied Cryptography
Handbook of Applied Cryptography
Low Complexity Word-Level Sequential Normal Basis Multipliers
IEEE Transactions on Computers
On the complexity of the dual basis of a type I optimal normal basis
Finite Fields and Their Applications
Low complexity normal bases in F2n
Finite Fields and Their Applications
Gauss periods as constructions of low complexity normal bases
Designs, Codes and Cryptography
Elliptic periods for finite fields
Finite Fields and Their Applications
Low complexity of a class of normal bases over finite fields
Finite Fields and Their Applications
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Let $${\mathbb{F}}_{q}$$ be a finite field and consider an extension $${\mathbb{F}}_{q^{n}}$$ where an optimal normal element exists. Using the trace of an optimal normal element in $${\mathbb{F}}_{q^{n}}$$ , we provide low complexity normal elements in $${\mathbb{F}}_{q^{m}}$$ , with m = n/k. We give theorems for Type I and Type II optimal normal elements. When Type I normal elements are used with m = n/2, m odd and q even, our construction gives Type II optimal normal elements in $${\mathbb{F}}_{q^{m}}$$ ; otherwise we give low complexity normal elements. Since optimal normal elements do not exist for every extension degree m of every finite field $${\mathbb{F}}_{q}$$ , our results could have a practical impact in expanding the available extension degrees for fast arithmetic using normal bases.