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The complexity of dual basis of a type I optimal normal basis of$\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$ was determined to be $3n-3$ or $3n-2$according as $q$ is even or odd, respectively, by Z.-X. Wan and K. Zhou in2007. We give a new proof to this result by clearly deriving the dual of a typeI optimal normal basis with the aid of a lemma on the dual of a polynomialbasis.

A new proof to complexity of dual basis of a type I optimal normal basis