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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On orders of optimal normal basis generators
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by Shuhong Gao and Scott A. Vanstone PDF
Math. Comp. 64 (1995), 1227-1233 Request permission


In this paper we give some experimental results on the multiplicative orders of optimal normal basis generators in ${F_{{2^n}}}$ over ${F_2}$ for $n \leq 1200$ whenever the complete factorization of ${2^n} - 1$ is known. Our results show that a subclass of optimal normal basis generators always have high multiplicative orders, at least $O(({2^n} - 1)/n)$, and are very often primitive. For a given optimal normal basis generator $\alpha$ in ${F_{{2^n}}}$ and an arbitrary integer e, we show that ${\alpha ^e}$ can be computed in $O(n \cdot v(e))$ bit operations, where $v(e)$ is the number of 1’s in the binary representation of e.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Math. Comp. 64 (1995), 1227-1233
  • MSC: Primary 11T30; Secondary 11Y05, 11Y16
  • DOI:
  • MathSciNet review: 1297469