Abstract
It is proven that any Dembowski–Ostrom polynomial is planar if and only if its evaluation map is 2-to-1, which can be used to explain some known planar Dembowski–Ostrom polynomials. A direct connection between a planar Dembowski–Ostrom polynomial and a permutation polynomial is established if the corresponding semifield is of odd dimension over its nucleus. In addition, all commutative semifields of order 35 are classified.
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Weng, G., Zeng, X. Further results on planar DO functions and commutative semifields. Des. Codes Cryptogr. 63, 413–423 (2012). https://doi.org/10.1007/s10623-011-9564-3
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DOI: https://doi.org/10.1007/s10623-011-9564-3