, Volume 3, Issue 1, pp 1-16
Date: 12 Mar 2010

New commutative semifields defined by new PN multinomials

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Abstract

We introduce two infinite classes of quadratic PN multinomials over \(\textbf{F}_{p^{2k}}\) where p is any odd prime. We prove that for k odd one of these classes defines a new family of commutative semifields (in part by studying the nuclei of these semifields). After the works of Dickson (Trans Am Math Soc 7:514–522, 1906) and Albert (Trans Am Math Soc 72:296–309, 1952), this is the firstly found infinite family of commutative semifields which is defined for all odd primes p. These results also imply that these PN functions are CCZ-inequivalent to all previously known PN mappings.