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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.

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Acknowledgements

This work was supported by Norwegian Research Council and partly by the grant NIL-I-004 from Iceland, Liechtenstein and Norway through the EEA and Norwegian Financial Mechanisms.

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  1. Department of Informatics, University of Bergen, PB 7803, 5020, Bergen, Norway

    Lilya Budaghyan & Tor Helleseth

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  1. Lilya Budaghyan
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  2. Tor Helleseth
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Correspondence to Lilya Budaghyan.

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Budaghyan, L., Helleseth, T. New commutative semifields defined by new PN multinomials. Cryptogr. Commun. 3, 1–16 (2011). https://doi.org/10.1007/s12095-010-0022-2

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  • DOI: https://doi.org/10.1007/s12095-010-0022-2

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