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New commutative semifields defined by new PN multinomials. (English) Zbl 1291.12006
Summary: We introduce two infinite classes of quadratic PN multinomials over \(\mathbb F_{p^{2k}}\) 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 L. E. Dickson [Trans. Am. Math. Soc. 7, 514–522 (1906; JFM 37.0112.01)] and A. A. Albert [Trans. Am. Math. Soc. 72, 296–309 (1952; Zbl 0046.03601)], 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.

MSC:
12K10 Semifields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94A60 Cryptography
51E99 Finite geometry and special incidence structures
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