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New semifields, PN and APN functions. (English) Zbl 1269.12006

Summary: We describe a method of proving that certain functions \(f: \mathbb F\to \mathbb F\) defined on a finite field \(\mathbb F\) are either PN-functions (in odd characteristic) or APN-functions (in characteristic 2). This method is illustrated by giving short proofs of the APN-respectively the PN-property for various families of functions. The main new contribution is the construction of a family of PN-functions and their corresponding commutative semifields of dimension \(4s\) in arbitrary odd characteristic. It is shown that a subfamily of order \(p^{4s}\) for odd \(s > 1\) is not isotopic to previously known examples.

MSC:

12K10 Semifields
11T06 Polynomials over finite fields
17A35 Nonassociative division algebras
51A35 Non-Desarguesian affine and projective planes
51A40 Translation planes and spreads in linear incidence geometry
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