New semifields, PN and APN functions

We describe a method of proving that certain functions $${f:F\longrightarrow F}$$ defined on a finite field 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 p4s for odd s > 1 is not isotopic to previously known examples.

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