Abstract
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 p 4s for odd s > 1 is not isotopic to previously known examples.
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Communicated by Simeon Ball.
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Bierbrauer, J. New semifields, PN and APN functions. Des. Codes Cryptogr. 54, 189–200 (2010). https://doi.org/10.1007/s10623-009-9318-7
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DOI: https://doi.org/10.1007/s10623-009-9318-7
Keywords
- Semifield
- PN function
- APN function
- Dembowski–Ostrom polynomial
- Middle nucleus
- Kernel
- Isotopy
- Dickson semifields
- Albert semifields