New semifields, PN and APN functions

作者：Bierbrauer Juergen^*

来源：Designs, Codes and Cryptography, 2010, 54(3): 189-200.

DOI：10.1007/s10623-009-9318-7

• Semifield
• PN function
• APN function
• Dembowski-Ostrom polynomial
• Middle nucleus
• Kernel
• Isotopy
• Dickson semifields
• Albert semifields

We describe a method of proving that certain functions f : F -> 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.

• 出版日期2010-3