The Construction Of Finite Semifields And Their Applications

Perfect nonlinear(PN) functions and almost perfect nonlinear(APN) functions have wide applications in cryptography, coding theory and finite geometry due to their resistance to differential analysis. Based on the correspondence between PN functions and commutative semifields, it is effective to investigate finite commutative semifields to enhance the development of cryptography and coding theory. The investigation of semifields was started by Dickson before Knuth giving the definition of characteristic of semifields. In 2013, Zhou and Pott came up with a family of finite presemifields(rank 2) based on multiplication of Albert presemifiled and multiplication of Conhen-Ganley presemifield(rank 2).According to the procedure of constructing the multiplication in Zhou-Pott presemifields and the form of multiplication of Albert twisted presemifield, this paper firstly construct a family of presemifields. With the property of automorphism of Finite Fields,it is proved that polynomial parameter in the presemifield must be induced by the non-square elments and automorphsim of Finite Field, which means it should be a permutation. Furthermore, we prove the uniqueness of polynomial parameter.Secondly, with the family of presemifield we just constructed and one PN function given by Zhou and Pott, we constructed two family of APN functions. The first family of APN functions is equivalent to the APN function associated with Zhou-Pott's presemifield. In the situation of m = 4 from the second family of APN functions, we tested it with the software magma, proving that our APN function is equivalent to a known APN function, but for situation m 4, it is still an open problem to test the equivalence between our APN function and known APN functions.