A new almost perfect nonlinear function (APN) on F 2 10 which is not equivalent to any of the previously known APN mappings is constructed. This is the first example of an APN mapping which is not equivalent to a power mapping.

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Methods for constructing large families of permutation polynomials of finite fields are introduced. For some of these permutations the cycle structure and the inverse mapping are determined. The results are applied to lift minimal blocking sets of PG(2, q) to those of PG(2, q n ).

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We study the Boolean functions f λ : F 2 n → F 2 , n = 6r, of the form f (x) = Tr(λx d ) with d = 2 2r + 2 r + 1 and λ ∈ F 2 n . Our main result is the characterization of those λ for which f λ are bent. We show also that the set of these cubic bent functions contains a subset, which with the constantly zero function forms a vector space of dimension 2r over F 2 . Further we determine the Walsh spectra of some related quadratic functions, the derivatives of the functions f λ .