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We study the problem of existence of APN functions of algebraic degree n over {\mathbb F}_{2^{n}} . We characterize such functions by means of derivatives and power moments of the Walsh transform. We deduce several non-existence results which imply, in particular, that for most of the known APN functions F over \mathbb F2^{n} the function x^{2^{n}-1}+F(x) is not APN, and changing a value of $F$ in a single point then results in non-APN functions. This leads us to conjectures that an APN function modified in one point cannot remain APN and that there exists no APN function of algebraic degree $n$ . [ABSTRACT FROM AUTHOR]

Abstract: We present a method for constructing new quadratic APN functions from known ones. Applying this method to the Gold power functions we construct an APN function over . It is proven that for this function is CCZ-inequivalent to the Gold functions, and in the case it is CCZ-inequivalent to any power mapping (and, therefore, to any APN function belonging to one of the families of APN functions known so far). [Copyright &y& Elsevier]

Abstract: We present results related to vectorial plateaued functions and mappings whose derivatives are -to-1 functions. The results in this note generalize facts about almost perfect nonlinear and almost bent functions. We investigate the connection between plateaued and -to-1 functions. We show that functions which are both plateaued and differentially uniform give rise to partial difference sets. [Copyright &y& Elsevier]

This paper introduces the first found infinite classes of almost perfect nonlinear (APN) polynomials which are not Carlet—Charpin—Zinoviev (CCZ)-equivalent to power functions (at least for some values of the number of variables). These are two classes of APN binomials from F2n to F2n (for n divisible by 3, resp., 4). We prove that these functions are extended affine (EA)-inequivalent to any power function and that they are CCZ-inequivalent to the Gold, Kasami, inverse, and Dobbertin functions when n ≥ 12. This means that for n even they are CCZ-inequivalent to any known APN function. In particular, for n = 12, 20, 24, they are therefore CCZ-inequivalent to any power function. [ABSTRACT FROM AUTHOR]

A method for constructing differentially 4-uniform quadratic hexanomials has been recently introduced by J. Dillon. We give various generalizations of this method and we deduce the constructions of new infinite classes of almost perfect nonlinear quadratic trinomials and hexanomials from F22m to F22m. We check for m = 3 that some of these functions are CCZ-inequivalent to power functions. [ABSTRACT FROM AUTHOR]

New infinite classes of almost bent and almost perfect nonlinear polynomials are constructed. It is shown that they are affine inequivalent to any sum of a power function and an affine function. [ABSTRACT FROM AUTHOR]

Almost perfect nonlinear (APN) and almost bent (AB) functions are integral components of modern block ciphers and play a fundamental role in symmetric cryptography. In this paper, we describe a procedure for searching for quadratic APN functions with coefficients in F 2 over the finite field F 2 n and apply this procedure to classify all such functions over F 2 n with n ≤ 9. We discover two new APN functions (which are also AB) over F 2 9 that are CCZ-inequivalent to any known APN function over this field. We also verify that there are no quadratic APN functions with coefficients in F 2 over F 2 n with 6 ≤ n ≤ 8 other than the currently known ones. [ABSTRACT FROM AUTHOR]