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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]

Boolean functions, and bent functions in particular, are considered up to so-called EA-equivalence, which is the most general known equivalence relation preserving bentness of functions. However, for a special type of bent functions, so-called Niho bent functions there is a more general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials. In the present work we study, for a given o-polynomial, a general construction which provides all possible o-equivalent Niho bent functions, and we considerably simplify it to a form which excludes EA-equivalent cases. That is, we identify all cases which can potentially lead to pairwise EA-inequivalent Niho bent functions derived from o-equivalence of any given Niho bent function. Furthermore, we determine all pairwise EA-inequivalent Niho bent functions arising from all known o-polynomials via o-equivalence. [ABSTRACT FROM AUTHOR]

This paper is dedicated to a question whether the currently known families of quadratic APN polynomials are pairwise different up to CCZ-equivalence. We reduce the list of these families to those CCZ-inequivalent to each other. In particular, we prove that the families of APN trinomials (constructed by Budaghyan and Carlet in 2008) and multinomials (constructed by Bracken et al. 2008) are contained in the APN hexanomial family introduced by Budaghyan and Carlet in 2008. We also prove that a generalization of these trinomial and multinomial families given by Duan et al. (2014) is contained in the family of hexanomials as well. [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]