A new almost perfect nonlinear function which is not quadratic.

*(English)*Zbl 1231.11140Summary: Following an example in [L. Budaghyan, C. Carlet and G. Leander, Finite Fields Appl. 15, No. 2, 150–159 (2009; Zbl 1184.94228)], we show how to change one coordinate function of an almost perfect nonlinear (APN) function in order to obtain new examples. It turns out that this is a very powerful method to construct new APN functions. In particular, we show that our approach can be used to construct a “non-quadratic” APN function. This new example is in remarkable contrast to all recently constructed functions which have all been quadratic. An equivalent function has been found independently by M. Brinkmann and G. Leander [Des. Codes Cryptography 49, No. 1–3, 273–288 (2008; Zbl 1184.94227)]. However, they claimed that their function is CCZ equivalent to a quadratic one. In this paper we give several reasons why this new function is not equivalent to a quadratic one.