A new almost perfect nonlinear function which is not quadratic

Coauthor A. Pott.

Advances in Mathematics of Communications, 3 No. 1 (2009), 59-81.


Erata in the printed version:

page 71 line 7: $f\in \tilde{V}$ should be $f\in \tilde{V}^{\perp}$.

page 79 function No 1.13: should be No. 1.10 + $u^{90}( \tr(u^{87} x^{3} + u^{141} x^{5} + u^{20} x^{9}) + \tr_{16/2}(u^{51} x^{17}))$.

page 80: table 10, for function no. 1.2 |M(GF)| should be 2113.


Following an example in [1], 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 the approach can be used to construct ``non-quadratic'' APN functions. 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 Brinkmann and Leander [2]. 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.

Download the preprint as pdf

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