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February  2009, 3(1): 59-81. doi: 10.3934/amc.2009.3.59

A new almost perfect nonlinear function which is not quadratic

Yves Edel 1, and  Alexander Pott 2,

1. 

Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, B-9000 Ghent, Belgium
2. 

Faculty of Mathematics, Otto-von-Guericke-University Magdeburg, D-39016 Magdeburg, Germany

Received  October 2008 Revised  January 2009 Published  January 2009

Following an example in [12], 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 Brinkmann and Leander [8]. 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.
Keywords: equivalence of functions, almost bent., Walsh spectrum, Almost perfect nonlinear.
Mathematics Subject Classification: Primary: 11T71; Secondary: 05E20, 94A6.
Citation: Yves Edel, Alexander Pott. A new almost perfect nonlinear function which is not quadratic. Advances in Mathematics of Communications, 2009, 3 (1) : 59-81. doi: 10.3934/amc.2009.3.59
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