A new almost perfect nonlinear function which is not quadratic

Yves Edel; Alexander Pott

doi:10.3934/amc.2009.3.59

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2009, Volume 3, Issue 1: 59-81. Doi: 10.3934/amc.2009.3.59

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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
2.
Faculty of Mathematics, Otto-von-Guericke-University Magdeburg, D-39016 Magdeburg

Received: September 30, 2008

Revised: December 31, 2008

Published: January 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 11T71; Secondary: 05E20, 94A60.

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