APN polynomials over GF(2^n) CCZ-inequivalent to quadratic functions and monomials

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n = 6

The polynomial

x3 + α17(x17 + x18 + x20 + x24) + α14( Tr( α52x3 + α6x5 + α19x7 + α28x11 + α2x13) + (α2x)9 + (α2x)18 + (α2x)36 + x21 + x42)

where is α is primitive in GF(2^6), is the only known example of an APN function CCZ-inequivalent to a monomial or quadratic function [1][2]. A Magma implementation of the polynomial is available.

  1. ↑ M. Brinkmann, G. Leander. On the classification of APN functions up to dimension five. Designs, Codes and Cryptography 49, pp. 273-288, 2008. https://doi.org/10.1007/s10623-008-9194-6
  2. ↑ Y. Edel, A. Pott. A new almost perfect nonlinear function which is not quadratic. Advances in Mathematics of Communications 3 (1), pp. 59-81, 2009. https://doi.org/10.3934/amc.2009.3.59
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