Bounds on the degree of APN polynomials: the case of x −1 + g(x)

Author： Leander Gregor   Rodier François  

Publisher： Springer Publishing Company

ISSN： 0925-1022

Source： Designs, Codes and Cryptography, Vol.59, Iss.1-3, 2011-04, pp. : 207-222

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Abstract

In this paper we consider APN functions $${f:mathcal{F}_{2^m}to mathcal{F}_{2^m}}$$ of the form f(x) = x −1 + g(x) where g is any non $${mathcal{F}_{2}}$$ -affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields $${mathcal{F}_{2^m}}$$ . Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ≤ 3 where these functions are equivalent to x 3.

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