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

Publication: Research - peer-reviewJournal article – Annual report year: 2011

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.
Original language English Designs, Codes and Cryptography 59 1-3 207-222 0925-1022 http://dx.doi.org/10.1007/s10623-010-9456-y Published - 2011
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