Low-degree planar polynomials over finite fields of characteristic two

Abstract

Planar functions are mappings from a finite field Fq to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even. We classify polynomials of degree at most q 1/4 that induce planar functions on Fq, by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over Fq that induce planar functions on infinitely many extensions of Fq. The proof strategy is to study the number of Fq-rational points of an algebraic curve attached to a putative planar function. Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions.

Towards the full classification of exceptional scattered polynomials

“…As in [1], the main tool is the use of branches and local quadratic transformations of a plane curve to obtain a better estimate for the intersection number of two components of a fixed curve at one of its singular points. Recently, an approach based on local quadratic transformations which uses implicitly branches has been applied in [5] to classify exceptional planar functions in characteristic two.…”

Polynomials with maximal differential uniformity and the exceptional APN conjecture

“…The first author, McGuire and Rodier have proved in [1] that if f is a polynomial of degree 2e with e odd and if f contains a term of odd degree then f is not exceptional APN. Moreover, Bartoli and Schmidt have stated in Proposition 1.4 in [4] that if a polynomial of even degree m is exceptional APN then m ≡ 0 (mod 4).…”