A mapping f: GF(pn)→GF(pn) is called differentially k-uniform if k is the maximum number of solutions x∊GF(pn) of f(x+a)−f(x)=b, where a,b∊GF(pn) and a≠0. A 1-uniform mapping is called perfect nonlinear (PN). In this paper we discuss some problems related to the equivalence of perfect nonlinear functions, and describe a class of perfect nonlinear binomials uxpk+1+x2 in GF(p2k). These are the first PN binomials known to us which are composed with inequivalent monomials. We show that this family of binomials is equivalent to the monomial x2. We survey some of the close connections between perfect nonlinear functions and finite affine planes, in particular those which are important for equivalence proofs.
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