Kim-type APN functions are affine equivalent to Gold functions

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The problem of finding APN permutations of \({\mathbb {F}}_{2^{n}}\) where n is even and n > 6 has been called the Big APN Problem. Li, Li, Helleseth and Qu recently characterized APN functions defined on \({\mathbb {F}}_{q^{2}}\) of the form f(x) = x3q + a1x2q+ 1 + a2xq+ 2 + a3x3, where q = 2m and m ≥ 4. We will call functions of this form Kim-type functions because they generalize the form of the Kim function that was used to construct an APN permutation of \({\mathbb {F}}_{2^{6}}\). We prove that Kim-type APN functions with m ≥ 4 (previously characterized by Li, Li, Helleseth, and Qu) are affine equivalent to one of two Gold functions G1(x) = x3 or \(G_{2}(x)=x^{2^{m-1}+1}\). Combined with the recent result of Göloğlu and Langevin who proved that, for even n, Gold APN functions are never CCZ equivalent to permutations, it follows that for m ≥ 4 Kim-type APN functions on \({\mathbb {F}}_{2^{2m}}\) are never CCZ equivalent to permutations.