Kim-type APN functions are affine equivalent to Gold functions
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The problem of finding APN permutations of 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 𝔽_q^2 of the form f(x)=x^3q+a_1x^2q+1+a_2x^q+2+a_3x^3, where q=2^m 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 𝔽_2^6. We extend the result of Li, Li, Helleseth and Qu by proving that if a Kim-type function f is APN and m≥ 4, then f is affine equivalent to one of two Gold functions G_1(x)=x^3 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 𝔽_2^2m are never CCZ equivalent to permutations.
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