Kim-type APN functions are affine equivalent to Gold functions.

To appear in Cryptography and Communications.

The problem of finding APN permutations of F_{2n} 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_{q2} of the form
f(x)=x^{3q}+a_{1}x^{2q+1}+a_{2}x^{q+2}+a_{3}x^{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_{26}.
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^{2m-1+1}. Combined with the recent result of Gologlu 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_{22m} are never CCZ equivalent to permutations.