B. Chase, P. Lisonek,
Kim-type APN functions are affine equivalent to Gold functions.
To appear in Cryptography and Communications.

The problem of finding APN permutations of F2n 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 Fq2 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 F26. 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 G1(x)=x3 or G2(x)=x2m-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 F22m are never CCZ equivalent to permutations.


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