# Some APN functions CCZ-equivalent to Gold functions and EA-inequivalent to power functions over GF(2^n)

Some APN functions CCZ-equivalent to Gold functions and EA-inequivalent to power functions over${\displaystyle \mathbb {F} _{2^{n}}}$ (constructed in [1])

Functions Conditions ${\displaystyle d^{\circ }}$
${\displaystyle x^{2^{i}+1}+(x^{2^{i}}+x+tr_{n}(1)+1)tr_{n}(x^{2^{i}+1}+x\ tr_{n}(1))}$ ${\displaystyle n\geqslant 4}$, ${\displaystyle \gcd(i,n)=1}$ ${\displaystyle 3}$
${\displaystyle [x+tr_{n/3}(x^{2(2^{i}+1)}+x^{4(2^{i}+1)})+tr_{n}(x)tr_{n/3}(x^{2^{i}+1}+x^{2^{2i}(2^{i}+1)})]^{2^{i}+1}}$ ${\displaystyle 6|n\ ,\gcd(i,n)=1}$ ${\displaystyle 4}$
${\displaystyle x^{2^{i}+1}+tr_{n/m}(x^{2^{i}+1})+x^{2^{i}}tr_{n/m}(x)+x\ tr_{n/m}(x)^{2^{i}}+[tr_{n/m}(x)^{2^{i}+1}+tr_{n/m}(x^{2^{i}+1})+tr_{n/m}(x)]^{\frac {1}{2^{i}+1}}(x^{2^{i}}+tr_{n/m}(x)^{2^{i}}+1)+[tr_{n/m}(x)^{2^{i}+1}+tr_{n/m}(x^{2^{i}+1})+tr_{n/m}(x)]^{\frac {2^{i}}{2^{i}+1}}(x+tr_{n/m}(x))}$ ${\displaystyle m\neq n\ ,n\ odd\ ,m|n\ ,\gcd(i,n)=1}$ ${\displaystyle m+2}$
1. Budaghyan L, Carlet C, Pott A. New classes of almost bent and almost perfect nonlinear polynomials. IEEE Transactions on Information Theory. 2006 Mar;52(3):1141-52.