# Difference between revisions of "CCZ-equivalence of Families of APN Polynomials over GF(2^n) from the table (for n between 6 and 11)"

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CCZ-equivalence of Families of APN Polynomials over GF(2^n) from the table (for n is equal or larger than 6 and equal or smaller than 11)

${\displaystyle N^{\circ }}$ Functions* ${\displaystyle n=6}$ ${\displaystyle n=7}$ ${\displaystyle n=8}$ ${\displaystyle n=9}$ ${\displaystyle n=10}$ ${\displaystyle n=11}$
${\displaystyle 1}$ ${\displaystyle x^{2^{s}+1}+u^{2^{k}-1}x^{2^{ik}+2^{mk+s}}}$

${\displaystyle p=3}$

Gold - - - - -
${\displaystyle 2}$ ${\displaystyle x^{2^{s}+1}+u^{2^{k}-1}x^{2^{ik}+2^{mk+s}}}$

${\displaystyle p=4}$

- - - - - -
${\displaystyle 3}$ ${\displaystyle x^{2^{2i}+2^{i}}+bx^{q+1}+cx^{q(2^{2i}+2^{i})}}$ New - - - New I (${\displaystyle i=1,c=u^{31},b=1)}$

New II (${\displaystyle i=3,c=u^{31},b=1)}$

-
${\displaystyle 4}$ ${\displaystyle x(x^{2^{i}}+x^{q}+cx^{2^{i}q})+x^{2^{i}}(c^{q}x^{q}+sx^{2^{i}q})+x^{(2^{i}+1)}q}$ ${\displaystyle N^{\circ }3}$ - New - ${\displaystyle N^{\circ }3}$: Case I ${\displaystyle (i=1,c=u^{3},s=u)}$

${\displaystyle N^{\circ }3}$: Case II ${\displaystyle (i=3,c=u^{3},s=w)}$

-
${\displaystyle 5}$ ${\displaystyle x^{3}+a^{-1}tr_{n}(a^{3}x^{9})}$ Gold${\displaystyle (a=1)}$

${\displaystyle N^{\circ }3(a=u)}$

New New I ${\displaystyle (a=1)}$

New II ${\displaystyle (a=u)}$

New New I ${\displaystyle (a=1)}$

New II ${\displaystyle (a=u)}$

New
${\displaystyle 6}$ ${\displaystyle x^{3}+a^{-1}tr_{n}^{3}(a^{3}x^{9}+a^{6}x^{18})}$ Gold${\displaystyle (a=1)}$

${\displaystyle N^{\circ }3(a=u)}$

- - New - -
${\displaystyle 7}$ ${\displaystyle x^{3}+a^{-1}tr_{n}^{3}(a^{6}x^{18}+a^{12}x^{36})}$ Gold${\displaystyle (a=1)}$

${\displaystyle N^{\circ }3(a=u)}$

- - New - -
${\displaystyle 8}$ ${\displaystyle ux^{2^{s}+1}+u^{2^{k}}x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^{k}+1}x^{2^{s}+2^{k+s}}}$

${\displaystyle v=0,w\neq 0}$

New - - - - -
${\displaystyle 9}$ ${\displaystyle ux^{2^{s}+1}+u^{2^{k}}x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^{k}+1}x^{2^{s}+2^{k+s}}}$

${\displaystyle v\neq 0,w=0}$

${\displaystyle N^{\circ }8}$ - - - - -
${\displaystyle 10}$ ${\displaystyle ux^{2^{s}+1}+u^{2^{k}}x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^{k}+1}x^{2^{s}+2^{k+s}}}$

${\displaystyle v\neq 0,w=0}$

${\displaystyle N^{\circ }8}$ - - - - -
${\displaystyle 11}$ ${\displaystyle (x+x^{2^{m}})^{2^{k}+1}+u^{(2^{n}-1)/(2^{m}-1)}(ux+u^{2^{m}}x^{2^{m}})^{(2^{k}+1)2^{i}}+u(x+x^{2^{m}})(ux+u^{2^{m}}x^{2^{m}})}$ ${\displaystyle -}$ - New ${\displaystyle (i=2)}$

${\displaystyle N^{\circ }4(i=0)}$

- - -