CCZ-equivalence of Families of APN Polynomials over GF(2^n) from the table (for n between 6 and 11)
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)
| [math]\displaystyle{ N^\circ }[/math] | Functions* | [math]\displaystyle{ n=6 }[/math] | [math]\displaystyle{ n=7 }[/math] | [math]\displaystyle{ n=8 }[/math] | [math]\displaystyle{ n=9 }[/math] | [math]\displaystyle{ n=10 }[/math] | [math]\displaystyle{ n=11 }[/math] |
|---|---|---|---|---|---|---|---|
| [math]\displaystyle{ 1 }[/math] | [math]\displaystyle{ x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}} }[/math]
[math]\displaystyle{ p=3 }[/math] | Gold | - | - | - | - | - |
| [math]\displaystyle{ 2 }[/math] | [math]\displaystyle{ x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}} }[/math]
[math]\displaystyle{ p=4 }[/math] | - | - | - | - | - | - |
| [math]\displaystyle{ 3 }[/math] | [math]\displaystyle{ x^{2^{2i}+2^i}+bx^{q+1}+cx^{q(2^{2i}+2^i)} }[/math] | New | - | - | - | New I ([math]\displaystyle{ i=1, c=u^{31}, b=1) }[/math]
New II ([math]\displaystyle{ i=3, c=u^{31}, b=1) }[/math] | - |
| [math]\displaystyle{ 4 }[/math] | [math]\displaystyle{ x(x^{2^i}+x^q+cx^{2^iq})+x^{2^i}(c^qx^q+sx^{2^iq})+x^{(2^i+1)}q }[/math] | [math]\displaystyle{ N^\circ 3 }[/math] | - | New | - | [math]\displaystyle{ N^\circ3 }[/math]: Case I [math]\displaystyle{ (i=1, c=u^3, s=u) }[/math]
[math]\displaystyle{ N^\circ3 }[/math]: Case II [math]\displaystyle{ (i=3, c=u^3, s=w) }[/math] | - |
| [math]\displaystyle{ 5 }[/math] | [math]\displaystyle{ x^3+a^{-1}tr_n(a^3x^9) }[/math] | Gold[math]\displaystyle{ (a=1) }[/math]
[math]\displaystyle{ N^\circ 3 (a=u) }[/math] | New | New I [math]\displaystyle{ (a=1) }[/math]
New II [math]\displaystyle{ (a=u) }[/math] | New | New I [math]\displaystyle{ (a=1) }[/math]
New II [math]\displaystyle{ (a=u) }[/math] | New |
| [math]\displaystyle{ 6 }[/math] | [math]\displaystyle{ x^3+a^{-1}tr_n^3(a^3x^9+a^6x^{18}) }[/math] | Gold[math]\displaystyle{ (a=1) }[/math]
[math]\displaystyle{ N^\circ 3 (a=u) }[/math] | - | - | New | - | - |
| [math]\displaystyle{ 7 }[/math] | [math]\displaystyle{ x^3+a^{-1}tr_n^3(a^6x^{18}+a^{12}x^{36}) }[/math] | Gold[math]\displaystyle{ (a=1) }[/math]
[math]\displaystyle{ N^\circ 3 (a=u) }[/math] | - | - | New | - | - |
| [math]\displaystyle{ 8 }[/math] | [math]\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}} }[/math]
[math]\displaystyle{ v=0, w\ne0 }[/math] | New | - | - | - | - | - |
| [math]\displaystyle{ 9 }[/math] | [math]\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}} }[/math]
[math]\displaystyle{ v\ne0, w=0 }[/math] | [math]\displaystyle{ N^\circ8 }[/math] | - | - | - | - | - |
| [math]\displaystyle{ 10 }[/math] | [math]\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}} }[/math]
[math]\displaystyle{ v\ne0, w=0 }[/math] | [math]\displaystyle{ N^\circ8 }[/math] | - | - | - | - | - |
| [math]\displaystyle{ 11 }[/math] | [math]\displaystyle{ (x+x^{2^m})^{2^k+1}+u^{(2^{n}-1)/(2^m-1)}(u x+u^{2^m}x^{2^m})^{(2^k+1)2^i}+u(x+x^{2^m})(u x+u ^{2^m}x^{2^m}) }[/math] | [math]\displaystyle{ - }[/math] | - | New [math]\displaystyle{ (i=2) }[/math]
[math]\displaystyle{ N^\circ4 (i=0) }[/math] | - | - | - |