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

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m (Nikolay moved page 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) to [[CCZ-equivalence of Families of APN Polynomials over GF(2^n) from the table (for n betw...)
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<tr>
<tr>
<td><math>1</math></td>
<td>1</td>
<td><math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}}</math>
<td><math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}}</math>
<math>p=3</math>
<math>p=3</math>
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<tr>
<tr>
<td><math>2</math></td>
<td>2</td>
<td><math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}}</math>
<td><math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}}</math>
<math>p=4</math>
<math>p=4</math>
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<tr>
<tr>
<td><math>3</math></td>
<td>3</td>
<td><math>x^{2^{2i}+2^i}+bx^{q+1}+cx^{q(2^{2i}+2^i)}</math></td>
<td><math>x^{2^{2i}+2^i}+bx^{q+1}+cx^{q(2^{2i}+2^i)}</math></td>
<td>New</td>
<td>New</td>
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<tr>
<tr>
<td><math>4</math></td>
<td>4</td>
<td><math>x(x^{2^i}+x^q+cx^{2^iq})+x^{2^i}(c^qx^q+sx^{2^iq})+x^{(2^i+1)}q</math></td>
<td><math>x(x^{2^i}+x^q+cx^{2^iq})+x^{2^i}(c^qx^q+sx^{2^iq})+x^{(2^i+1)}q</math></td>
<td><math>N^\circ 3</math></td>
<td><math>N^\circ 3</math></td>
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<tr>
<tr>
<td><math>5</math></td>
<td>5</td>
<td><math>x^3+a^{-1}tr_n(a^3x^9)</math></td>
<td><math>x^3+a^{-1}tr_n(a^3x^9)</math></td>
<td>Gold<math>(a=1)</math>
<td>Gold<math>(a=1)</math>
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<tr>
<tr>
<td><math>6</math></td>
<td>6</td>
<td><math>x^3+a^{-1}tr_n^3(a^3x^9+a^6x^{18})</math></td>
<td><math>x^3+a^{-1}tr_n^3(a^3x^9+a^6x^{18})</math></td>
<td>Gold<math>(a=1)</math>
<td>Gold<math>(a=1)</math>
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<tr>
<tr>
<td><math>7</math></td>
<td>7</td>
<td><math>x^3+a^{-1}tr_n^3(a^6x^{18}+a^{12}x^{36})</math></td>
<td><math>x^3+a^{-1}tr_n^3(a^6x^{18}+a^{12}x^{36})</math></td>
<td>Gold<math>(a=1)</math>
<td>Gold<math>(a=1)</math>
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<tr>
<tr>
<td><math>8</math></td>
<td>8</td>
<td><math>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>
<td><math>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>v=0, w\ne0</math>
<math>v=0, w\ne0</math>
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<tr>
<tr>
<td><math>9</math></td>
<td>9</td>
<td><math>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>
<td><math>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>v\ne0, w=0</math>
<math>v\ne0, w=0</math>
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<tr>
<tr>
<td><math>10</math></td>
<td>10</td>
<td><math>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>
<td><math>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>v\ne0, w=0</math>
<math>v\ne0, w=0</math>
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<tr>
<tr>
<td><math>11</math></td>
<td>11</td>
<td><math>(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>
<td><math>(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}</math>
<td><math>-</math></td>
<math>+u(x+x^{2^m})(u x+u ^{2^m}x^{2^m})</math>
<td>-</td>
<td>-</td>
<td>-</td>
<td>New <math>(i=2)</math>
<td>New <math>(i=2)</math>

Revision as of 14:45, 5 November 2019

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]
1 [math]\displaystyle{ x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}} }[/math]

[math]\displaystyle{ p=3 }[/math]

Gold - - - - -
2 [math]\displaystyle{ x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}} }[/math]

[math]\displaystyle{ p=4 }[/math]

- - - - - -
3 [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]

-
4 [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]

-
5 [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
6 [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 - -
7 [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 - -
8 [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 - - - - -
9 [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] - - - - -
10 [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] - - - - -
11 [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} }[/math]

[math]\displaystyle{ +u(x+x^{2^m})(u x+u ^{2^m}x^{2^m}) }[/math]

- - New [math]\displaystyle{ (i=2) }[/math]

[math]\displaystyle{ N^\circ4 (i=0) }[/math]

- - -