Classification of Quadratic APN Trinomials, Quadrinomials, Pentanomials, Hexanomials (CCZ-inequivalent with infinite monomial families) in Small Dimensions with all Coefficients equal to 1: Difference between revisions
Jump to navigation
Jump to search
No edit summary |
No edit summary |
||
| Line 500: | Line 500: | ||
<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>q=2^{m},n=2m, \gcd(i,m)=1, c\in\mathbb{F}_{2^n}, s\in\mathbb{F}_{2^n}\backslash \mathbb{F}_q, X^{2^i+1}+cX^{2^i}+c^qX+1</math> is irreducible over <math>\mathbb{F}_{2^n}</math></td> | <td><math>q=2^{m},n=2m, \gcd(i,m)=1, c\in\mathbb{F}_{2^n}, s\in\mathbb{F}_{2^n}\backslash \mathbb{F}_q, X^{2^i+1}+cX^{2^i}+c^qX+1</math> is irreducible over <math>\mathbb{F}_{2^n}</math></td> | ||
<td><ref>Budaghyan L, Carlet C, Leander G. Constructing new APN functions from known ones. Finite Fields and Their Applications. 2009 Apr 1;15(2):150-9.</ref></td> | <td><ref>Budaghyan L, Carlet C. Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Transactions on Information Theory. 2008 May;54(5):2354-7.</ref></td> | ||
</tr> | |||
<tr> | |||
<td><math>5</math></td> | |||
<td><math>x^3+a^{-1}tr_n(a^3x^9)</math></td> | |||
<td><math>a\ne0</math></td> | |||
<td><ref>Budaghyan L, Carlet C, Leander G. Constructing new APN functions from known ones. Finite Fields and Their Applications. 2009 Apr 1;15(2):150-9.</ref><ref name="workshope">Budaghyan L, Carlet C, Leander G. On a construction of quadratic APN functions. In2009 IEEE Information Theory Workshop 2009 Oct 11 (pp. 374-378). IEEE.</ref></td> | |||
</tr> | |||
<tr> | |||
<td><math>6</math></td> | |||
<td><math>x^3+a^{-1}tr_n^3(a^3x^9+a^6x^{18})</math></td> | |||
<td><math>3|n, a\ne0</math></td> | |||
<td><ref name="workshope"/></td> | |||
</tr> | |||
<tr> | |||
<td><math>7</math></td> | |||
<td><math>x^3+a^{-1}tr_n^3(a^6x^{18}+a^{12}x^{36})</math></td> | |||
<td><math>3|n, a\ne0</math></td> | |||
<td><ref name="workshope"/></td> | |||
</tr> | |||
<tr> | |||
<td><math>8-10</math></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> | |||
<td><math>n=3k, 3|(k+s), \gcd(k,3)=\gcd(s,3k)=1,v, w\in\mathbb{F}_{2^k}, vw \ne 1</math></td> | |||
<td><ref>Bracken C, Byrne E, Markin N, Mcguire G. New families of quadratic almost perfect nonlinear trinomials and multinomials. Finite Fields and Their Applications. 2008 Jul 1;14(3):703-14.</ref></td> | |||
</tr> | |||
<tr> | |||
<td><math>11</math></td> | |||
<td><math>(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})</math></td> | |||
<td><math>m\geqslant 2, 2|m,n=2m,\gcd(k, m)=1, i</math> is even</td> | |||
<td><ref>Zhou Y, Pott A. A new family of semifields with 2 parameters. Advances in Mathematics. 2013 Feb 15;234:43-60.</ref></td> | |||
</tr> | </tr> | ||
Revision as of 13:45, 22 February 2019
Table 1: Classification of Quadratic APN Trinomials (CCZ-inequivalent to infinite monomial families) in Small Dimensions with Coefficients in [math]\displaystyle{ \mathbb{F}_2 }[/math]
| [math]\displaystyle{ n }[/math] | [math]\displaystyle{ N^\circ }[/math] | Functions | Families from tables 5 | Relation to [6] |
|---|---|---|---|---|
| [math]\displaystyle{ 6 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 7 }[/math] | [math]\displaystyle{ 7.1 }[/math] | [math]\displaystyle{ x^{20} + x^6 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ Table 7: N^\circ 8.1 }[/math] |
| [math]\displaystyle{ 7.2 }[/math] | [math]\displaystyle{ x^{34} + x^{18} + x^5 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7: N^\circ 2.1 }[/math] | |
| [math]\displaystyle{ 8 }[/math] | [math]\displaystyle{ 8.1 }[/math] | [math]\displaystyle{ x^{72} + x^6 + x^3 }[/math] | [math]\displaystyle{ N^\circ5 }[/math] | [math]\displaystyle{ Table 9: N^\circ1.3 }[/math] |
| [math]\displaystyle{ 8.2 }[/math] | [math]\displaystyle{ x^{72} + x^{36} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 9:N^\circ1.4 }[/math] | |
| [math]\displaystyle{ 9 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 10 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 11 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
Table 2: Classification of Quadratic APN Quadrinomials (CCZ-inequivalent with infinite monomial
families) in Small Dimensions with Coefficients as 1
| [math]\displaystyle{ n }[/math] | [math]\displaystyle{ N^\circ }[/math] | Functions | Families from tables 5 | Relation to [6] |
|---|---|---|---|---|
| [math]\displaystyle{ 6 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 7 }[/math] | [math]\displaystyle{ 7.1 }[/math] | [math]\displaystyle{ x^{72} + x^{40} + x^{12} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ Table 7: N^\circ12.1 }[/math] |
| [math]\displaystyle{ 7.2 }[/math] | [math]\displaystyle{ x^{33} + x^{17} + x^{12} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ2.2 }[/math] | |
| [math]\displaystyle{ 7.3 }[/math] | [math]\displaystyle{ x^{34} + x^{33} + x^{10} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ10.1 }[/math] | |
| [math]\displaystyle{ 7.4 }[/math] | [math]\displaystyle{ x^{66} + x^{34} + x^{20} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ11.1 }[/math] | |
| [math]\displaystyle{ 7.5 }[/math] | [math]\displaystyle{ x^{68} + x^{18} + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ8.1 }[/math] | |
| [math]\displaystyle{ 7.6 }[/math] | [math]\displaystyle{ x^{66} + x^{18} + x^9 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ9.1 }[/math] | |
| [math]\displaystyle{ 8 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 9 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 10 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 11 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
Table 3: Classification of Quadratic APN Quadrinomials (CCZ-inequivalent with infinite monomial
families) in Small Dimensions with Coefficients as 1
| [math]\displaystyle{ n }[/math] | [math]\displaystyle{ N^\circ }[/math] | Functions | Families from tables 5 | Relation to [6] |
|---|---|---|---|---|
| [math]\displaystyle{ 6 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 7 }[/math] | [math]\displaystyle{ 7.1 }[/math] | [math]\displaystyle{ x^{68} + x^{40} + x^{24} + x^6 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ Table 7: N^\circ13.1 }[/math] |
| [math]\displaystyle{ 7.2 }[/math] | [math]\displaystyle{ x^{65} + x^{20} + x^{18} + x^6 + x^3 }[/math] | [math]\displaystyle{ N^\circ5 }[/math] | [math]\displaystyle{ 7:N^\circ1.2 }[/math] | |
| [math]\displaystyle{ 7.3 }[/math] | [math]\displaystyle{ x^{40} + x^{34} + x^{18} + x^{10} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ12.1 }[/math] | |
| [math]\displaystyle{ 7.4 }[/math] | [math]\displaystyle{ x^{48} + x^{40} + x^{10} + x^9 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ2.1 }[/math] | |
| [math]\displaystyle{ 7.5 }[/math] | [math]\displaystyle{ x^{33} + x^9 + x^6 + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ11.1 }[/math] | |
| [math]\displaystyle{ 7.6 }[/math] | [math]\displaystyle{ x^{40} + x^{36} + x^{34} + x^{24} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ10.1 }[/math] | |
| [math]\displaystyle{ 7.7 }[/math] | [math]\displaystyle{ x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ2.1 }[/math] | |
| [math]\displaystyle{ 7.8 }[/math] | [math]\displaystyle{ x^{65} + x^{36} + x^{20} + x^{17} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ14.1 }[/math] | |
| [math]\displaystyle{ 7.9 }[/math] | [math]\displaystyle{ x^{40} + x^{33} + x^{17} + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ8.1 }[/math] | |
| [math]\displaystyle{ 7.10 }[/math] | [math]\displaystyle{ x^{36} + x^{33} + x^{18} + x^9 + x^5 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ10.1 }[/math] | |
| [math]\displaystyle{ 8 }[/math] | [math]\displaystyle{ 8.1 }[/math] | [math]\displaystyle{ x^{36} + x^{33} + x^9 + x^6 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ Table 9: N^\circ1.4 }[/math] |
| [math]\displaystyle{ 8.2 }[/math] | [math]\displaystyle{ x^{72} + x^{66} + x^{12} + x^6 + x^3 }[/math] | [math]\displaystyle{ N^\circ5 }[/math] | [math]\displaystyle{ 9:N^\circ1.3 }[/math] | |
| [math]\displaystyle{ 8.3 }[/math] | [math]\displaystyle{ x^{130} + x^{66} + x^{40} + x^{12} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 9:N^\circ6.1 }[/math] | |
| [math]\displaystyle{ 8.4 }[/math] | [math]\displaystyle{ x^{66} + x^{40} + x^{18} + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 9:N^\circ5.1 }[/math] | |
| [math]\displaystyle{ 9 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 10 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 11 }[/math] | [math]\displaystyle{ 11.1 }[/math] | [math]\displaystyle{ x^{12} + x^{10} + x^9 + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 11.2 }[/math] | [math]\displaystyle{ x^{258} + x^{257} + x^{18} + x^{17} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | |
| [math]\displaystyle{ 11.3 }[/math] | [math]\displaystyle{ x^{96} + x^{66} + x^{34} + x^{33} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | |
| [math]\displaystyle{ 11.4 }[/math] | [math]\displaystyle{ x^{80} + x^{68} + x^{65} + x^{17} + x^5 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | |
| [math]\displaystyle{ 11.5 }[/math] | [math]\displaystyle{ x^{260} + x^{257} + x^{36} + x^{33} + x^5 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
Table 4: Classification of Quadratic APN Hexanomial (CCZ-inequivalent with infinite monomial
families) in Small Dimensions with Coefficients as 1
| [math]\displaystyle{ n }[/math] | [math]\displaystyle{ N^\circ }[/math] | Functions | Families from tables 5 | Relation to [6] |
|---|---|---|---|---|
| [math]\displaystyle{ 6 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 7 }[/math] | [math]\displaystyle{ 7.1 }[/math] | [math]\displaystyle{ x^{34} + x^{33} + x^{12} + x^6 + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ Table 7: N^\circ14.2 }[/math] |
| [math]\displaystyle{ 7.2 }[/math] | [math]\displaystyle{ x^{40} + x^{24} + x^{20} + x^9 + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ14.1 }[/math] | |
| [math]\displaystyle{ 7.3 }[/math] | [math]\displaystyle{ x^{33} + x^{24} + x^{20} + x^{18} + x^{12} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ12.1 }[/math] | |
| [math]\displaystyle{ 7.4 }[/math] | [math]\displaystyle{ x^{24} + x^{17} + x^{12} + x^{10} + x^6 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ2.1 }[/math] | |
| [math]\displaystyle{ 7.5 }[/math] | [math]\displaystyle{ x^{40} + x^{34} + x^{18} + x^{17} + x^5 + x^3 }[/math] | [math]\displaystyle{ N^\circ5 }[/math] | [math]\displaystyle{ 7:N^\circ1.2 }[/math] | |
| [math]\displaystyle{ 7.6 }[/math] | [math]\displaystyle{ x^{48} + x^{40} + x^{18} + x^{10} + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ11.1 }[/math] | |
| [math]\displaystyle{ 7.7 }[/math] | [math]\displaystyle{ x^{40} + x^{12} + x^{10} + x^9 + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ2.2 }[/math] | |
| [math]\displaystyle{ 7.8 }[/math] | [math]\displaystyle{ x^{34} + x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ9.1 }[/math] | |
| [math]\displaystyle{ 7.9 }[/math] | [math]\displaystyle{ x^{34} + x^{33} + x^{20} + x^{17} + x^{10} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ13.1 }[/math] | |
| [math]\displaystyle{ 7.10 }[/math] | [math]\displaystyle{ x^{36} + x^{33} + x^{24} + x^9 + x^6 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ10.1 }[/math] | |
| [math]\displaystyle{ 7.11 }[/math] | [math]\displaystyle{ x^{40} + x^{36} + x^{20} + x^{10} + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ10.2 }[/math] | |
| [math]\displaystyle{ 7.12 }[/math] | [math]\displaystyle{ x^{36} + x^{34} + x^{20} + x^{10} + x^9 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 7:N^\circ8.1 }[/math] | |
| [math]\displaystyle{ 8 }[/math] | [math]\displaystyle{ 8.1 }[/math] | [math]\displaystyle{ x^{68} + x^{34} + x^{17} + x^{12} + x^9 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ Table 9: N^\circ5.1 }[/math] |
| [math]\displaystyle{ 8.2 }[/math] | [math]\displaystyle{ x^{72} + x^{40} + x^{34} + x^{20} + x^{12} + x^3 }[/math] | [math]\displaystyle{ N^\circ5 }[/math] | [math]\displaystyle{ 9:N^\circ6.1 }[/math] | |
| [math]\displaystyle{ 8.3 }[/math] | [math]\displaystyle{ x^{72} + x^{66} + x^{34} + x^{18} + x^{10} + x^5 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ 9:N^\circ4.1 }[/math] | |
| [math]\displaystyle{ 9 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 10 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
| [math]\displaystyle{ 11 }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] | [math]\displaystyle{ - }[/math] |
Table 5: Known classes of quadratic APN polynomials CCZ-inequivalent to APN monomials on [math]\displaystyle{ \mathbb{F}_{2^n} }[/math] [math]\displaystyle{ u }[/math] is primitive in [math]\displaystyle{ \mathbb{F}_{2^n}^* }[/math]}
- ↑ Budaghyan L, Carlet C, Leander G. Two classes of quadratic APN binomials inequivalent to power functions. IEEE Transactions on Information Theory. 2008 Sep;54(9):4218-29.
- ↑ Budaghyan L, Carlet C. Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Transactions on Information Theory. 2008 May;54(5):2354-7.
- ↑ Budaghyan L, Carlet C. Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Transactions on Information Theory. 2008 May;54(5):2354-7.
- ↑ Budaghyan L, Carlet C, Leander G. Constructing new APN functions from known ones. Finite Fields and Their Applications. 2009 Apr 1;15(2):150-9.
- ↑ 5.0 5.1 5.2 Budaghyan L, Carlet C, Leander G. On a construction of quadratic APN functions. In2009 IEEE Information Theory Workshop 2009 Oct 11 (pp. 374-378). IEEE.
- ↑ Bracken C, Byrne E, Markin N, Mcguire G. New families of quadratic almost perfect nonlinear trinomials and multinomials. Finite Fields and Their Applications. 2008 Jul 1;14(3):703-14.
- ↑ Zhou Y, Pott A. A new family of semifields with 2 parameters. Advances in Mathematics. 2013 Feb 15;234:43-60.
| [math]\displaystyle{ N^\circ }[/math] | Fanctions | Conditions | Reference |
|---|---|---|---|
| [math]\displaystyle{ 1-2 }[/math] | [math]\displaystyle{ x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}} }[/math] | [math]\displaystyle{ n=pk, p\in \{3,4\}, \gcd(k,3)=\gcd(s,3k)=1, i=sk \bmod p, m=p-i, n\geqslant12 }[/math] | [1] |
| [math]\displaystyle{ 3 }[/math] | [math]\displaystyle{ x^{2^{2i}+2^i}+bx^{q+1}+cx^{q(2^{2i}+2^i)} }[/math] | [math]\displaystyle{ q=2^{m}, n=2m, cb^{q}+b\neq 0, \gcd(i,m)=1, \gcd(2^{i}+1,q+1)\neq 1, c \not\in \{\lambda^{(2^{i}+1)(q-1)}, \lambda\in \mathbb{F}_{2^{n}}\}, c^{q+1}=1 }[/math] | [2] |
| [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{ q=2^{m},n=2m, \gcd(i,m)=1, c\in\mathbb{F}_{2^n}, s\in\mathbb{F}_{2^n}\backslash \mathbb{F}_q, X^{2^i+1}+cX^{2^i}+c^qX+1 }[/math] is irreducible over [math]\displaystyle{ \mathbb{F}_{2^n} }[/math] | [3] |
| [math]\displaystyle{ 5 }[/math] | [math]\displaystyle{ x^3+a^{-1}tr_n(a^3x^9) }[/math] | [math]\displaystyle{ a\ne0 }[/math] | [4][5] |
| [math]\displaystyle{ 6 }[/math] | [math]\displaystyle{ x^3+a^{-1}tr_n^3(a^3x^9+a^6x^{18}) }[/math] | [math]\displaystyle{ 3|n, a\ne0 }[/math] | [5] |
| [math]\displaystyle{ 7 }[/math] | [math]\displaystyle{ x^3+a^{-1}tr_n^3(a^6x^{18}+a^{12}x^{36}) }[/math] | [math]\displaystyle{ 3|n, a\ne0 }[/math] | [5] |
| [math]\displaystyle{ 8-10 }[/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{ n=3k, 3|(k+s), \gcd(k,3)=\gcd(s,3k)=1,v, w\in\mathbb{F}_{2^k}, vw \ne 1 }[/math] | [6] |
| [math]\displaystyle{ 11 }[/math] | [math]\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}) }[/math] | [math]\displaystyle{ m\geqslant 2, 2|m,n=2m,\gcd(k, m)=1, i }[/math] is even | [7] |