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 |
||
(One intermediate revision by one other user not shown) | |||
Line 5: | Line 5: | ||
<table class="borderless"> | <table class="borderless"> | ||
<tr> | <tr> | ||
<th> | <th>Dimension</th> | ||
<th><math>N^\circ</math></th> | <th><math>N^\circ</math></th> | ||
<th>Functions</th> | <th>Functions</th> | ||
Line 81: | Line 81: | ||
<table class="borderless"> | <table class="borderless"> | ||
<tr> | <tr> | ||
<th> | <th>Dimension</th> | ||
<th><math>N^\circ</math></th> | <th><math>N^\circ</math></th> | ||
<th>Functions</th> | <th>Functions</th> | ||
Line 173: | Line 173: | ||
<table class="borderless"> | <table class="borderless"> | ||
<tr> | <tr> | ||
<th> | <th>Dimension</th> | ||
<th><math>N^\circ</math></th> | <th><math>N^\circ</math></th> | ||
<th>Functions</th> | <th>Functions</th> | ||
Line 293: | Line 293: | ||
<tr class="divider"> | <tr class="divider"> | ||
<td | <td><math>11</math></td> | ||
<td><math>-</math></td> | <td><math>-</math></td> | ||
<td><math>-</math></td> | <td><math>-</math></td> | ||
<td><math>-</math></td> | <td><math>-</math></td> | ||
<td><math>-</math></td> | <td><math>-</math></td> | ||
Line 331: | Line 307: | ||
<table class="borderless"> | <table class="borderless"> | ||
<tr> | <tr> | ||
<th> | <th>Dimension</th> | ||
<th><math>N^\circ</math></th> | <th><math>N^\circ</math></th> | ||
<th>Functions</th> | <th>Functions</th> |
Latest revision as of 13:57, 19 August 2019
The following tables list CCZ-inequivalent representatives found by systematically searching for APN functions among all trinomials, quadrinomials, pentanomials and hexanomials with coefficients in [math]\displaystyle{ \mathbb{F}_{2} }[/math] over [math]\displaystyle{ \mathbb{F}_{2^n} }[/math] with [math]\displaystyle{ 6 \le n \le 11 }[/math] [1]. The tables also list which equivalence class from [2] the functions belong to. Only polynomials inequivalent to power functions are considered. If the polynomial is equivalent to a family from the table of infinite families, this is also listed.
Trinomials
Dimension | [math]\displaystyle{ N^\circ }[/math] | Functions | Familiy | Relation to [2] |
---|---|---|---|---|
[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] | Table 7: № 8.1 |
[math]\displaystyle{ 7.2 }[/math] | [math]\displaystyle{ x^{34} + x^{18} + x^5 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 2.1 | |
[math]\displaystyle{ 8 }[/math] | [math]\displaystyle{ 8.1 }[/math] | [math]\displaystyle{ x^{72} + x^6 + x^3 }[/math] | [math]\displaystyle{ N^\circ5 }[/math] | Table 9: № 1.3 |
[math]\displaystyle{ 8.2 }[/math] | [math]\displaystyle{ x^{72} + x^{36} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 9: № 1.4 | |
[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] |
Quadrinomials
Dimension | [math]\displaystyle{ N^\circ }[/math] | Functions | Families | Relation to [2] |
---|---|---|---|---|
[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] | Table 7: № 12.1 |
[math]\displaystyle{ 7.2 }[/math] | [math]\displaystyle{ x^{33} + x^{17} + x^{12} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 2.2 | |
[math]\displaystyle{ 7.3 }[/math] | [math]\displaystyle{ x^{34} + x^{33} + x^{10} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 10.1 | |
[math]\displaystyle{ 7.4 }[/math] | [math]\displaystyle{ x^{66} + x^{34} + x^{20} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 11.1 | |
[math]\displaystyle{ 7.5 }[/math] | [math]\displaystyle{ x^{68} + x^{18} + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 8.1 | |
[math]\displaystyle{ 7.6 }[/math] | [math]\displaystyle{ x^{66} + x^{18} + x^9 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 9.1 | |
[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] |
Pentanomials
Dimension | [math]\displaystyle{ N^\circ }[/math] | Functions | Families | Relation to [2] |
---|---|---|---|---|
[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] | Table 7: № 13.1 |
[math]\displaystyle{ 7.2 }[/math] | [math]\displaystyle{ x^{65} + x^{20} + x^{18} + x^6 + x^3 }[/math] | [math]\displaystyle{ N^\circ5 }[/math] | Table 7: № 1.2 | |
[math]\displaystyle{ 7.3 }[/math] | [math]\displaystyle{ x^{40} + x^{34} + x^{18} + x^{10} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 12.1 | |
[math]\displaystyle{ 7.4 }[/math] | [math]\displaystyle{ x^{48} + x^{40} + x^{10} + x^9 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 1.2 | |
[math]\displaystyle{ 7.5 }[/math] | [math]\displaystyle{ x^{33} + x^9 + x^6 + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 11.1 | |
[math]\displaystyle{ 7.6 }[/math] | [math]\displaystyle{ x^{40} + x^{36} + x^{34} + x^{24} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 10.1 | |
[math]\displaystyle{ 7.7 }[/math] | [math]\displaystyle{ x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 2.1 | |
[math]\displaystyle{ 7.8 }[/math] | [math]\displaystyle{ x^{65} + x^{36} + x^{20} + x^{17} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 14.1 | |
[math]\displaystyle{ 7.9 }[/math] | [math]\displaystyle{ x^{40} + x^{33} + x^{17} + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 8.1 | |
[math]\displaystyle{ 7.10 }[/math] | [math]\displaystyle{ x^{36} + x^{33} + x^{18} + x^9 + x^5 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 10.1 | |
[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] | Table 9: № 1.4 |
[math]\displaystyle{ 8.2 }[/math] | [math]\displaystyle{ x^{72} + x^{66} + x^{12} + x^6 + x^3 }[/math] | [math]\displaystyle{ N^\circ5 }[/math] | Table 9: № 1.3 | |
[math]\displaystyle{ 8.3 }[/math] | [math]\displaystyle{ x^{130} + x^{66} + x^{40} + x^{12} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 9: № 6.1 | |
[math]\displaystyle{ 8.4 }[/math] | [math]\displaystyle{ x^{66} + x^{40} + x^{18} + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 9: № 5.1 | |
[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] |
Hexanomials
Dimension | [math]\displaystyle{ N^\circ }[/math] | Functions | Families | Relation to [2] |
---|---|---|---|---|
[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] | Table 7: № 14.2 |
[math]\displaystyle{ 7.2 }[/math] | [math]\displaystyle{ x^{40} + x^{24} + x^{20} + x^9 + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 14.1 | |
[math]\displaystyle{ 7.3 }[/math] | [math]\displaystyle{ x^{33} + x^{24} + x^{20} + x^{18} + x^{12} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 12.1 | |
[math]\displaystyle{ 7.4 }[/math] | [math]\displaystyle{ x^{24} + x^{17} + x^{12} + x^{10} + x^6 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 2.1 | |
[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] | Table 7: № 1.2 | |
[math]\displaystyle{ 7.6 }[/math] | [math]\displaystyle{ x^{48} + x^{40} + x^{18} + x^{10} + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 11.1 | |
[math]\displaystyle{ 7.7 }[/math] | [math]\displaystyle{ x^{40} + x^{12} + x^{10} + x^9 + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 2.2 | |
[math]\displaystyle{ 7.8 }[/math] | [math]\displaystyle{ x^{34} + x^{24} + x^{10} + x^9 + x^6 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 9.1 | |
[math]\displaystyle{ 7.9 }[/math] | [math]\displaystyle{ x^{34} + x^{33} + x^{20} + x^{17} + x^{10} + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 13.1 | |
[math]\displaystyle{ 7.10 }[/math] | [math]\displaystyle{ x^{36} + x^{33} + x^{24} + x^9 + x^6 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 10.1 | |
[math]\displaystyle{ 7.11 }[/math] | [math]\displaystyle{ x^{40} + x^{36} + x^{20} + x^{10} + x^5 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 10.2 | |
[math]\displaystyle{ 7.12 }[/math] | [math]\displaystyle{ x^{36} + x^{34} + x^{20} + x^{10} + x^9 + x^3 }[/math] | [math]\displaystyle{ - }[/math] | Table 7: № 8.1 | |
[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] | Table 9: № 5.1 |
[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] | Table 9: № 6.1 | |
[math]\displaystyle{ 8.3 }[/math] | [math]\displaystyle{ x^{72} + x^{66} + x^{34} + x^{18} + x^{10} + x^5 }[/math] | [math]\displaystyle{ - }[/math] | Table 9: № 4.1 | |
[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] |