Difference between revisions of "Classification of Quadratic APN Trinomials, Quadrinomials, Pentanomials, Hexanomials (CCZ-inequivalent with infinite monomial families) in Small Dimensions with all Coefficients equal to 1"

Revision as of 13:19, 26 February 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 $\mathbb {F} _{2}$ over $\mathbb {F} _{2^{n}}$ with $6\leq n\leq 11$ ^[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

$n$	$N^{\circ }$	Functions	Familiy	Relation to ^[2]
$6$	$-$	$-$	$-$	$-$
$7$	$7.1$	$x^{20}+x^{6}+x^{3}$	$-$	$Table7:N^{\circ }8.1$
$7$	$7.2$	$x^{34}+x^{18}+x^{5}$	$-$	$7:N^{\circ }2.1$
$8$	$8.1$	$x^{72}+x^{6}+x^{3}$	$N^{\circ }5$	$Table9:N^{\circ }1.3$
$8$	$8.2$	$x^{72}+x^{36}+x^{3}$	$-$	$9:N^{\circ }1.4$
$9$	$-$	$-$	$-$	$-$
$10$	$-$	$-$	$-$	$-$
$11$	$-$	$-$	$-$	$-$

Quadrinomials

$n$	$N^{\circ }$	Functions	Families	Relation to ^[2]
$6$	$-$	$-$	$-$	$-$
$7$	$7.1$	$x^{72}+x^{40}+x^{12}+x^{3}$	$-$	$Table7:N^{\circ }12.1$
	$7.2$	$x^{33}+x^{17}+x^{12}+x^{3}$	$-$	$7:N^{\circ }2.2$
	$7.3$	$x^{34}+x^{33}+x^{10}+x^{3}$	$-$	$7:N^{\circ }10.1$
	$7.4$	$x^{66}+x^{34}+x^{20}+x^{3}$	$-$	$7:N^{\circ }11.1$
	$7.5$	$x^{68}+x^{18}+x^{5}+x^{3}$	$-$	$7:N^{\circ }8.1$
	$7.6$	$x^{66}+x^{18}+x^{9}+x^{3}$	$-$	$7:N^{\circ }9.1$
$8$	$-$	$-$	$-$	$-$
$9$	$-$	$-$	$-$	$-$
$10$	$-$	$-$	$-$	$-$
$11$	$-$	$-$	$-$	$-$

Pentanomials

$n$	$N^{\circ }$	Functions	Families	Relation to ^[2]
$6$	$-$	$-$	$-$	$-$
$7$	$7.1$	$x^{68}+x^{40}+x^{24}+x^{6}+x^{3}$	$-$	$Table7:N^{\circ }13.1$
	$7.2$	$x^{65}+x^{20}+x^{18}+x^{6}+x^{3}$	$N^{\circ }5$	$7:N^{\circ }1.2$
	$7.3$	$x^{40}+x^{34}+x^{18}+x^{10}+x^{3}$	$-$	$7:N^{\circ }12.1$
	$7.4$	$x^{48}+x^{40}+x^{10}+x^{9}+x^{3}$	$-$	$7:N^{\circ }2.1$
	$7.5$	$x^{33}+x^{9}+x^{6}+x^{5}+x^{3}$	$-$	$7:N^{\circ }11.1$
	$7.6$	$x^{40}+x^{36}+x^{34}+x^{24}+x^{3}$	$-$	$7:N^{\circ }10.1$
	$7.7$	$x^{24}+x^{10}+x^{9}+x^{6}+x^{3}$	$-$	$7:N^{\circ }2.1$
	$7.8$	$x^{65}+x^{36}+x^{20}+x^{17}+x^{3}$	$-$	$7:N^{\circ }14.1$
	$7.9$	$x^{40}+x^{33}+x^{17}+x^{5}+x^{3}$	$-$	$7:N^{\circ }8.1$
	$7.10$	$x^{36}+x^{33}+x^{18}+x^{9}+x^{5}$	$-$	$7:N^{\circ }10.1$
$8$	$8.1$	$x^{36}+x^{33}+x^{9}+x^{6}+x^{3}$	$-$	$Table9:N^{\circ }1.4$
	$8.2$	$x^{72}+x^{66}+x^{12}+x^{6}+x^{3}$	$N^{\circ }5$	$9:N^{\circ }1.3$
	$8.3$	$x^{130}+x^{66}+x^{40}+x^{12}+x^{3}$	$-$	$9:N^{\circ }6.1$
	$8.4$	$x^{66}+x^{40}+x^{18}+x^{5}+x^{3}$	$-$	$9:N^{\circ }5.1$
$9$	$-$	$-$	$-$	$-$
$10$	$-$	$-$	$-$	$-$
$11$	$11.1$	$x^{12}+x^{10}+x^{9}+x^{5}+x^{3}$	$-$	$-$
	$11.2$	$x^{258}+x^{257}+x^{18}+x^{17}+x^{3}$	$-$	$-$
	$11.3$	$x^{96}+x^{66}+x^{34}+x^{33}+x^{3}$	$-$	$-$
	$11.4$	$x^{80}+x^{68}+x^{65}+x^{17}+x^{5}$	$-$	$-$
	$11.5$	$x^{260}+x^{257}+x^{36}+x^{33}+x^{5}$	$-$	$-$

Hexanomials

$n$	$N^{\circ }$	Functions	Families	Relation to ^[2]
$6$	$-$	$-$	$-$	$-$
$7$	$7.1$	$x^{34}+x^{33}+x^{12}+x^{6}+x^{5}+x^{3}$	$-$	$Table7:N^{\circ }14.2$
	$7.2$	$x^{40}+x^{24}+x^{20}+x^{9}+x^{5}+x^{3}$	$-$	$7:N^{\circ }14.1$
	$7.3$	$x^{33}+x^{24}+x^{20}+x^{18}+x^{12}+x^{3}$	$-$	$7:N^{\circ }12.1$
	$7.4$	$x^{24}+x^{17}+x^{12}+x^{10}+x^{6}+x^{3}$	$-$	$7:N^{\circ }2.1$
	$7.5$	$x^{40}+x^{34}+x^{18}+x^{17}+x^{5}+x^{3}$	$N^{\circ }5$	$7:N^{\circ }1.2$
	$7.6$	$x^{48}+x^{40}+x^{18}+x^{10}+x^{5}+x^{3}$	$-$	$7:N^{\circ }11.1$
	$7.7$	$x^{40}+x^{12}+x^{10}+x^{9}+x^{5}+x^{3}$	$-$	$7:N^{\circ }2.2$
	$7.8$	$x^{34}+x^{24}+x^{10}+x^{9}+x^{6}+x^{3}$	$-$	$7:N^{\circ }9.1$
	$7.9$	$x^{34}+x^{33}+x^{20}+x^{17}+x^{10}+x^{3}$	$-$	$7:N^{\circ }13.1$
	$7.10$	$x^{36}+x^{33}+x^{24}+x^{9}+x^{6}+x^{3}$	$-$	$7:N^{\circ }10.1$
	$7.11$	$x^{40}+x^{36}+x^{20}+x^{10}+x^{5}+x^{3}$	$-$	$7:N^{\circ }10.2$
	$7.12$	$x^{36}+x^{34}+x^{20}+x^{10}+x^{9}+x^{3}$	$-$	$7:N^{\circ }8.1$
$8$	$8.1$	$x^{68}+x^{34}+x^{17}+x^{12}+x^{9}+x^{3}$	$-$	$Table9:N^{\circ }5.1$
	$8.2$	$x^{72}+x^{40}+x^{34}+x^{20}+x^{12}+x^{3}$	$N^{\circ }5$	$9:N^{\circ }6.1$
	$8.3$	$x^{72}+x^{66}+x^{34}+x^{18}+x^{10}+x^{5}$	$-$	$9:N^{\circ }4.1$
$9$	$-$	$-$	$-$	$-$
$10$	$-$	$-$	$-$	$-$
$11$	$-$	$-$	$-$	$-$

↑ Sun B. On Classification and Some Properties of APN Functions.
↑ ^2.0 ^2.1 ^2.2 ^2.3 ^2.4 Edel Y, Pott A. A new almost perfect nonlinear function which is not quadratic. Adv. in Math. of Comm.. 2009 Mar;3(1):59-81.

[1] Sun B. On Classification and Some Properties of APN Functions.

[edelPott-2] 2.0 ^2.1 ^2.2 ^2.3 ^2.4 Edel Y, Pott A. A new almost perfect nonlinear function which is not quadratic. Adv. in Math. of Comm.. 2009 Mar;3(1):59-81.

[1]

[2]

@@ Line 1: / Line 1: @@
-Table 1: Classification of Quadratic APN Trinomials (CCZ-inequivalent to infinite monomial
+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>\mathbb{F}_{2}</math> over <math>\mathbb{F}_{2^n}</math> with <math>6 \le n \le 11</math> <ref>Sun B. On Classification and Some Properties of APN Functions.</ref>. The tables also list which equivalence class from <ref name="edelPott">Edel Y, Pott A. A new almost perfect nonlinear function which is not quadratic. Adv. in Math. of Comm.. 2009 Mar;3(1):59-81.</ref> the functions belong to. Only polynomials inequivalent to power functions are considered. If the polynomial is equivalent to a family from the [[Known infinite families of quadratic APN polynomials over GF(2^n)|table of infinite families]], this is also listed.
-families) in Small Dimensions with Coefficients in <math>\mathbb{F}_2</math>
+== Trinomials ==
 <table class="borderless">
@@ Line 7: / Line 8: @@
 <th><math>N^\circ</math></th>
 <th>Functions</th>
-<th>Families from tables 5</th>
+<th>Familiy</th>
-<th>Relation to [6]</th>
+<th>Relation to <ref name="edelPott"></ref></th>
 </tr>
@@ Line 77: / Line 78: @@
-Table 2: Classification of Quadratic APN Quadrinomials (CCZ-inequivalent with infinite monomial
+== Quadrinomials ==
-families) in Small Dimensions with Coefficients as 1
 <table class="borderless">
@@ Line 85: / Line 85: @@
 <th><math>N^\circ</math></th>
 <th>Functions</th>
-<th>Families from tables 5</th>
+<th>Families</th>
-<th>Relation to [6]</th>
+<th>Relation to <ref name="edelPott"></ref></th>
 </tr>
@@ Line 170: / Line 170: @@
-Table 3: Classification of Quadratic APN Quadrinomials (CCZ-inequivalent with infinite monomial
+== Pentanomials ==
-families) in Small Dimensions with Coefficients as 1
@@ Line 179: / Line 178: @@
 <th><math>N^\circ</math></th>
 <th>Functions</th>
-<th>Families from tables 5</th>
+<th>Families</th>
-<th>Relation to [6]</th>
+<th>Relation to <ref name="edelPott"></ref></th>
 </tr>
@@ Line 330: / Line 329: @@
-Table 4: Classification of Quadratic APN Hexanomial (CCZ-inequivalent with infinite monomial
+== Hexanomials ==
-families) in Small Dimensions with Coefficients as 1
 <table>
@@ Line 339: / Line 337: @@
 <th><math>N^\circ</math></th>
 <th>Functions</th>
-<th>Families from tables 5</th>
+<th>Families</th>
-<th>Relation to [6]</th>
+<th>Relation to <ref name="edelPott"></ref></th>
 </tr>
@@ Line 470: / Line 468: @@
 </table>
-Table 5: Known classes of quadratic APN polynomials CCZ-inequivalent to APN monomials on <math>\mathbb{F}_{2^n}</math> <math>u</math> is primitive in <math>\mathbb{F}_{2^n}^*</math>}
-<table>
-<tr>
-<th><math>N^\circ</math></th>
-<th>Fanctions</th>
-<th>Conditions</th>
-<th>Reference</th>
-</tr>
-<tr>
-<td><math>1-2</math></td>
-<td><math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}}</math></td>
-<td><math>n=pk,  p\in \{3,4\},  \gcd(k,3)=\gcd(s,3k)=1,  i=sk  \bmod p,  m=p-i,  n\geqslant12</math></td>
-<td><ref>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.</ref></td>
-</tr>
-<tr>
-<td><math>3</math></td>
-<td><math>x^{2^{2i}+2^i}+bx^{q+1}+cx^{q(2^{2i}+2^i)}</math></td>
-<td><math>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></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>4</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><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>

Difference between revisions of "Classification of Quadratic APN Trinomials, Quadrinomials, Pentanomials, Hexanomials (CCZ-inequivalent with infinite monomial families) in Small Dimensions with all Coefficients equal to 1"

Revision as of 13:19, 26 February 2019

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