Known infinite families of quadratic APN polynomials over GF(2^n): Difference between revisions
Jump to navigation
Jump to search
(Created page with "<table> <tr> <th><math>N^\circ</math></th> <th>Functions</th> <th>Conditions</th> <th>References</th> </tr> <tr> <td>C1-C2</td> <td><math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s...") |
No edit summary |
||
Line 10: | Line 10: | ||
<td>C1-C2</td> | <td>C1-C2</td> | ||
<td><math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}}</math></td> | <td><math>x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}}</math></td> | ||
<td><math>n = pk, \gcd(k,3) = \gcd(s,3k) = 1, p \in \{3,4\}, i = sk \ | <td><math>n = pk, \gcd(k,3) = \gcd(s,3k) = 1, p \in \{3,4\}, i = sk\bmod p, m = p -i, n \ge 12, u \text{ primitive in } \mathbb{F}_{2^n}^*</math></td> | ||
<td><ref>L. | <td><ref>Budaghyan, L., Carlet, C. and Leander, G., 2008. Two classes of quadratic APN binomials inequivalent to power functions. IEEE Transactions on Information Theory, 54(9), pp.4218-4229.</ref></td> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 18: | Line 18: | ||
<td>C3</th> | <td>C3</th> | ||
<td><math>sx^{q+1}+x^{2^i+1}+x^{q(2^i+1)}+cx^{2^iq+1}+c^qx^{2^i+q}</math></td> | <td><math>sx^{q+1}+x^{2^i+1}+x^{q(2^i+1)}+cx^{2^iq+1}+c^qx^{2^i+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} \setminus \mathbb{F}_{q}, X^{2^i+1}+cX^{2^i}+c^{q}X+1 \text{ has no solution } x</math> s.t. <math>x^{q+1}=1</math></td> | <td><math>q=2^m, n=2m,</math> <math>gcd(i,m)=1</math>, <math>c\in \mathbb{F}_{2^n}, s \in \mathbb F_{2^n} \setminus \mathbb{F}_{q}, X^{2^i+1}+cX^{2^i}+c^{q}X+1 \text{ has no solution } x</math> s.t. <math>x^{q+1}=1</math></td> | ||
<td><ref>L. | <td><ref>Budaghyan, L. and Carlet, C., 2008. Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Transactions on Information Theory, 54(5), pp.2354-2357.</ref></td> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 27: | Line 27: | ||
<td><math>x^3+a^{-1} \mathrm {Tr}_n (a^3x^9)</math></td> | <td><math>x^3+a^{-1} \mathrm {Tr}_n (a^3x^9)</math></td> | ||
<td><math>a\neq 0</math></td> | <td><math>a\neq 0</math></td> | ||
<td><ref>L. | <td><ref>Budaghyan, L., Carlet, C. and Leander, G., 2009. Constructing new APN functions from known ones. Finite Fields and Their Applications, 15(2), pp.150-159.</ref></td> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 35: | Line 35: | ||
<td><math>x^3+a^{-1} \mathrm {Tr}_n^3 (a^3x^9+a^6x^{18})</math></td> | <td><math>x^3+a^{-1} \mathrm {Tr}_n^3 (a^3x^9+a^6x^{18})</math></td> | ||
<td><math>3|n </math>, <math>a\ne0</math></td> | <td><math>3|n </math>, <math>a\ne0</math></td> | ||
<td><ref name="2_ref">L. | <td><ref name="2_ref">Budaghyan, L., Carlet, C. and Leander, G., 2009, October. On a construction of quadratic APN functions. In Information Theory Workshop, 2009. ITW 2009. IEEE (pp. 374-378). IEEE.</ref></td> | ||
</tr> | </tr> | ||
Line 47: | Line 47: | ||
<td>C7-C9</td> | <td>C7-C9</td> | ||
<td><math>ux^{2^s+1}+u^{2^k} x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^k+1} | <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, \gcd(k,3)=\gcd(s,3k)=1, v, w\in\mathbb{F}_{2^k}, vw \ne 1, 3|(k+s) u \text{ primitive in } \mathbb{F}_{2^n}^* </math></td> | <td><math>n=3k, \gcd(k,3)=\gcd(s,3k)=1, v, w\in\mathbb{F}_{2^k}, vw \ne 1, 3|(k+s), u \text{ primitive in } \mathbb{F}_{2^n}^* </math></td> | ||
<td><ref>Bracken, C., Byrne, E., Markin, N., & Mcguire, G. (2011). ''A few more quadratic APN functions. Cryptography and Communications'', 3(1), 43-53.</ref></td> | <td><ref>Bracken, C., Byrne, E., Markin, N., & Mcguire, G. (2011). ''A few more quadratic APN functions. Cryptography and Communications'', 3(1), 43-53.</ref></td> | ||
</tr> | </tr> | ||
Line 54: | Line 54: | ||
<tr> | <tr> | ||
<td>C10</td> | <td>C10</td> | ||
<td><math>(x+x^{ | <td><math>(x+x^{2{^m}})^{2^k+1}+u'(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>n=2m, m\geqslant 2</math> even, <math>\gcd(k, m)=1,</math> | <td><math>n=2m, m\geqslant 2</math> even, <math>\gcd(k, m)=1,</math>, <math> i \geqslant 2</math> even, <math>u\text{ primitive in } \mathbb{F}_{2^n}^*, u' \in \mathbb{F}_{2^m} \text{ not a cube }</math></td> | ||
<td><ref>Göloğlu, | <td><ref>Göloğlu, F., 2015. Almost perfect nonlinear trinomials and hexanomials. Finite Fields and Their Applications, 33, pp.258-282.</ref></td> | ||
</tr> | </tr> | ||
Line 62: | Line 62: | ||
<td>C11</td> | <td>C11</td> | ||
<td><math>a^2x^{2^{2m+1}+1}+b^2x^{2^{m+1}+1}+ax^{2^{2m}+2}+bx^{2^{m}+2}+(c^2+c)x^3</math></td> | <td><math>a^2x^{2^{2m+1}+1}+b^2x^{2^{m+1}+1}+ax^{2^{2m}+2}+bx^{2^{m}+2}+(c^2+c)x^3</math></td> | ||
<td><math>n=3m, m \ \text{odd} | <td><math>n=3m, m \ \text{odd}, L(x)=ax^{2^{m}}+bx^{2^{m}}+cx</math> satisfies the conditions in lemma 8 of [7]</td> | ||
<td><ref>Villa, I., Budaghyan, L., Calderini, M., Carlet, C., & Coulter, R. ''On Isotopic Construction of APN Functions.'' SETA 2018</ref></td> | <td><ref>Villa, I., Budaghyan, L., Calderini, M., Carlet, C., & Coulter, R. ''On Isotopic Construction of APN Functions.'' SETA 2018</ref></td> | ||
</tr> | </tr> |
Revision as of 13:29, 11 January 2019
[math]\displaystyle{ N^\circ }[/math] | Functions | Conditions | References |
---|---|---|---|
C1-C2 | [math]\displaystyle{ x^{2^s+1}+u^{2^k-1}x^{2^{ik}+2^{mk+s}} }[/math] | [math]\displaystyle{ n = pk, \gcd(k,3) = \gcd(s,3k) = 1, p \in \{3,4\}, i = sk\bmod p, m = p -i, n \ge 12, u \text{ primitive in } \mathbb{F}_{2^n}^* }[/math] | [1] |
C3 | [math]\displaystyle{ sx^{q+1}+x^{2^i+1}+x^{q(2^i+1)}+cx^{2^iq+1}+c^qx^{2^i+q} }[/math] | [math]\displaystyle{ q=2^m, n=2m, }[/math] [math]\displaystyle{ gcd(i,m)=1 }[/math], [math]\displaystyle{ c\in \mathbb{F}_{2^n}, s \in \mathbb F_{2^n} \setminus \mathbb{F}_{q}, X^{2^i+1}+cX^{2^i}+c^{q}X+1 \text{ has no solution } x }[/math] s.t. [math]\displaystyle{ x^{q+1}=1 }[/math] | [2] |
C4 | [math]\displaystyle{ x^3+a^{-1} \mathrm {Tr}_n (a^3x^9) }[/math] | [math]\displaystyle{ a\neq 0 }[/math] | [3] |
C5 | [math]\displaystyle{ x^3+a^{-1} \mathrm {Tr}_n^3 (a^3x^9+a^6x^{18}) }[/math] | [math]\displaystyle{ 3|n }[/math], [math]\displaystyle{ a\ne0 }[/math] | [4] |
C6 | [math]\displaystyle{ x^3+a^{-1} \mathrm{Tr}_n^3(a^6x^{18}+a^{12}x^{36}) }[/math] | [math]\displaystyle{ 3|n, a \ne 0 }[/math] | [4] |
C7-C9 | [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, \gcd(k,3)=\gcd(s,3k)=1, v, w\in\mathbb{F}_{2^k}, vw \ne 1, 3|(k+s), u \text{ primitive in } \mathbb{F}_{2^n}^* }[/math] | [5] |
C10 | [math]\displaystyle{ (x+x^{2{^m}})^{2^k+1}+u'(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{ n=2m, m\geqslant 2 }[/math] even, [math]\displaystyle{ \gcd(k, m)=1, }[/math], [math]\displaystyle{ i \geqslant 2 }[/math] even, [math]\displaystyle{ u\text{ primitive in } \mathbb{F}_{2^n}^*, u' \in \mathbb{F}_{2^m} \text{ not a cube } }[/math] | [6] |
C11 | [math]\displaystyle{ a^2x^{2^{2m+1}+1}+b^2x^{2^{m+1}+1}+ax^{2^{2m}+2}+bx^{2^{m}+2}+(c^2+c)x^3 }[/math] | [math]\displaystyle{ n=3m, m \ \text{odd}, L(x)=ax^{2^{m}}+bx^{2^{m}}+cx }[/math] satisfies the conditions in lemma 8 of [7] | [7] |
- ↑ Budaghyan, L., Carlet, C. and Leander, G., 2008. Two classes of quadratic APN binomials inequivalent to power functions. IEEE Transactions on Information Theory, 54(9), pp.4218-4229.
- ↑ Budaghyan, L. and Carlet, C., 2008. Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Transactions on Information Theory, 54(5), pp.2354-2357.
- ↑ Budaghyan, L., Carlet, C. and Leander, G., 2009. Constructing new APN functions from known ones. Finite Fields and Their Applications, 15(2), pp.150-159.
- ↑ 4.0 4.1 Budaghyan, L., Carlet, C. and Leander, G., 2009, October. On a construction of quadratic APN functions. In Information Theory Workshop, 2009. ITW 2009. IEEE (pp. 374-378). IEEE.
- ↑ Bracken, C., Byrne, E., Markin, N., & Mcguire, G. (2011). A few more quadratic APN functions. Cryptography and Communications, 3(1), 43-53.
- ↑ Göloğlu, F., 2015. Almost perfect nonlinear trinomials and hexanomials. Finite Fields and Their Applications, 33, pp.258-282.
- ↑ Villa, I., Budaghyan, L., Calderini, M., Carlet, C., & Coulter, R. On Isotopic Construction of APN Functions. SETA 2018