Known inifinte families of quadratic APN polynomials over GF(2^n): Difference between revisions

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<th>References</th>
<th>References</th>
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<tr>
<tr>
<td>C1-C2</td>
<td>C1-C2</td>
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<td><math>n = pk, \gcd(k,3) = \gcd(s,3k) = 1, p \in \{3,4\}, i = sk \mod p, m = p -i, n \ge 12, u \text{ primitive in } \mathbb{F}_{2^n}^*</math></td>
<td><math>n = pk, \gcd(k,3) = \gcd(s,3k) = 1, p \in \{3,4\}, i = sk \mod p, m = p -i, n \ge 12, u \text{ primitive in } \mathbb{F}_{2^n}^*</math></td>
<td><ref>L. Budaghyan, C. Carlet, G. Leander, ''Two Classes of Quadratic APN Binomials Inequivalent to Power Functions'', IEEE Trans. Inform. Theory 54(9), 2008, pp. 4218-4229</ref>
<td><ref>L. Budaghyan, C. Carlet, G. Leander, ''Two Classes of Quadratic APN Binomials Inequivalent to Power Functions'', IEEE Trans. Inform. Theory 54(9), 2008, pp. 4218-4229</ref>
</td>
</tr>
</tr>
<tr>
<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>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</math> has no solution <math>x</math> s.t. <math>x^{q+1}=1</math><td>
<td><ref>L. Budaghyan and C. Carlet. Classes of Quadratic APN Trinomials and Hexanomials and Related Structures. {\em IEEE Trans. Inform. Theory}, vol. 54, no. 5, pp. 2354-2357, 2008.</ref>
</td>
</tr>
<tr>
<td>C4</td>
<td><math>x^3+a^{-1} \mathrm {Tr}_n (a^3x^9)</math></td>
<td><math>a\neq 0</math></td>
<td><ref>L. Budaghyan, C. Carlet and G.Leander, Constructinig new APN functions from known ones, Finite Fields and their applications, vol.15, issue 2, Apr. 2009, pp. 150-159.</ref>
</td>
</tr>
<tr>
<td>C5</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><ref name="2_ref">L. Budaghyan, C. Carlet and G.Leander, On a Construction of quadratic APN functions, Proceedings of IEEE information Theory workshop ITW'09, Oct. 2009, 374-378.</ref></td>
</tr>
<tr>
<td>C6</td>
<td><math>x^3+a^{-1} \mathrm{Tr}_n^3(a^6x^{18}+a^{12}x^{36})</math></td>
<td> <math>3|n, a \ne 0</math></td>
<td><ref name="2_ref" />
</tr>
<tr>
<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},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><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>
</table>
</table>

Revision as of 14:16, 14 December 2018

[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 \mod 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, 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 }[/math] has no solution [math]\displaystyle{ 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]
  1. L. Budaghyan, C. Carlet, G. Leander, Two Classes of Quadratic APN Binomials Inequivalent to Power Functions, IEEE Trans. Inform. Theory 54(9), 2008, pp. 4218-4229
  2. L. Budaghyan and C. Carlet. Classes of Quadratic APN Trinomials and Hexanomials and Related Structures. {\em IEEE Trans. Inform. Theory}, vol. 54, no. 5, pp. 2354-2357, 2008.
  3. L. Budaghyan, C. Carlet and G.Leander, Constructinig new APN functions from known ones, Finite Fields and their applications, vol.15, issue 2, Apr. 2009, pp. 150-159.
  4. 4.0 4.1 L. Budaghyan, C. Carlet and G.Leander, On a Construction of quadratic APN functions, Proceedings of IEEE information Theory workshop ITW'09, Oct. 2009, 374-378.
  5. Bracken, C., Byrne, E., Markin, N., & Mcguire, G. (2011). A few more quadratic APN functions. Cryptography and Communications, 3(1), 43-53.