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

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<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\bmod 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\}</math>, <math>i = sk\bmod p, m = p -i, n \ge 12, u \text{ primitive in } \mathbb{F}_{2^n}^*</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>
 
<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>
 
</td>
 
</td>
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<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,</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><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}</math>, <math>X^{2^i+1}+cX^{2^i}+c^{q}X+1  \text{ has no solution } x \text{ s.t. }x^{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>
 
<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>
 
</td>
 
</td>
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<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}x^{2^{s}+2^{k+s}}</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, \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}</math>, <math>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. A few more quadratic APN functions. Cryptography and Communications. 2011 Mar 1;3(1):43-53.</ref></td>
 
<td><ref>Bracken C, Byrne E, Markin N, Mcguire G. A few more quadratic APN functions. Cryptography and Communications. 2011 Mar 1;3(1):43-53.</ref></td>
 
</tr>
 
</tr>
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<td>C12</td>
 
<td>C12</td>
 
<td><math>ut(x)(x^q+x)+t(x)^{2^{2i}+2^{3i}}+at(x)^{2^{2i}}(x^q+x)^{2^i}+b(x^q+x)^{2^i+1}</math></td>
 
<td><math>ut(x)(x^q+x)+t(x)^{2^{2i}+2^{3i}}+at(x)^{2^{2i}}(x^q+x)^{2^i}+b(x^q+x)^{2^i+1}</math></td>
<td><math>n=2m, q=2^m, \gcd(m,i)=1, t(x)=u^qx+x^qu, X^{2^i+1}+aX+b \mbox{ has no solution over }\mathbb{F}_{2^m}</math></td>
+
<td><math>n=2m, q=2^m, \gcd(m,i)=1, t(x)=u^qx+x^qu</math>, <math> X^{2^i+1}+aX+b \mbox{ has no solution over }\mathbb{F}_{2^m}</math></td>
 
<td><ref>Taniguchi H. On some quadratic APN functions.  Des. Codes Cryptogr. 2019, https://doi.org/10.1007/s10623-018-00598-2</ref></td>
 
<td><ref>Taniguchi H. On some quadratic APN functions.  Des. Codes Cryptogr. 2019, https://doi.org/10.1007/s10623-018-00598-2</ref></td>
 
</tr>
 
</tr>
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<tr>
 
<tr>
<td><math>n = 2m, m\ odd, 3 \nmid m, (a,b,c) = (\beta, \beta^2, 1), \beta \text{ primitive in } \mathbb{F}_{2^2}, i \in \{ m-2, m, 2m-1, (m-2)^{-1} \mod n \}</math></td>
+
<td><math>n = 2m, m\ odd, 3 \nmid m, (a,b,c) = (\beta, \beta^2, 1), \beta \text{ primitive in } \mathbb{F}_{2^2}</math>, <math>i \in \{ m-2, m, 2m-1, (m-2)^{-1} \mod n \}</math></td>
 
</tr>
 
</tr>
  
  
 
</table>
 
</table>

Revision as of 15:50, 5 November 2019

Functions Conditions References
C1-C2 , [1]
C3 , [2]
C4 [3]
C5 , [4]
C6 [4]
C7-C9 , [5]
C10 even, and even, [6]
C11 satisfies the conditions in Theorem 3.6 of [7] [7]
C12 , [8]
C13 [9]
,
  1. 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.
  2. 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.
  3. 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.
  4. 4.0 4.1 Budaghyan L, Carlet C, Leander G. On a construction of quadratic APN functions. InInformation Theory Workshop, 2009. ITW 2009. IEEE 2009 Oct 11 (pp. 374-378). IEEE.
  5. Bracken C, Byrne E, Markin N, Mcguire G. A few more quadratic APN functions. Cryptography and Communications. 2011 Mar 1;3(1):43-53.
  6. Zhou Y, Pott A. A new family of semifields with 2 parameters. Advances in Mathematics. 2013 Feb 15;234:43-60.
  7. Villa I, Budaghyan L, Calderini M, Carlet C, Coulter R. Constructing APN functions through isotopic shift. Cryptology ePrint Archive, Report 2018/769
  8. Taniguchi H. On some quadratic APN functions. Des. Codes Cryptogr. 2019, https://doi.org/10.1007/s10623-018-00598-2
  9. Budaghyan L, Helleseth T, Kaleyski N. A new family of APN quadrinomials. Cryptology ePrint Archive, Report 2019/994