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

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<td>C10</td>
 
<td>C10</td>
 
<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>(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>, <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>
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<td><math>n=2m, m\geqslant 2</math> even, <math>\gcd(k, m)=1</math> and <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, F., 2015. Almost perfect nonlinear trinomials and hexanomials. Finite Fields and Their Applications, 33, pp.258-282.</ref></td>
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<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>
 
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<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}, L(x)=ax^{2^{m}}+bx^{2^{m}}+cx</math> satisfies the conditions in lemma 8 of [7]</td>
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<td><math>n=3m, m \ \text{odd}, L(x)=ax^{2^{2m}}+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>
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<td><ref>Villa, I., Budaghyan, L., Calderini, M., Carlet, C., & Coulter, R. On Isotopic Construction of APN Functions. SETA 2018</ref></td>
 
</tr>
 
</tr>
  
  
 
</table>
 
</table>

Revision as of 11:38, 18 January 2019

Functions Conditions References
C1-C2 [1]
C3 , s.t. [2]
C4 [3]
C5 , [4]
C6 [4]
C7-C9 [5]
C10 even, and even, [6]
C11 satisfies the conditions in Lemma 8 of [7] [7]
  1. 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.
  2. 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.
  3. 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. 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.
  5. Bracken, C., Byrne, E., Markin, N., & Mcguire, G. (2011). A few more quadratic APN functions. Cryptography and Communications, 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. On Isotopic Construction of APN Functions. SETA 2018