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

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(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...")
 
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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 \mod p, m = p -i, n \ge 12, u \text{ primitive in } \mathbb{F}_{2^n}^*</math></td>
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<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. 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>
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<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>
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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,  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>
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<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. 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>
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<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>
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<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. 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>
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<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>
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<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. 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>
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<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>
  
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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>
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<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>
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<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>
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<tr>
 
<tr>
 
<td>C10</td>
 
<td>C10</td>
<td><math>(x+x^{2m})^{2^k+1}+u'(ux+u^{2m} x^{2m})^{(2^k+1)2^i}+u(x+x^{2m})(ux+u^{2m} x^{2m})</math></td>
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<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> 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>
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<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, Faruk. ''Almost perfect nonlinear trinomials and hexanomials.'' Finite Fields and Their Applications 33 (2015): 258-282.</ref></td>
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<td><ref>Göloğlu, F., 2015. Almost perfect nonlinear trinomials and hexanomials. Finite Fields and Their Applications, 33, pp.258-282.</ref></td>
 
</tr>
 
</tr>
  
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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^{2m}}+bx^{2m}+cx \ \text{satisfies the conditions in lemma 8 of}\ [3]</math></td>
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<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 14:29, 11 January 2019

Functions Conditions References
C1-C2 [1]
C3 , s.t. [2]
C4 [3]
C5 , [4]
C6 [4]
C7-C9 [5]
C10 even, , 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. Göloğlu, F., 2015. Almost perfect nonlinear trinomials and hexanomials. Finite Fields and Their Applications, 33, pp.258-282.
  7. Villa, I., Budaghyan, L., Calderini, M., Carlet, C., & Coulter, R. On Isotopic Construction of APN Functions. SETA 2018