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. 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><ref>L. Budaghyan, C. Carlet, G. Leander. Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inform. Theory, 54(9), pp. 4218-4229, 2008. https://doi.org/10.1109/TIT.2008.928275</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,</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. 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><ref>L. Budaghyan and C. Carlet. Classes of Quadratic APN Trinomials and Hexanomials and Related Structures. IEEE Trans. Inform. Theory, vol. 54, no. 5, pp. 2354-2357, May 2008. https://doi.org/10.1109/TIT.2008.920246</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>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><ref>L. Budaghyan, C. Carlet, G. Leander. Constructing new APN functions from known ones. Finite Fields and Their Applications, v. 15, issue 2, pp. 150-159, April 2009. https://doi.org/10.1016/j.ffa.2008.10.001</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">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>
+
<td><ref name="2_ref">L. Budaghyan, C. Carlet, G. Leander. On a construction of quadratic APN functions. Proceedings of IEEE Information Theory Workshop, ITW’09, pp. 374-378, Taormina, Sicily, Oct. 2009. https://doi.org/10.1109/ITW.2009.5351383</ref></td>
 
</tr>
 
</tr>
  
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<td><math>x^3+a^{-1} \mathrm{Tr}_n^3(a^6x^{18}+a^{12}x^{36})</math></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> <math>3|n, a \ne 0</math></td>
<td><ref name="2_ref" />
+
<td><ref name="2_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>
 
<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. (2011). ''A few more quadratic APN functions. Cryptography and Communications'', 3(1), 43-53.</ref></td>
+
<td><ref>C. Bracken, E. Byrne, N. Markin, G. McGuire. A few more quadratic APN functions. Cryptography and Communications 3, pp. 45-53, 2008. https://doi.org/10.1007/s12095-010-0038-7</ref></td>
 
</tr>
 
</tr>
  
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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>
+
<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>
+
<td><ref>Y. Zhou, A. Pott. A new family of semifields with 2 parameters. Advances in Mathematics, v. 234, pp. 43-60, 2013. https://doi.org/10.1016/j.aim.2012.10.014</ref></td>
 
</tr>
 
</tr>
  
 
<tr>
 
<tr>
 
<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>L(x)^{2^i}x+L(x)x^{2^i}</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>
+
<td><math>n=km, m>1, \gcd(n,i)=1, L(x)=\sum_{j=0}^{k-1}a_jx^{2^{jm}}</math> satisfies the conditions in Theorem 6.3 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>L. Budaghyan, M. Calderini, C. Carlet, R. S. Coulter, I. Villa. Constructing APN Functions through Isotopic Shifts. IEEE Trans. Inform. Theory, early access article. https://doi.org/10.1109/TIT.2020.2974471</ref></td>
 +
</tr>
 +
 
 +
<tr>
 +
<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>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>H. Taniguchi. On some quadratic APN functions. Designs, Codes and Cryptography 87, pp. 1973-1983, 2019. https://doi.org/10.1007/s10623-018-00598-2</ref></td>
 +
</tr>
 +
 
 +
<tr>
 +
<td rowspan="2">C13</td>
 +
<td rowspan="2"><math>x^3 + a (x^{2^i + 1})^{2^k} + b x^{3 \cdot 2^m} + c (x^{2^{i+m} + 2^m})^{2^k}</math></td>
 +
<td><math>n = 2m = 10, (a,b,c) = (\beta,0,0), i = 3, k = 2, \beta \text{ primitive in } \mathbb{F}_{2^2}</math></td>
 +
<td rowspan="2"><ref>L. Budaghyan, T. Helleseth, N. Kaleyski. A new family of APN quadrinomials. IEEE Trans. Inf. Theory, early access article. https://doi.org/10.1109/TIT.2020.3007513</ref></td>
 +
</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}</math>, <math>i \in \{ m-2, m, 2m-1, (m-2)^{-1} \mod n \}</math></td>
 
</tr>
 
</tr>
  
  
 
</table>
 
</table>

Latest revision as of 13:34, 24 August 2020

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 6.3 of [7] [7]
C12 , [8]
C13 [9]
,
  1. L. Budaghyan, C. Carlet, G. Leander. Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inform. Theory, 54(9), pp. 4218-4229, 2008. https://doi.org/10.1109/TIT.2008.928275
  2. L. Budaghyan and C. Carlet. Classes of Quadratic APN Trinomials and Hexanomials and Related Structures. IEEE Trans. Inform. Theory, vol. 54, no. 5, pp. 2354-2357, May 2008. https://doi.org/10.1109/TIT.2008.920246
  3. L. Budaghyan, C. Carlet, G. Leander. Constructing new APN functions from known ones. Finite Fields and Their Applications, v. 15, issue 2, pp. 150-159, April 2009. https://doi.org/10.1016/j.ffa.2008.10.001
  4. 4.0 4.1 L. Budaghyan, C. Carlet, G. Leander. On a construction of quadratic APN functions. Proceedings of IEEE Information Theory Workshop, ITW’09, pp. 374-378, Taormina, Sicily, Oct. 2009. https://doi.org/10.1109/ITW.2009.5351383
  5. C. Bracken, E. Byrne, N. Markin, G. McGuire. A few more quadratic APN functions. Cryptography and Communications 3, pp. 45-53, 2008. https://doi.org/10.1007/s12095-010-0038-7
  6. Y. Zhou, A. Pott. A new family of semifields with 2 parameters. Advances in Mathematics, v. 234, pp. 43-60, 2013. https://doi.org/10.1016/j.aim.2012.10.014
  7. L. Budaghyan, M. Calderini, C. Carlet, R. S. Coulter, I. Villa. Constructing APN Functions through Isotopic Shifts. IEEE Trans. Inform. Theory, early access article. https://doi.org/10.1109/TIT.2020.2974471
  8. H. Taniguchi. On some quadratic APN functions. Designs, Codes and Cryptography 87, pp. 1973-1983, 2019. https://doi.org/10.1007/s10623-018-00598-2
  9. L. Budaghyan, T. Helleseth, N. Kaleyski. A new family of APN quadrinomials. IEEE Trans. Inf. Theory, early access article. https://doi.org/10.1109/TIT.2020.3007513