Known infinite families of APN power functions over GF(2^n): Difference between revisions

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The following table provides a summary of all known infinite families of power APN functions of the form <math>F(x) = x^d</math>.
The following table provides a summary of all known infinite families of power APN functions of the form <span class="htmlMath">F(x) = x<sup>d</sup></span>.


<table>
<table>
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<th>Exponent</th>
<th>Exponent</th>
<th>Conditions</th>
<th>Conditions</th>
<th><math>\deg(x^d)</math></th>
<th><span class="htmlMath"><span class="latexCommand">deg</span>(x<sup>d</sup>)</span></th>
<th>Reference</th>
<th>Reference</th>
</tr>
</tr>
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<tr>
<tr>
<td>Gold</td>
<td>Gold</td>
<td><math>2^i + 1</math></td>
<td><span class="htmlMath">2<sup>i</sup> + 1</span></td>
<td><math>\gcd(i,n) = 1</math></td>
<td><span class="htmlMath"><span class="latexCommand">gcd</span>(i,n) = 1</span></td>
<td><math>2</math></td>
<td><span class="htmlMath">2</span></td>
<td> <ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory. 1968;14(1):154-6.</ref><ref name="kaisa_ref">Nyberg K. Differentially uniform mappings for cryptography. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 55-64).</ref>
<td> <ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory. 1968;14(1):154-6.</ref><ref name="kaisa">Nyberg K. Differentially uniform mappings for cryptography. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 55-64).</ref>
  </td>
  </td>
</tr>
</tr>
<tr>
<tr>


<tr>
<tr>
<td>Kasami</td>
<td>Kasami</td>
<td><math>2^{2i} - 2^i + 1</math></td>
<td><span class="htmlMath">2<sup>2i</sup> - 2<sup>i</sup> + 1</span></td>
<td><math>\gcd(i,n) = 1</math></td>
<td><span class="htmlMath"><span class="latexCommand">gcd</span>(i,n) = 1</span></td>
<td><math>i + 1</math></td>
<td><span class="htmlMath">i + 1</span></td>
<td><ref>Janwa H, Wilson RM. Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes. InInternational Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes 1993 May 10 (pp. 180-194).</ref><ref>Kasami T. The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes. Information and Control. 1971 May 1;18(4):369-94.</ref></td>
<td><ref>Janwa H, Wilson RM. Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes. InInternational Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes 1993 May 10 (pp. 180-194).</ref><ref>Kasami T. The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes. Information and Control. 1971 May 1;18(4):369-94.</ref></td>
</tr>
</tr>


<tr>
<tr>
<td>Welch</td>
<td>Welch</td>
<td><math>2^t + 3</math></td>
<td><span class="htmlMath">2<sup>t</sup> + 3</span></td>
<td><math>n = 2t + 1</math></td>
<td><span class="htmlMath">n = 2t + 1</span></td>
<td><math>3</math></td>
<td><span class="htmlMath">3</span></td>
<td><ref>Dobbertin H. Almost perfect nonlinear power functions on <math>GF(2^n)</math>: the Welch case. IEEE Transactions on Information Theory. 1999 May;45(4):1271-5.</ref></td>
<td><ref>Dobbertin H. Almost perfect nonlinear power functions on <span class="htmlMath">GF(2<sup>n</sup>)</span>: the Welch case. IEEE Transactions on Information Theory. 1999 May;45(4):1271-5.</ref></td>
</tr>
</tr>


<tr>
<tr>
<td rowspan="2">Niho</td>
<td rowspan="2">Niho</td>
<td><math>2^t + 2^{t/2} - 1, t</math> even</td>
<td><span class="htmlMath">2<sup>t</sup> + 2<sup>t/2</sup> - 1, t</span> even</td>
<td rowspan="2"><math>n = 2t + 1</math></td>
<td rowspan="2"><span class="htmlMath">n = 2t + 1</span></td>
<td><math>(t+2)/2</math></td>
<td><span class="htmlMath">(t+2)/2</span></td>
<td rowspan="2"><ref>Dobbertin H. Almost perfect nonlinear power functions on <math>GF(2^n)</math>: the Niho case. Information and Computation. 1999 May 25;151(1-2):57-72.</ref></td>
<td rowspan="2"><ref>Dobbertin H. Almost perfect nonlinear power functions on <span class="htmlMath">GF(2<sup>n</sup>)</span>: the Niho case. Information and Computation. 1999 May 25;151(1-2):57-72.</ref></td>
</tr>
</tr>


<tr>
<tr>
<td><math>2^t + 2^{(3t+1)/2} - 1, t</math> odd</td>
<td><span class="htmlMath">2<sup>t</sup> + 2<sup>(3t+1)/2</sup> - 1, t</span> odd</td>
<td><math>t + 1</math></td>
<td><span class="htmlMath">t + 1</span></td>
</tr>
</tr>


<tr>
<tr>
<td>Inverse</td>
<td>Inverse</td>
<td><math>2^{2t} - 1</math></td>
<td><span class="htmlMath">2<sup>2t</sup> - 1</span></td>
<td><math>n = 2t + 1</math></td>
<td><span class="htmlMath">n = 2t + 1</span></td>
<td><math>n-1</math></td>
<td><span class="htmlMath">n-1</span></td>
<td><ref name="kaisa_ref" /><ref>Beth T, Ding C. On almost perfect nonlinear permutations. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 65-76). </ref>
<td><ref name="kaisa" /><ref>Beth T, Ding C. On almost perfect nonlinear permutations. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 65-76). </ref>
</tr>
</tr>


<tr>
<tr>
<td>Dobbertin</td>
<td>Dobbertin</td>
<td><math>2^{4i} + 2^{3i} + 2^{2i} + 2^i - 1</math></td>
<td><span class="htmlMath">2<sup>4i</sup> + 2<sup>3i</sup> + 2<sup>2i</sup> + 2<sup>i</sup> - 1</span></td>
<td><math>n = 5i</math></td>
<td><span class="htmlMath">n = 5i</span></td>
<td><math>i + 3</math></td>
<td><span class="htmlMath">i + 3</span></td>
<td><ref>Dobbertin H. Almost perfect nonlinear power functions on <math>GF(2^n)</math>: a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121).</ref></td>
<td><ref>Dobbertin H. Almost perfect nonlinear power functions on <span class="htmlMath">GF(2<sup>n</sup>)</span>: a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121).</ref></td>
</tr>
</tr>


</table>
</table>

Revision as of 10:17, 12 September 2019

The following table provides a summary of all known infinite families of power APN functions of the form F(x) = xd.

Family Exponent Conditions deg(xd) Reference
Gold 2i + 1 gcd(i,n) = 1 2 [1][2]
Kasami 22i - 2i + 1 gcd(i,n) = 1 i + 1 [3][4]
Welch 2t + 3 n = 2t + 1 3 [5]
Niho 2t + 2t/2 - 1, t even n = 2t + 1 (t+2)/2 [6]
2t + 2(3t+1)/2 - 1, t odd t + 1
Inverse 22t - 1 n = 2t + 1 n-1 [2][7]
Dobbertin 24i + 23i + 22i + 2i - 1 n = 5i i + 3 [8]
  1. ↑ Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory. 1968;14(1):154-6.
  2. ↑ 2.0 2.1 Nyberg K. Differentially uniform mappings for cryptography. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 55-64).
  3. ↑ Janwa H, Wilson RM. Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes. InInternational Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes 1993 May 10 (pp. 180-194).
  4. ↑ Kasami T. The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes. Information and Control. 1971 May 1;18(4):369-94.
  5. ↑ Dobbertin H. Almost perfect nonlinear power functions on GF(2n): the Welch case. IEEE Transactions on Information Theory. 1999 May;45(4):1271-5.
  6. ↑ Dobbertin H. Almost perfect nonlinear power functions on GF(2n): the Niho case. Information and Computation. 1999 May 25;151(1-2):57-72.
  7. ↑ Beth T, Ding C. On almost perfect nonlinear permutations. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 65-76).
  8. ↑ Dobbertin H. Almost perfect nonlinear power functions on GF(2n): a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121).
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