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 < | 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>< | <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>< | <td><span class="htmlMath">2<sup>i</sup> + 1</span></td> | ||
<td>< | <td><span class="htmlMath"><span class="latexCommand">gcd</span>(i,n) = 1</span></td> | ||
<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=" | <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>< | <td><span class="htmlMath">2<sup>2i</sup> - 2<sup>i</sup> + 1</span></td> | ||
<td>< | <td><span class="htmlMath"><span class="latexCommand">gcd</span>(i,n) = 1</span></td> | ||
<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>< | <td><span class="htmlMath">2<sup>t</sup> + 3</span></td> | ||
<td>< | <td><span class="htmlMath">n = 2t + 1</span></td> | ||
<td>< | <td><span class="htmlMath">3</span></td> | ||
<td><ref>Dobbertin H. Almost perfect nonlinear power functions on < | <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>< | <td><span class="htmlMath">2<sup>t</sup> + 2<sup>t/2</sup> - 1, t</span> even</td> | ||
<td rowspan="2">< | <td rowspan="2"><span class="htmlMath">n = 2t + 1</span></td> | ||
<td>< | <td><span class="htmlMath">(t+2)/2</span></td> | ||
<td rowspan="2"><ref>Dobbertin H. Almost perfect nonlinear power functions on < | <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>< | <td><span class="htmlMath">2<sup>t</sup> + 2<sup>(3t+1)/2</sup> - 1, t</span> odd</td> | ||
<td>< | <td><span class="htmlMath">t + 1</span></td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>Inverse</td> | <td>Inverse</td> | ||
<td>< | <td><span class="htmlMath">2<sup>2t</sup> - 1</span></td> | ||
<td>< | <td><span class="htmlMath">n = 2t + 1</span></td> | ||
<td>< | <td><span class="htmlMath">n-1</span></td> | ||
<td><ref name=" | <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>< | <td><span class="htmlMath">2<sup>4i</sup> + 2<sup>3i</sup> + 2<sup>2i</sup> + 2<sup>i</sup> - 1</span></td> | ||
<td>< | <td><span class="htmlMath">n = 5i</span></td> | ||
<td>< | <td><span class="htmlMath">i + 3</span></td> | ||
<td><ref>Dobbertin H. Almost perfect nonlinear power functions on < | <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] |
- ↑ Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory. 1968;14(1):154-6.
- ↑ 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).
- ↑ 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).
- ↑ 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.
- ↑ Dobbertin H. Almost perfect nonlinear power functions on GF(2n): the Welch case. IEEE Transactions on Information Theory. 1999 May;45(4):1271-5.
- ↑ Dobbertin H. Almost perfect nonlinear power functions on GF(2n): the Niho case. Information and Computation. 1999 May 25;151(1-2):57-72.
- ↑ 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).
- ↑ 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).