Known infinite families of APN power functions over GF(2^n): Difference between revisions
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<td><span class="htmlMath"><span class="latexCommand">gcd</span>(i,n) = 1</span></td> | <td><span class="htmlMath"><span class="latexCommand">gcd</span>(i,n) = 1</span></td> | ||
<td><span class="htmlMath">2</span></td> | <td><span class="htmlMath">2</span></td> | ||
<td> <ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. | <td> <ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inform. Theory, 14, pp. 154-156, 1968. https://doi.org/10.1109/TIT.1968.1054106</ref><ref name="kaisa">Nyberg K. Differentially uniform mappings for cryptography. Advances in Cryptography, EUROCRYPT’93, Lecture Notes in Computer Science 765, pp. 55-64, 1994. Lecture Notes in Computer Science, vol 765. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48285-7_6</ref> | ||
</td> | </td> | ||
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<td><span class="htmlMath"><span class="latexCommand">gcd</span>(i,n) = 1</span></td> | <td><span class="htmlMath"><span class="latexCommand">gcd</span>(i,n) = 1</span></td> | ||
<td><span class="htmlMath">i + 1</span></td> | <td><span class="htmlMath">i + 1</span></td> | ||
<td><ref>Janwa H, Wilson | <td><ref>Janwa H., Wilson R.M. Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes. Proceedings of AAECC-10, LNCS, vol. 673, Berlin, Springer-Verlag, pp. 180-194, 1993. https://doi.org/10.1007/3-540-56686-4_43</ref><ref>Kasami T. The weight enumerators for several classes of subcodes of the second order binary Reed-Muller codes. Inform. and Control, 18, pp. 369-394, 1971. https://doi.org/10.1016/S0019-9958(71)90473-6</ref></td> | ||
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<td><span class="htmlMath">n = 2t + 1</span></td> | <td><span class="htmlMath">n = 2t + 1</span></td> | ||
<td><span class="htmlMath">3</span></td> | <td><span class="htmlMath">3</span></td> | ||
<td><ref>Dobbertin H. Almost perfect nonlinear power functions | <td><ref>Dobbertin H. Almost perfect nonlinear power functions over GF (2n): the Welch case. IEEE Trans. Inform. Theory, 45, pp. 1271-1275, 1999. https://doi.org/10.1109/18.761283</ref></td> | ||
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<td rowspan="2"><span class="htmlMath">n = 2t + 1</span></td> | <td rowspan="2"><span class="htmlMath">n = 2t + 1</span></td> | ||
<td><span class="htmlMath">(t+2)/2</span></td> | <td><span class="htmlMath">(t+2)/2</span></td> | ||
<td rowspan="2"><ref>Dobbertin H. Almost perfect nonlinear power functions | <td rowspan="2"><ref>Dobbertin H. Almost perfect nonlinear power functions over GF (2n): the Niho case. Inform. and Comput., 151, pp. 57-72, 1999. https://doi.org/10.1006/inco.1998.2764</ref></td> | ||
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<td><span class="htmlMath">n = 2t + 1</span></td> | <td><span class="htmlMath">n = 2t + 1</span></td> | ||
<td><span class="htmlMath">n-1</span></td> | <td><span class="htmlMath">n-1</span></td> | ||
<td><ref name="kaisa" /><ref>Beth T, Ding C. On almost perfect nonlinear permutations. | <td><ref name="kaisa" /><ref>Beth T., Ding C. On almost perfect nonlinear permutations. Advances in Cryptology-EUROCRYPT’93, Lecture Notes in Computer Science, 765, Springer-Verlag, New York, pp. 65-76, 1993. https://doi.org/10.1007/3-540-48285-7_7</ref> | ||
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<td><span class="htmlMath">n = 5i</span></td> | <td><span class="htmlMath">n = 5i</span></td> | ||
<td><span class="htmlMath">i + 3</span></td> | <td><span class="htmlMath">i + 3</span></td> | ||
<td><ref>Dobbertin H. Almost perfect nonlinear power functions | <td><ref>Dobbertin H. Almost perfect nonlinear power functions over GF (2n): a new case for n divisible by 5. Proceedings of Finite Fields and Applications FQ5, pp. 113-121, 2000. https://doi.org/10.1007/978-3-642-56755-1_11</ref></td> | ||
</tr> | </tr> | ||
</table> | </table> |
Revision as of 16:22, 9 July 2020
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. Inform. Theory, 14, pp. 154-156, 1968. https://doi.org/10.1109/TIT.1968.1054106
- ↑ 2.0 2.1 Nyberg K. Differentially uniform mappings for cryptography. Advances in Cryptography, EUROCRYPT’93, Lecture Notes in Computer Science 765, pp. 55-64, 1994. Lecture Notes in Computer Science, vol 765. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48285-7_6
- ↑ Janwa H., Wilson R.M. Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes. Proceedings of AAECC-10, LNCS, vol. 673, Berlin, Springer-Verlag, pp. 180-194, 1993. https://doi.org/10.1007/3-540-56686-4_43
- ↑ Kasami T. The weight enumerators for several classes of subcodes of the second order binary Reed-Muller codes. Inform. and Control, 18, pp. 369-394, 1971. https://doi.org/10.1016/S0019-9958(71)90473-6
- ↑ Dobbertin H. Almost perfect nonlinear power functions over GF (2n): the Welch case. IEEE Trans. Inform. Theory, 45, pp. 1271-1275, 1999. https://doi.org/10.1109/18.761283
- ↑ Dobbertin H. Almost perfect nonlinear power functions over GF (2n): the Niho case. Inform. and Comput., 151, pp. 57-72, 1999. https://doi.org/10.1006/inco.1998.2764
- ↑ Beth T., Ding C. On almost perfect nonlinear permutations. Advances in Cryptology-EUROCRYPT’93, Lecture Notes in Computer Science, 765, Springer-Verlag, New York, pp. 65-76, 1993. https://doi.org/10.1007/3-540-48285-7_7
- ↑ Dobbertin H. Almost perfect nonlinear power functions over GF (2n): a new case for n divisible by 5. Proceedings of Finite Fields and Applications FQ5, pp. 113-121, 2000. https://doi.org/10.1007/978-3-642-56755-1_11