Known infinite families of APN power functions over GF(2^n)

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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. ↑ R. Gold. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory, 14, pp. 154-156, 1968. https://doi.org/10.1109/TIT.1968.1054106
  2. ↑ 2.0 2.1 K. Nyberg. 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
  3. ↑ H. Janwa, R. M. Wilson. Hyperplane sections of Fermat varieties in P3 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
  4. ↑ 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
  5. ↑ H. Dobbertin. 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
  6. ↑ H. Dobbertin. 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
  7. ↑ T. Beth, C. Ding. 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
  8. ↑ H. Dobbertin. 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
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