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 <math>F(x) = x^d</math>.

Family Exponent Conditions <math>\deg(x^d)</math> Reference
Gold <math>2^i + 1</math> <math>\gcd(i,n) = 1</math> 2 [1][2]
Kasami <math>2^{2i} - 2^i + 1</math> <math>\gcd(i,n) = 1</math> <math>i + 1</math> [3][4]
Welch <math>2^t + 3</math> <math>n = 2t + 1</math> <math>3</math> [5]
Niho <math>2^t + 2^{t/2} - 1, t</math> even <math>n = 2t + 1</math> <math>(t+2)/2</math> [6]
<math>2^t + 2^{(3t+1)/2} - 1, t</math> odd <math>t + 1</math>
Inverse <math>2^{2t} - 1</math> <math>n = 2t + 1</math> <math>n-1</math> [2][7]
Dobbertin <math>2^{4i} + 2^{3i} + 2^{2i} + 2^i - 1</math> <math>n = 5i</math> <math>i + 3</math> [8]
  1. ↑ Robert Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE transactions on Information Theory, 14(1):154-156, 1968
  2. ↑ 2.0 2.1 Kaisa Nyberg, Differentially uniform mappings for cryptography, Workshop on the Theory and Application of Cryptographic Techniques, pp. 55-64, Springer, 1993
  3. ↑ Heeralal Janwa and Richard M Wilson, Hyperplane sections of Fermat varieties in <math>P^3</math> in char. 2 and some applications to cyclic codes, International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pp. 180-194, Springer, 1993
  4. ↑ Tadao Kasami, The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes, Information and Control, 18(4):369-394, 1971
  5. ↑ Hans Dobbertin, Almost perfect nonlinear power functions on <math>GF(2^n)</math>: the Welch case, IEEE Transactions on Information Theory, 45(4):1271-1275, 1999
  6. ↑ Hans Dobbertin, Almost perfect nonlinear power functions on <math>GF(2^n)</math>: the Niho case, Information and Computation, 151(1-2):57-72, 1999
  7. ↑ Thomas Beth and Cunsheng Ding, On almost perfect nonlinear permutations, Workshop on the Theory and Application of Cryptographic Techniques, pp. 65-76, Springer, 1993
  8. ↑ Hans Dobbertin, Almost perfect nonlinear power functions over <math>GF(2^n)</math>: a new case for <math>n</math> divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121
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