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]\displaystyle{ F(x) = x^d }[/math].

Family Exponent Conditions [math]\displaystyle{ \deg(x^d) }[/math] Reference
Gold [math]\displaystyle{ 2^i + 1 }[/math] [math]\displaystyle{ \gcd(i,n) = 1 }[/math] 2 [1][2]
Kasami [math]\displaystyle{ 2^{2i} - 2^i + 1 }[/math] [math]\displaystyle{ \gcd(i,n) = 1 }[/math] [math]\displaystyle{ i + 1 }[/math] [3][4]
Welch [math]\displaystyle{ 2^t + 3 }[/math] [math]\displaystyle{ n = 2t + 1 }[/math] [math]\displaystyle{ 3 }[/math] [5]
Niho [math]\displaystyle{ 2^t + 2^{t/2} - 1, t }[/math] even [math]\displaystyle{ n = 2t + 1 }[/math] [math]\displaystyle{ (t+2)/2 }[/math] [6]
[math]\displaystyle{ 2^t + 2^{(3t+1)/2} - 1, t }[/math] odd [math]\displaystyle{ t + 1 }[/math]
Inverse [math]\displaystyle{ 2^{2t} - 1 }[/math] [math]\displaystyle{ n = 2t + 1 }[/math] [math]\displaystyle{ n-1 }[/math] [2][7]
Dobbertin [math]\displaystyle{ 2^{4i} + 2^{3i} + 2^{2i} + 2^i - 1 }[/math] [math]\displaystyle{ n = 5i }[/math] [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ GF(2^n) }[/math]: a new case for [math]\displaystyle{ n }[/math] divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121