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]
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  2. 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).
  3. 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).
  4. 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.
  5. Dobbertin H. Almost perfect nonlinear power functions on GF(2n): the Welch case. IEEE Transactions on Information Theory. 1999 May;45(4):1271-5.
  6. Dobbertin H. Almost perfect nonlinear power functions on GF(2n): the Niho case. Information and Computation. 1999 May 25;151(1-2):57-72.
  7. 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).
  8. 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).