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 .

Family Exponent Conditions Reference
Gold 2 [1][2]
Kasami [3][4]
Welch [5]
Niho even [6]
odd
Inverse [2][7]
Dobbertin [8]
  1. Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory. 1988;14(1):154-6.
  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). Springer, Berlin, Heidelberg.
  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). Springer, Berlin, Heidelberg.
  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. Hans Dobbertin, Almost perfect nonlinear power functions on : the Welch case, IEEE Transactions on Information Theory, 45(4):1271-1275, 1999
  6. Hans Dobbertin, Almost perfect nonlinear power functions on : 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 : a new case for divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121