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

The following table provides a summary of all known infinite families of power APN functions of the form ${\displaystyle F(x)=x^{d}}$.

Family Exponent Conditions ${\displaystyle \deg(x^{d})}$ Reference
Gold ${\displaystyle 2^{i}+1}$ ${\displaystyle \gcd(i,n)=1}$ ${\displaystyle 2}$ [1][2]
Kasami ${\displaystyle 2^{2i}-2^{i}+1}$ ${\displaystyle \gcd(i,n)=1}$ ${\displaystyle i+1}$ [3][4]
Welch ${\displaystyle 2^{t}+3}$ ${\displaystyle n=2t+1}$ ${\displaystyle 3}$ [5]
Niho ${\displaystyle 2^{t}+2^{t/2}-1,t}$ even ${\displaystyle n=2t+1}$ ${\displaystyle (t+2)/2}$ [6]
${\displaystyle 2^{t}+2^{(3t+1)/2}-1,t}$ odd ${\displaystyle t+1}$
Inverse ${\displaystyle 2^{2t}-1}$ ${\displaystyle n=2t+1}$ ${\displaystyle n-1}$ [2][7]
Dobbertin ${\displaystyle 2^{4i}+2^{3i}+2^{2i}+2^{i}-1}$ ${\displaystyle n=5i}$ ${\displaystyle i+3}$ [8]
1. Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory. 1968;14(1):154-6.
2. 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 ${\displaystyle GF(2^{n})}$: the Welch case. IEEE Transactions on Information Theory. 1999 May;45(4):1271-5.
6. Dobbertin H. Almost perfect nonlinear power functions on ${\displaystyle GF(2^{n})}$: 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 ${\displaystyle GF(2^{n})}$: a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121).