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

Family Exponent Conditions ${\displaystyle \deg(x^{d})}$ Reference
Gold ${\displaystyle 2^{i}+1}$ ${\displaystyle \gcd(i,n)=1}$ 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 ${\displaystyle 2^{2t}-1}$ ${\displaystyle n=2t+1}$ ${\displaystyle n-1}$ [7][2]
Dobbertin ${\displaystyle 2^{4i}+2^{3i}+2^{2i}+2^{i}-1}$ ${\displaystyle n=5i}$ ${\displaystyle i+3}$ [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. 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 ${\displaystyle P^{3}}$ 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 ${\displaystyle GF(2^{n})}$: the Welch case, IEEE Transactions on Information Theory, 45(4):1271-1275, 1999
6. Hans Dobbertin, Almost perfect nonlinear power functions on ${\displaystyle GF(2^{n})}$: 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 ${\displaystyle GF(2^{n})}$: a new case for ${\displaystyle n}$ divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121