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 $F(x)=x^{d}$ .

Family	Exponent	Conditions	$\deg(x^{d})$	Reference
Gold	$2^{i}+1$	$\gcd(i,n)=1$	2	^[1]^[2]
Kasami	$2^{2i}-2^{i}+1$	$\gcd(i,n)=1$	$i+1$	^[3]^[4]
Welch	$2^{t}+3$	$n=2t+1$	$3$	^[5]
Niho	$2^{t}+2^{t/2}-1,t$ even	$n=2t+1$	$(t+2)/2$	^[6]
Niho	$2^{t}+2^{(3t+1)/2}-1,t$ odd	$n=2t+1$	$t+1$	^[6]
Inverse	$2^{2t}-1$	$n=2t+1$	$n-1$	^[2]^[7]
Dobbertin	$2^{4i}+2^{3i}+2^{2i}+2^{i}-1$	$n=5i$	$i+3$	^[8]

↑ Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory. 1988;14(1):154-6.
↑ ^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.
↑ 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.
↑ 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.
↑ Hans Dobbertin, Almost perfect nonlinear power functions on $GF(2^{n})$ : the Welch case, IEEE Transactions on Information Theory, 45(4):1271-1275, 1999
↑ Hans Dobbertin, Almost perfect nonlinear power functions on $GF(2^{n})$ : the Niho case, Information and Computation, 151(1-2):57-72, 1999
↑ Thomas Beth and Cunsheng Ding, On almost perfect nonlinear permutations, Workshop on the Theory and Application of Cryptographic Techniques, pp. 65-76, Springer, 1993
↑ Hans Dobbertin, Almost perfect nonlinear power functions over $GF(2^{n})$ : a new case for $n$ divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121

Navigation menu