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]} |
2^{t} + 2^{(3t+1)/2} - 1, t odd | t + 1 | |||
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. 1968;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).
- ↑ 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).
- ↑ 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.
- ↑ Dobbertin H. Almost perfect nonlinear power functions on GF(2^{n}): the Welch case. IEEE Transactions on Information Theory. 1999 May;45(4):1271-5.
- ↑ Dobbertin H. Almost perfect nonlinear power functions on GF(2^{n}): the Niho case. Information and Computation. 1999 May 25;151(1-2):57-72.
- ↑ 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).
- ↑ Dobbertin H. Almost perfect nonlinear power functions on GF(2^{n}): a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121).