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 <math>F(x) = x^d</math>.
| Family | Exponent | Conditions | <math>\deg(x^d)</math> | Reference |
|---|---|---|---|---|
| Gold | <math>2^i + 1</math> | <math>\gcd(i,n) = 1</math> | 2 | [1][2] |
| Kasami | <math>2^{2i} - 2^i + 1</math> | <math>\gcd(i,n) = 1</math> | <math>i + 1</math> | [3][4] |
| Welch | <math>2^t + 3</math> | <math>n = 2t + 1</math> | <math>3</math> | [5] |
| Niho | <math>2^t + 2^{t/2} - 1, t</math> even | <math>n = 2t + 1</math> | <math>(t+2)/2</math> | [6] |
| <math>2^t + 2^{(3t+1)/2} - 1, t</math> odd | <math>t + 1</math> | |||
| Inverse | <math>2^{2t} - 1</math> | <math>n = 2t + 1</math> | <math>n-1</math> | [2]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). Springer, Berlin, Heidelberg.</ref> |
| Dobbertin | <math>2^{4i} + 2^{3i} + 2^{2i} + 2^i - 1</math> | <math>n = 5i</math> | <math>i + 3</math> | [7] |
- ↑ 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.
- ↑ Dobbertin H. Almost perfect nonlinear power functions on <math>GF(2^n)</math>: the Welch case. IEEE Transactions on Information Theory. 1999 May;45(4):1271-5.
- ↑ Dobbertin H. Almost perfect nonlinear power functions on <math>GF(2^n)</math>: the Niho case. Information and Computation. 1999 May 25;151(1-2):57-72.
- ↑ Dobbertin H. Almost perfect nonlinear power functions on <math>GF(2^n)</math>: a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121). Springer, Berlin, Heidelberg.