Known infinite families of APN power functions over GF(2^n) html
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The following table provides a summary of all known infinite families of power APN functions of the form F(x) = xd.
| Family | Exponent | Conditions | deg(xd) | Reference |
|---|---|---|---|---|
| Gold | 2i + 1 | gcd(i,n) = 1 | 2 | [1][2] |
| Kasami | 22i - 2i + 1 | gcd(i,n) = 1 | i + 1 | [3][4] |
| Welch | 2t + 3 | n = 2t + 1 | 3 | [5] |
| Niho | 2t + 2t/2 - 1, t even | n = 2t + 1 | (t+2)/2 | [6] |
| 2t + 2(3t+1)/2 - 1, t odd | t + 1 | |||
| Inverse | 22t - 1 | n = 2t + 1 | n-1 | [2][7] |
| Dobbertin | 24i + 23i + 22i + 2i - 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(2n): the Welch case. IEEE Transactions on Information Theory. 1999 May;45(4):1271-5.
- ↑ Dobbertin H. Almost perfect nonlinear power functions on GF(2n): 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(2n): a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121).