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]\displaystyle{ F(x) = x^d }[/math].
| Family | Exponent | Conditions | [math]\displaystyle{ \deg(x^d) }[/math] | Reference |
|---|---|---|---|---|
| Gold | [math]\displaystyle{ 2^i + 1 }[/math] | [math]\displaystyle{ \gcd(i,n) = 1 }[/math] | [math]\displaystyle{ 2 }[/math] | [1][2] |
| Kasami | [math]\displaystyle{ 2^{2i} - 2^i + 1 }[/math] | [math]\displaystyle{ \gcd(i,n) = 1 }[/math] | [math]\displaystyle{ i + 1 }[/math] | [3][4] |
| Welch | [math]\displaystyle{ 2^t + 3 }[/math] | [math]\displaystyle{ n = 2t + 1 }[/math] | [math]\displaystyle{ 3 }[/math] | [5] |
| Niho | [math]\displaystyle{ 2^t + 2^{t/2} - 1, t }[/math] even | [math]\displaystyle{ n = 2t + 1 }[/math] | [math]\displaystyle{ (t+2)/2 }[/math] | [6] |
| [math]\displaystyle{ 2^t + 2^{(3t+1)/2} - 1, t }[/math] odd | [math]\displaystyle{ t + 1 }[/math] | |||
| Inverse | [math]\displaystyle{ 2^{2t} - 1 }[/math] | [math]\displaystyle{ n = 2t + 1 }[/math] | [math]\displaystyle{ n-1 }[/math] | [2][7] |
| Dobbertin | [math]\displaystyle{ 2^{4i} + 2^{3i} + 2^{2i} + 2^i - 1 }[/math] | [math]\displaystyle{ n = 5i }[/math] | [math]\displaystyle{ i + 3 }[/math] | [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 [math]\displaystyle{ 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]\displaystyle{ GF(2^n) }[/math]: 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 [math]\displaystyle{ GF(2^n) }[/math]: a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121).