Known infinite families of APN power functions over GF(2^n)
From Boolean
The following table provides a summary of all known infinite families of power APN functions of the form .
| Family | Exponent | Conditions | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \deg(x^{d})} | Reference |
|---|---|---|---|---|
| Gold | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2^{i}+1} | [1][2] | ||
| Kasami | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2^{2i}-2^{i}+1} | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i+1} | [3][4] | |
| Welch | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2^{t}+3} | [5] | ||
| Niho | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2^{t}+2^{t/2}-1,t} even | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (t+2)/2} | [6] | |
| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2^{t}+2^{(3t+1)/2}-1,t} odd | ||||
| Inverse | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2^{2t}-1} | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n-1} | [2][7] | |
| Dobbertin | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 2^{4i}+2^{3i}+2^{2i}+2^{i}-1} | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n=5i} | [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).
- ↑ 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 : the Welch case. IEEE Transactions on Information Theory. 1999 May;45(4):1271-5.
- ↑ Dobbertin H. Almost perfect nonlinear power functions on : 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 : a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121).
