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} | 2 | [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 | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle t+1} | |||
| 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} | Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i+3} | [8] |
- ↑ Robert Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE transactions on Information Theory, 14(1):154-156, 1968
- ↑ 2.0 2.1 Kaisa Nyberg, Differentially uniform mappings for cryptography, Workshop on the Theory and Application of Cryptographic Techniques, pp. 55-64, Springer, 1993
- ↑ Heeralal Janwa and Richard M Wilson, Hyperplane sections of Fermat varieties in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P^{3}} in char. 2 and some applications to cyclic codes, International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pp. 180-194, Springer, 1993
- ↑ Tadao Kasami, The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes, Information and Control, 18(4):369-394, 1971
- ↑ Hans Dobbertin, Almost perfect nonlinear power functions on : the Welch case, IEEE Transactions on Information Theory, 45(4):1271-1275, 1999
- ↑ Hans Dobbertin, Almost perfect nonlinear power functions on : the Niho case, Information and Computation, 151(1-2):57-72, 1999
- ↑ Thomas Beth and Cunsheng Ding, On almost perfect nonlinear permutations, Workshop on the Theory and Application of Cryptographic Techniques, pp. 65-76, Springer, 1993
- ↑ Hans Dobbertin, Almost perfect nonlinear power functions over : a new case for divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121
