Known infinite families of APN power functions over GF(2^n)
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Family | Exponent | Conditions | [math]\displaystyle{ \deg(x^d) }[/math] | Reference |
---|---|---|---|---|
Gold | [math]\displaystyle{ 2^i + 1 }[/math] | [math]\displaystyle{ \gcd(i,n) = 1 }[/math] | 2 | [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] | [7][2] |
Dobbertin | [math]\displaystyle{ 2^{4i} + 2^{3i} + 2^{2i} + 2^i - 1 }[/math] | [math]\displaystyle{ n = 5i }[/math] | [math]\displaystyle{ i + 3 }[/math] | [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 [math]\displaystyle{ P^3 }[/math] 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 [math]\displaystyle{ GF(2^n) }[/math]: the Welch case, IEEE Transactions on Information Theory, 45(4):1271-1275, 1999
- ↑ Hans Dobbertin, Almost perfect nonlinear power functions on [math]\displaystyle{ GF(2^n) }[/math]: 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 [math]\displaystyle{ GF(2^n) }[/math]: a new case for [math]\displaystyle{ n }[/math] divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121