幂等与p-幂等二次函数：非线性和余维数的分布
数论
2016-03-16 v1
摘要
二次函数 Q : F p n → F p Q:F_{p^n}\rightarrow F_p Q : F p n → F p Q ^ \widehat{Q} Q b ∈ F p n b\in F_{p^n} b ∈ F p n ∣ Q ^ ( b ) ∣ ∈ { 0 , p n + s 2 } |\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\} ∣ Q ( b ) ∣ ∈ { 0 , p 2 n + s } 0 ≤ s ≤ n − 1 0\le s\le n-1 0 ≤ s ≤ n − 1 Q Q Q C 1 \mathcal{C}_1 C 1 n n n C 1 = { Q ( x ) = T r ( ∑ i = 1 ⌊ ( n − 1 ) / 2 ⌋ a i x 2 i + 1 ) : a i ∈ F 2 } \mathcal{C}_1 = \{Q(x) = Tr(\sum_{i=1}^{\lfloor (n-1)/2\rfloor}a_ix^{2^i+1})\;:\; a_i \in F_2\} C 1 = { Q ( x ) = T r ( ∑ i = 1 ⌊( n − 1 ) /2 ⌋ a i x 2 i + 1 ) : a i ∈ F 2 } C 2 \mathcal{C}_2 C 2 n n n C 2 = { Q ( x ) = T r ( ∑ i = 1 ( n / 2 ) − 1 a i x 2 i + 1 ) + T r n / 2 ( a n / 2 x 2 n / 2 + 1 ) : a i ∈ F 2 } \mathcal{C}_2 = \{Q(x) = Tr(\sum_{i=1}^{(n/2)-1}a_ix^{2^i+1}) + {\rm Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in F_2\} C 2 = { Q ( x ) = T r ( ∑ i = 1 ( n /2 ) − 1 a i x 2 i + 1 ) + T r n/2 ( a n /2 x 2 n /2 + 1 ) : a i ∈ F 2 } p p p p p p D \mathcal{D} D D = { Q ( x ) = T r ( ∑ i = 0 ⌊ n / 2 ⌋ a i x p i + 1 ) : a i ∈ F p } \mathcal{D} = \{Q(x) = Tr(\sum_{i=0}^{\lfloor n/2\rfloor}a_ix^{p^i+1})\;:\; a_i \in F_p\} D = { Q ( x ) = T r ( ∑ i = 0 ⌊ n /2 ⌋ a i x p i + 1 ) : a i ∈ F p } C 1 \mathcal{C}_1 C 1 C 2 \mathcal{C}_2 C 2 D \mathcal{D} D s s s p > 2 p > 2 p > 2 p p p C 1 \mathcal{C}_1 C 1 C 2 \mathcal{C}_2 C 2
引用
@article{arxiv.1603.04685,
title = {Idempotent and p-potent quadratic functions: Distribution of nonlinearity and co-dimension},
author = {Nurdagül Anbar and Wilfried Meidl and Alev Topuzoglu},
journal= {arXiv preprint arXiv:1603.04685},
year = {2016}
}