源自 Coulter-Matthews 弯曲函数的新三元弯曲函数类
信息论
2017-07-18 v1 math.IT
摘要
构造新的弯曲函数多年来一直是活跃的研究课题。对于满足 gcd ( n , k ) = 1 \gcd(n, k)=1 g cd( n , k ) = 1 k k k a ∈ F 3 n ∗ a\in\mathbb{F}_{3^n}^{*} a ∈ F 3 n ∗ f ( x ) = T r ( a x 3 k + 1 2 ) f(x)=Tr(ax^{\frac{3^k+1}{2}}) f ( x ) = T r ( a x 2 3 k + 1 ) F 3 n \mathbb{F}_{3^n} F 3 n T r ( ⋅ ) : F 3 n → F 3 Tr(\cdot):\mathbb{F}_{3^n}\rightarrow\mathbb{F}_3 T r ( ⋅ ) : F 3 n → F 3 f ( x ) f(x) f ( x ) k > 1 k>1 k > 1 ( ( 1 + 5 2 ) w + 1 − ( 1 − 5 2 ) w + 1 ) / 5 \left(\left(\frac{1+\sqrt{5}}{2}\right)^{w+1}-\right.\left.\left(\frac{1-\sqrt{5}}{2}\right)^{w+1}\right)/\sqrt{5} ( ( 2 1 + 5 ) w + 1 − ( 2 1 − 5 ) w + 1 ) / 5 ( ( 1 + 5 2 ) n − w + 1 − ( 1 − 5 2 ) n − w + 1 ) / 5 \left(\left(\frac{1+\sqrt{5}}{2}\right)^{n-w+1}-\right.\left.\left(\frac{1-\sqrt{5}}{2}\right)^{n-w+1}\right)/\sqrt{5} ( ( 2 1 + 5 ) n − w + 1 − ( 2 1 − 5 ) n − w + 1 ) / 5 0 < w < n 0<w<n 0 < w < n w k ≡ 1 ( m o d n ) wk\equiv 1\ (\bmod\;n) w k ≡ 1 ( mod n ) k = ( n + 1 ) / 2 k=(n+1)/2 k = ( n + 1 ) /2 ( ( 1 + 5 2 ) n − 1 − ( 1 − 5 2 ) n − 1 ) / 5 \left(\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}-\right.\left.\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\right)/\sqrt{5} ( ( 2 1 + 5 ) n − 1 − ( 2 1 − 5 ) n − 1 ) / 5 k = n − 1 k=n-1 k = n − 1 ( ( 1 + 5 2 ) n − ( 1 − 5 2 ) n ) / 5 \left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\right.\left.\left(\frac{1-\sqrt{5}}{2}\right)^n\right)/\sqrt{5} ( ( 2 1 + 5 ) n − ( 2 1 − 5 ) n ) / 5 k = ( n − 1 ) / 2 k=(n-1)/2 k = ( n − 1 ) /2 ( ( 1 + 5 2 ) n − 1 − ( 1 − 5 2 ) n − 1 ) / 5 \left(\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}-\right.\left.\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\right)/\sqrt{5} ( ( 2 1 + 5 ) n − 1 − ( 2 1 − 5 ) n − 1 ) / 5 ( n , k ) = ( 5 t + 4 , 4 t + 3 ) (n, k)=(5t+4, 4t+3) ( n , k ) = ( 5 t + 4 , 4 t + 3 ) ( 5 t + 1 , 4 t + 1 ) (5t+1, 4t+1) ( 5 t + 1 , 4 t + 1 ) t ≥ 1 t\geq 1 t ≥ 1 ( n , k ) = ( 7 t + 6 , 6 t + 5 ) (n, k)=(7t+6, 6t+5) ( n , k ) = ( 7 t + 6 , 6 t + 5 ) ( 7 t + 1 , 6 t + 1 ) (7t+1, 6t+1) ( 7 t + 1 , 6 t + 1 ) t ≥ 1 t\geq 1 t ≥ 1
引用
@article{arxiv.1707.04783,
title = {New Classes of Ternary Bent Functions from the Coulter-Matthews Bent Functions},
author = {Honggang Hu and Xiaolong Yang and Shaohua Tang},
journal= {arXiv preprint arXiv:1707.04783},
year = {2017}
}