3 problems

Let $F:{\mathbb F}_{2^n}\to{\mathbb F}_{2^n}$ be a function, and let its nonlinearity be … where $\mathcal W_F(a,b)$ is the Walsh transform of $F$. Even-dimensional nonlinearity bo…

Li–Li–Helleseth–Qu conjecture. The function $f_\alpha$ is bent if and only if $\alpha$ is not a cube in $\mathbb F_q$.

Let $n=2k>8$, let $F(x)=x^{d(2^k+1)}$ over $\mathbb{F}_{2^n}$, and assume ${\rm gcd}(d,2^k-1)=1$. For $\alpha\in\mathbb{F}_{2^n}\setminus\mathbb{F}_{2^k}$, write…