9 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…

Let $f^n$ be the cubic rotation symmetric Boolean function in $n$ variables generated by the monomial $x_0x_1x_2$. Its weight $wt(f^n)$ is the number of inputs in $\mathbb{F}_2^n$…

Conjecture.

Let $m$ and $n$ be the parameters of the vectorial function $F$ under consideration, and let $u$ satisfy … with $u\not\equiv 2^k\pmod{2^m-1}$ for every $k\geq 0$. Define … where…

Let $RM(2,n)$ and $RM(3,n)$ denote the binary Reed–Muller codes of orders $2$ and $3$, respectively. The cubic-function conjecture. The exact value of the maximum second-order nonl…

Let $n$ be even and let $F:\mathbb{F}_{2^n}\to\mathbb{F}_{2^n}$ be a function. Its nonlinearity is … The known maximum nonlinearity is $2^{n-1}-2^{\frac n2}$. Even-dimensional nonl…

Let $k$ be odd, and consider the butterfly structures over $\mathbb{F}_{2^{2k}}$ with exponent $e=3\times 2^t$, whose nonlinearity has been verified experimentally to equal…

Let $F:F_{2^n}\to F_{2^n}$ be an $(n,n)$-function, and let … be its nonlinearity. Even-dimensional nonlinearity conjecture. When $n$ is even, … For odd $n$, the corresponding bound…

Let $l\geq 4$ be an integer, and let $f$ be the rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}\cdots x_{le}$, with the parameter $e$ as in the paper's…