Need literature or link to family of cryptographic invertible polynomials which has distribution similar to inverted bell curve Please help me identify a family of polynomials P(x,k) with the following properties:


$x\in Z_{2^n} $
$\forall k, \forall p_k\in P,\ \exists\ p_k^{-1} |\ p_k^{-1}(p_k(x)\equiv x\ mod\ 2^n $
Frequency distribution of $y = p_k(x)\ mod\ 2^n$ is an inverted bell curve


Thanks for suggestions
These requirements are impossible. If $p_k$ is a permutation, then the distribution induced by "choose $x$ uniformly in $\mathbb{Z}_n$, output $y=p_k(x)$" is also uniform in  $\mathbb{Z}_n$. It can't follow a normal distribution. For every possible $y$, there is exactly one $x$ that makes it happen, and that $x$ is assigned probability $1/2^n$.