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Please help me identify a family of polynomials P(x,k) with the following properties:

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

Thanks for suggestions

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  • $\begingroup$ So why would you name it "cryptographic invertible", not jut "invertible"? $\endgroup$ – Vadym Fedyukovych Sep 2 '18 at 7:42
  • $\begingroup$ I need pseudo-randomness on $y$ for any $x$ $\endgroup$ – Baruntar Sep 2 '18 at 7:45
  • $\begingroup$ How would you measure/define "frequency"? It would be natural to expect just a single "inverse" for any given polynomial. $\endgroup$ – Vadym Fedyukovych Sep 2 '18 at 15:37
  • $\begingroup$ In #3, what is the underlying random process? Uniform choice of x and fixed k? Uniform choice of k and fixed x? $\endgroup$ – Mikero Sep 2 '18 at 16:17
  • $\begingroup$ @Mikero, Uniform choice of x for fixed k. $k$ is chosen randomly once to define $p_k(x)$ i.e. $p_K(x)=p(x,k)|_{k=K}$ $\endgroup$ – Baruntar Sep 3 '18 at 5:42
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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 $1/2^n$.

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