Permutations of the elements of $\mathbb Z_p$ Let $p$ be prime. Describe all permutations $\sigma$ of the elements of $\mathbb Z_p$, having the property that $\{\sigma(i)-i: i\in\mathbb Z_p\}=\mathbb Z_p$

(Added by Robert Lewis in an attempt to provide background, motivation, and other context for this engaging problem)
This problem essentially asks for a method of representing permutations $\sigma$ of the finite field $\Bbb Z_p$ which respects the algebraic structure/computations inherent in such fields.

These are known as complete mappings of the finite field $\mathbb{F}_{p}$. I think you will find that it is quite a difficult problem to describe all such permutations.
Here is a good overview:
Niederreiter, Harald; Robinson, Karl H., Complete mappings of finite fields, J. Aust. Math. Soc., Ser. A 33, 197-212 (1982). ZBL0495.12018.
You will see that this is very dependent on $p$, for example when $p=7$
we have $\pm {(x+a)}^{4} +3x + b$
giving examples for any choice of $a,b \in \mathbb{F}_{7}$; but finding and describing such polynomials relies on ad-hoc methods.
