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.

Take all linear polynomials of the form  $f(x) = ax+b$ with $a=2,3,\ldots p-1$ and $b=0,1,2,\ldots p-1$. Then $f(x)-x$ is also a non-constant linear polynomial and hence has a unique zero, hence fixed point for $f$. As $a$ is coprime to $p$ these functions are bijective and provide permutations as desired.
