KöMaL 2026 B5517

For which prime numbers $p$ does there exist a permutation $a_1$, $a_2$, $\dots$, $a_{p-1}$ of numbers 1, 2, $\ldots$, $p-1$ such that numbers $a_1^1$, $a_2^2$, $\ldots$, $a_{p-1}^{p-1}$ have pairwise different remainders modulo $p$?