A non-trivial Finite field map.

Question

Is there a map $\sigma$ such that at some $\tau\in\Bbb F_p^\times\backslash\{1\}$ we have at every $\alpha,\beta\in\Bbb F_p$ $$\sigma(\alpha+\tau\cdot\beta)=\alpha-\tau\cdot\beta$$ holding true?

Answer 1

Setting $\beta=0$ gives $\sigma(\alpha)=\alpha$ for all $\alpha$. Setting $\alpha=0$ gives $\sigma(\tau\beta)=-\tau\beta$ so letting $\gamma=\tau\beta$, $\sigma(\gamma)=-\gamma$ for all $\gamma$ ($\gamma$ can be anything since $\tau\neq 0$). So this is only possible for $p=2$, but then $\mathbb{F}_p^\times\setminus\{1\}$ is empty so it is trivially impossible.