为什么我们不能使用Z / pZ组进行加密？

Sorry if this is a noob question, but for instance in ECDSA, we start by considering the field $\mathbb F_p$, whose the elements also form a group under multiplication. Why don't we just use this group instead of the one that is generated by the solutions of an elliptic curve equation lying in this field? I.E. choose an element lying in this field $g$ and generate a number $n$, and let $g^n$ be the private key. Is is just because of the reduction in key size? It seems like a high price to pay for the additional structure imposed by the curve equation.Sorry if this is a noob question, but for insta