The restricted sumsets in finite abelian groups

Suppose that $k\geq 2$ and $A$ is a non-empty subset of a finite abelian group $G$ with $|G|>1$. Then the cardinality of the restricted sumset $$k^\wedge A:=\{a_1+\cdots+a_k:\,a_1,\ldots,a_k\in A,\ a_i\neq a_j\text{ for }i\neq j\}$$ is at least $$\min\{p(G), k|A|-k^2+1\},$$ where $p(G)$ denotes the least prime divisor of $|G|$.