Discrete logarithm problem in subgroup of index 2. ElGamal
As an example, algorithms, such as baby step giant step  with time and memory complexity $$T=M=O(\sqrt{N})$$ or Pollard's rho  with time and memory complexity $$T=O(\sqrt{N}),\quad M=O(1)$$have complexities that depend on the size $N$ of the group over which the DL is defined.

So $N=\mathbb{Z}_p^{\ast}=p-1,$ while a subgroup of index 2 has size $N'=(p-1)/2$ and the complexity improves accordingly.