In 2000, Biehl et al. proposed a fault-based attack on elliptic curve cryptography. In this paper, we refined the fault attack method. An elliptic curve E is defined over prime field ${⊩}_p$ with base point $P\in E\left({⊩}_p\right)$. Applying the fault attack on these curves, the discrete logarithm on the curve can be computed in subexponential time of $L_p(1/\mathrm{2,1}+o(1\left)\right)$. The runtime bound relies on heuristics conjecture about smooth numbers similar to the ones used by Lenstra, 1987.