The Grain v1 stream cipher is one of the seven finalists in the final e STREAM portfolio. Though many attacks have been published,no recovery attack better than exhaustive key search on full Grain v1 in the single key setting has been found yet. In this paper,new state recovery attacks on Grain v1 utilizing the weak normality order of the employed keystream output function in the cipher are proposed. These attacks have remarkable advantages in the offline time,online time and memory complexities,which are all better than exhaustive key search. The success probability of each new attack is 0.632. The proposed attack primarily depends on the order of weak normality of the employed keystream output function. This shows that the weak normality order should be carefully considered when designing the keystream output functions of Grain-like stream ciphers. The Grain v1 stream cipher is one of the seven finalists in the final e STREAM portfolio. Though many attacks have been published, no recovery attack better than exhaustive key search on full Grain v1 in the single key setting has been yet yet. , new state recovery attacks on Grain v1 utilizing the weak normality order of the employed keystream output function in the cipher are proposed. These attacks have Dominant advantages in the offline time, online time and memory complexities, which are all better than exhaustive key search. The success probability of each new attack is 0.632. The proposed attack first depends on the order of weak normality of the employed keystream output function. This shows that the weak normality order should be carefully considered when designing the keystream output functions of Grain-like stream ciphers.