The side channel attacks will lead to the destruction of the security of the traditional cryptographic scheme. Leakage-resilient identity-based signature has attracted great attention. Based on the dual system encryption technology, we construct an identity-based signature scheme that can resist continuous private key leakage. In the standard model, the security of the scheme is proved. The key points of our leakage-resilient signature scheme are as follows: (1) The private key can be extended according to the security requirements. In other words, when the leakage is serious, we can select a bigger value n, where n is a parameter related to the leakage rate. (2) An elaborate key update algorithm makes the scheme resist continuous leakage attacks. Furthermore, the updated private key has the same distribution as the previous private key. (3) The proposed scheme is fully secure in the standard model rather than in the random oracle model or in the general group model. In order to achieve this goal, we use dual system encryption technology. Thus, the security of the constructed scheme does not depend on the number of queries of the attacker.