As an important primitive, attribute-based encryption (ABE) has attracted much attention in the relative-leakage model. However, the leakage rate in almost all of the existing ABE schemes is restricted with a leakage parameter. It implicitly suggests that higher leakage rate results in larger ciphertexts and keys. Aiming at tackling the challenge, novel leakage-resilient ciphertext-policy attribute-based encryption (CP-ABE) and key-policy attribute-based encryption (KP-ABE) are designed in this paper. It achieves shorter secret key size and simultaneously does not suffer from the undesirable drawback above. To realize this, the concepts of CP-AB-HPS and KP-AB-HPS are introduced, which generalize the notion of hash proof system (HPS) to attribute-based setting. Under some static assumptions, the proposed schemes are proved adaptively secure in the standard model. In addition, the results in simulation experiments indicate that the proposed scheme is efficient and practical.