Attribute-based encryption mechanism can achieve very flexible access control, so it has a wide range of applications in the distributed environment, such as fine-grained access control, audit log applications, cloud storage systems. Key-policy attribute-based encryption (KP-ABE) scheme is especially suitable for video on demand, pay TV, etc. Most of the existing KP-ABE schemes do not consider the side channel attacks which probably leak some secret information about the cryptosystems. In the paper, we present the formal definition and security model of key-policy attribute-based encryption scheme which is resilient to continual auxiliary input (CAI) leakage. What is more, we present a concrete KP-ABE scheme. The proposed scheme is proved secure under the static assumptions.