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Lossy trapdoor functions (LTFs) are very useful tools in constructing complex cryptographic primitives in a black-box manner, such as injective trapdoor functions, collision-resistant hashes, CCA secure public-key encryption, and so on. However, the trapdoor is very sensitive in lossy trapdoor function systems, and the attacker can obtain partial sensitive information of trapdoor by the side-channel attacks, which leads to not only the leakage of sensitive information but also the impossibility of provable security. In this paper, we present the new model of updatable lossy trapdoor functions in presence of consecutive and continual leakage-resilient, to provide a more efficient mechanism in solving the sensitive trapdoor leakage problem in LTF systems. Our contribution has threefold: 1) we give the definition and model of consecutive and continual leakage-resilient LTFs, and provide the concrete construction to achieve the lossiness of 50%; 2) using the proposed LTF scheme as a primitive, we present a updatable public-key encryption in the presence of consecutive and continual leakage-resilience, in which the leakage of secret key can occur during the updates that can simulate the real leakage scenarios; and 3) We provide a secure application deployment in sensitive-data revealing environments that employ the proposed CCLR-PKE scheme as a building block, in which a side-channel analyzer might obtain some sensitive information by controlling the secret channel, watching the private memory and detecting the algorithm executing and so on.