For the leakage of secret keying material under side-channel attacks in attribute-based encryption, these existing solutions only allow attackers to get length-bounded leakage on the secret key. First of all, we combine the model of continuous auxiliary-inputs leakage with dual system encryption to achieve strong leakage resilience. Secondly, we reasonably design the generation of secret key to reduce its size, and then devise the first attribute-based encryption that remains secure even if the attacker can obtain the leakage of continuous auxiliary inputs. In the end, our scheme is proved fully secure in the standard model based on reasonable assumptions, even if the attacker gets the leakage of the secret key from an auxiliary-input function. Our scheme achieves the continuous and unbounded leakage of both the master key and the private key, and does not assume fully erasure of old secret keys during the key-update query. In addition, it has a desirable composition feature. Compared with relevant solutions, this scheme not only benefits the best leakage resilience, but also has shorter length of master keys and private keys.