Masking is one of the most effective countermeasures for securely implementing cryptographic algorithms against power side-channel attacks, the design of which however turns out to be intricate and error-prone. While techniques have been proposed to rigorously verify implementations of cryptographic algorithms, currently they are limited in scalability. To address this issue, compositional approaches have been investigated, but insofar they fail to prove the security of recent efficient implementations. To fill this gap, we propose a novel compositional verification approach. In particular, we introduce two new language-level security notions based on which we propose composition strategies and verification algorithms. Our approach is able to prove efficient implementations, which cannot be done by prior compositional approaches. We implement our approach as a tool CONVINCE and conduct extensive experiments to confirm its efficacy. We also use CONVINCE to further explore the design space of the AES Sbox with least refreshing by replacing its implementation for finite-field multiplication with more efficient counterparts. We automatically prove leakage-freeness of these new versions. As a result, we can effectively reduce 1,600 randomness and 3,200 XOR-operations of the state-of-the-art AES implementation.