<p>Dynamic multi-key FHE (MKFHE), an extension of standard FHE, not only supports computation on ciphertexts under different keys but also allows the results to be used in further computation involving new keys. This capability aligns well with cloud computing application scenarios. However, existing dynamic MKFHE implementations exhibit prohibitively high computational complexity and lack efficient ciphertext extension mechanisms. To address this, we propose a new “Request-Response” working mode and construct a dynamic MKFHE scheme in asymmetric key setting from learning with errors (LWE) under this mode. Our approach enables users and the server to perform the ciphertext extension operation interactively, which improves the efficiency of ciphertext extension on the server and reduces the fees paid by users to the cloud server provider at the expense of a small amount of computation burden on users. The comparative analysis results show that our scheme no longer eliminates the need for public extension key and further reduces the public key size from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\tilde{O}\left( {n^{2} (K + L)^{2} } \right)\)</EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\tilde{O}\left( {n(K + L)^{2} } \right)\)</EquationSource> </InlineEquation>. In the process of ciphertext extension, the upper bound noise of the extended ciphertext remains unchanged (the noise expansion rate is 1), effectively inhibiting noise growth and allowing for smaller public parameters. Additionally, our scheme no longer uses the mainstream linear combination algorithm, reducing the computational overhead from <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(O\left( {n^{4} \ell^{4} } \right)\)</EquationSource> </InlineEquation> to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(O\left( {n^{2} \ell^{2} } \right)\)</EquationSource> </InlineEquation>. Our scheme is proven to be secure under the standard LWE assumption</p>