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Multi-Key Fully Homomorphic Encryption (MKFHE) enables arbitrary operations on data encrypted with different public keys, allowing all participating users jointly decrypting the final ciphertext. The multi-key BFV FHE scheme inherits BFV\u2019s advantages in ring element encryption and scale invariance. Nonetheless, it also has some disadvantages, such as additional noise generated during the relinearization process, the need for costly transformations during the external product process, and the requirement for a Common Reference String (CRS). In this paper, we investigate the MKFHE scheme for RLWE-based BFV. Firstly, we improve the modulus size of the evaluation key and the public key to construct a modulus enchancement relinearization method, which can significantly reduce the noise generated during the relinearization process. Secondly, we propose to use an inner product via Gadget decomposition in the relinearization based on the MK-BFV scheme instead of the original outer product operation, which can reduce the complexity of the NTT operation to <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\left( {d + 2\\tilde{d}} \\right)r^{\\prime } \/\\left( {d + 2} \\right)\\tilde{l}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mfenced>\n                      <mml:mrow>\n                        <mml:mi>d<\/mml:mi>\n                        <mml:mo>+<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                        <mml:mover>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:mo>~<\/mml:mo>\n                        <\/mml:mover>\n                      <\/mml:mrow>\n                    <\/mml:mfenced>\n                    <mml:msup>\n                      <mml:mi>r<\/mml:mi>\n                      <mml:mo>\u2032<\/mml:mo>\n                    <\/mml:msup>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mfenced>\n                      <mml:mrow>\n                        <mml:mi>d<\/mml:mi>\n                        <mml:mo>+<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:mfenced>\n                    <mml:mover>\n                      <mml:mi>l<\/mml:mi>\n                      <mml:mo>~<\/mml:mo>\n                    <\/mml:mover>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> of the original one. Finally, we propose a MK-BFV without CRS on the basis of the previous ones, which enhances the user's control over his own key.<\/jats:p>","DOI":"10.1186\/s42400-024-00337-2","type":"journal-article","created":{"date-parts":[[2025,6,26]],"date-time":"2025-06-26T01:01:19Z","timestamp":1750899679000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An efficient multi-key BFV fully homomorphic encryption scheme with optimized relinearization"],"prefix":"10.1186","volume":"8","author":[{"given":"Sai","family":"Hu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0006-7463-6193","authenticated-orcid":false,"given":"Ruwei","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,6,26]]},"reference":[{"key":"337_CR1","doi-asserted-by":"crossref","unstructured":"Bajard J-C, Eynard J, Hasan MA, Zucca V (2016) A full RNS variant of FV like somewhat homomorphic encryption schemes. 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