{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T18:05:17Z","timestamp":1775585117100,"version":"3.50.1"},"reference-count":23,"publisher":"Institution of Engineering and Technology (IET)","issue":"1","license":[{"start":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T00:00:00Z","timestamp":1753833600000},"content-version":"vor","delay-in-days":210,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2020YFA0712300"],"award-info":[{"award-number":["2020YFA0712300"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005230","name":"Natural Science Foundation of Chongqing Municipality","doi-asserted-by":"publisher","award":["cstc2021jcyj-msxmX0821"],"award-info":[{"award-number":["cstc2021jcyj-msxmX0821"]}],"id":[{"id":"10.13039\/501100005230","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002367","name":"Chinese Academy of Sciences","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002367","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["ietresearch.onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["IET Information Security"],"published-print":{"date-parts":[[2025,1]]},"abstract":"<jats:p>\n                    The current multikey fully homomorphic encryption (MKFHE) needs to add exponential noise in the distributed decryption phase to ensure the simulatability of partial decryption. Such a large noise causes the ciphertext modulus of the scheme to increase exponentially compared to the single\u2010key fully homomorphic encryption (FHE), further reducing the efficiency of the scheme and making the hardness problem on the lattice on which the scheme relies have a subexponential approximation factor  (which means that the security of the scheme is reduced). To address this problem, this paper analyzes in detail the noise in partial decryption of the MKFHE based on the learning with error (LWE) problem. It points out that this part of the noise is composed of private key and the noise in initial ciphertext. Therefore, as long as the encryption scheme is leak\u2010resistant and the noise in partial decryption is independent of the noise in the initial ciphertext, the semantic security of the ciphertext can be guaranteed. In order to make the noise in the initial ciphertext independent of the noise in the partial decryption, this paper proves the smudging lemma on discrete Gaussian distribution and achieves this goal by multiplying the initial ciphertext by a \u201cdummy\u201d ciphertext with a plaintext of 1. Based on the above method, this paper removes the exponential noise in the distributed decryption phase for the first time and reduces the ciphertext modulus of MKFHE from 2\n                    <jats:sup>\n                      <jats:italic>\u03c9<\/jats:italic>\n                      (\n                      <jats:italic>\u03bb<\/jats:italic>\n                      <jats:italic>L<\/jats:italic>\n                      \u2009log\n                      <jats:italic>\u03bb<\/jats:italic>\n                      )\n                    <\/jats:sup>\n                    to 2\n                    <jats:sup>\n                      <jats:italic>O<\/jats:italic>\n                      (\n                      <jats:italic>\u03bb<\/jats:italic>\n                      +\n                      <jats:italic>L<\/jats:italic>\n                      )\n                    <\/jats:sup>\n                    as the same level as the FHE.\n                  <\/jats:p>","DOI":"10.1049\/ise2\/7550044","type":"journal-article","created":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T09:08:25Z","timestamp":1753866505000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Multikey Fully Homomorphic Encryption: Removing Noise Flooding in Distributed Decryption via the Smudging Lemma on Discrete Gaussian Distribution"],"prefix":"10.1049","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0009-0006-8479-7046","authenticated-orcid":false,"given":"Xiaokang","family":"Dai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1169-379X","authenticated-orcid":false,"given":"Wenyuan","family":"Wu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0530-1155","authenticated-orcid":false,"given":"Yong","family":"Feng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"265","published-online":{"date-parts":[[2025,7,30]]},"reference":[{"key":"e_1_2_13_1_2","doi-asserted-by":"crossref","unstructured":"L\u00f3pez-AltA. TromerE. andVaikuntanathanV. On-the-Fly Multiparty Computation on the Cloud via Multikey Fully Homomorphic Encryption STOC\u201912: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing 2012 New York NY USA Association for Computing Machinery 1219\u20131234 https:\/\/doi.org\/10.1145\/2213977.2214086 2-s2.0-84862629748.","DOI":"10.1145\/2213977.2214086"},{"key":"e_1_2_13_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10623-022-01160-x"},{"key":"e_1_2_13_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-70500-2_22"},{"key":"e_1_2_13_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-49896-5_26"},{"key":"e_1_2_13_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-53018-4_8"},{"key":"e_1_2_13_6_2","unstructured":"ClearM.andMcGoldrickC. Multi-Identity and Multi-Key Leveled FHE From Learning With Errors Cryptology ePrint Archive 2014 Paper 2014\/798https:\/\/eprint.iacr.org\/2014\/798."},{"key":"e_1_2_13_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-53644-5_9"},{"key":"e_1_2_13_8_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-77287-1_2"},{"key":"e_1_2_13_9_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-40041-4_5"},{"key":"e_1_2_13_10_2","unstructured":"BensimhounM. N-Dimensional Cumulative Function and Other Useful Facts about Gaussians and Normal Densities 2009 Jerusalem Israel 1\u20138 Technical Report."},{"key":"e_1_2_13_11_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02868568"},{"key":"e_1_2_13_12_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-981-96-0875-1_10"},{"key":"e_1_2_13_13_2","doi-asserted-by":"publisher","DOI":"10.1049\/ise2.12036"},{"key":"e_1_2_13_14_2","doi-asserted-by":"publisher","DOI":"10.1049\/iet-ifs.2018.5341"},{"key":"e_1_2_13_15_2","unstructured":"BourseF. PinoR. D. MinelliM. andWeeH. FHE Circuit Privacy Almost for Free Cryptology ePrint Archive 2016 Paper 2016\/381https:\/\/eprint.iacr.org\/2016\/381."},{"key":"e_1_2_13_16_2","unstructured":"DucasL.andStehl\u00e9D. Sanitization of FHE Ciphertexts Cryptology ePrint Archive 2016 Paper 2016\/164https:\/\/eprint.iacr.org\/2016\/164."},{"key":"e_1_2_13_17_2","article-title":"Circuit Privacy for FHEW\/TFHE-Style Fully Homomorphic Encryption in Practice","author":"Kluczniak K.","year":"2022","journal-title":"Cryptology ePrint Archive"},{"key":"e_1_2_13_18_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-03810-6_14"},{"key":"e_1_2_13_19_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01445125"},{"key":"e_1_2_13_20_2","doi-asserted-by":"crossref","unstructured":"RegevO. On Lattices Learning With Errors Random Linear Codes and Cryptography STOC \u201905: Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing 2005 New York NY USA Association for Computing Machinery 84\u201393 https:\/\/doi.org\/10.1145\/1060590.1060603 2-s2.0-33745571012.","DOI":"10.1145\/1060590.1060603"},{"key":"e_1_2_13_21_2","doi-asserted-by":"crossref","unstructured":"PeikertC. Public-Key Cryptosystems From the Worst-Case Shortest Vector Problem: Extended Abstract STOC\u201909: Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing 2009 New York NY USA Association for Computing Machinery 333\u2013342 https:\/\/doi.org\/10.1145\/1536414.1536461 2-s2.0-70350642078.","DOI":"10.1145\/1536414.1536461"},{"key":"e_1_2_13_22_2","doi-asserted-by":"publisher","DOI":"10.1515\/jmc-2015-0016"},{"key":"e_1_2_13_23_2","unstructured":"DaiX. WuW. andFengY. Key Lifting: Multi-Key Fully Homomorphic Encryption in Plain Model Without Noise Flooding Cryptology ePrint Archive 2022 Paper 2022\/055https:\/\/eprint.iacr.org\/2022\/055."}],"container-title":["IET Information Security"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/ietresearch.onlinelibrary.wiley.com\/doi\/pdf\/10.1049\/ise2\/7550044","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/ietresearch.onlinelibrary.wiley.com\/doi\/full-xml\/10.1049\/ise2\/7550044","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/ietresearch.onlinelibrary.wiley.com\/doi\/pdf\/10.1049\/ise2\/7550044","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T22:35:51Z","timestamp":1773009351000},"score":1,"resource":{"primary":{"URL":"https:\/\/ietresearch.onlinelibrary.wiley.com\/doi\/10.1049\/ise2\/7550044"}},"subtitle":[],"editor":[{"given":"Xueqin","family":"Liang","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2025,1]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["10.1049\/ise2\/7550044"],"URL":"https:\/\/doi.org\/10.1049\/ise2\/7550044","archive":["Portico"],"relation":{},"ISSN":["1751-8709","1751-8717"],"issn-type":[{"value":"1751-8709","type":"print"},{"value":"1751-8717","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1]]},"assertion":[{"value":"2025-02-24","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-06-25","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-07-30","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"7550044"}}