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But their scheme did not work, since the noise of homomorphic multiplication is complicated and large, which leads to failure of decryption. More specifically, they chose LWR instances as a public key and the private key therein as a secret key and then used the tensor product to implement homomorphic multiplication, which resulted in a tangly modulus problem. Recall that there are two moduli in the LWR instances, and then the moduli will tangle together due to the tensor product. Inspired by their work, we built the first workable<jats:sans-serif> LWR<\/jats:sans-serif>-based FHE scheme eliminating the tangly modulus problem by cleverly adopting the celebrated<jats:italic> approximate eigenvector<\/jats:italic> method proposed by Gentry et al. at Crypto 2013. Roughly speaking, we use a specific matrix multiplication to perform the homomorphic multiplication, hence no tangly modulus problem. Furthermore, we also extend the<jats:sans-serif> LWR<\/jats:sans-serif>-based FHE scheme to the multikey setting using the tricks used to construct<jats:sans-serif> LWE<\/jats:sans-serif>-based multikey FHE by Mukherjee and Wichs at Eurocrypt 2016. Our<jats:sans-serif> LWR<\/jats:sans-serif>-based multikey FHE construction provides an alternative to the existing multikey FHEs and can also be applied to multiparty computation with higher efficiency.<\/jats:p>","DOI":"10.1155\/2018\/5967635","type":"journal-article","created":{"date-parts":[[2018,4,23]],"date-time":"2018-04-23T23:33:09Z","timestamp":1524526389000},"page":"1-12","source":"Crossref","is-referenced-by-count":13,"title":["LWR-Based Fully Homomorphic Encryption, Revisited"],"prefix":"10.1155","volume":"2018","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3082-4495","authenticated-orcid":true,"given":"Fucai","family":"Luo","sequence":"first","affiliation":[{"name":"School of Cyber Security, University of Chinese Academy of Sciences, Beijing, China"},{"name":"State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China"},{"name":"Data Assurance and Communication Security Research Center, Chinese Academy of Sciences, Beijing, China"}]},{"given":"Fuqun","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Science, Hangzhou Normal University, Hangzhou, China"},{"name":"Westone Cryptologic Research Center, Beijing, China"}]},{"given":"Kunpeng","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Cyber Security, University of Chinese Academy of Sciences, Beijing, China"},{"name":"State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China"},{"name":"Data Assurance and Communication Security Research Center, Chinese Academy of Sciences, Beijing, China"}]},{"given":"Jie","family":"Li","sequence":"additional","affiliation":[{"name":"School of Cyber Security, University of Chinese Academy of Sciences, Beijing, China"},{"name":"State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China"},{"name":"Data Assurance and Communication Security Research Center, Chinese Academy of Sciences, Beijing, China"}]},{"given":"Kefei","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Science, Hangzhou Normal University, Hangzhou, China"},{"name":"Westone Cryptologic Research Center, Beijing, China"}]}],"member":"311","reference":[{"issue":"11","key":"29","first-page":"169","volume":"4","year":"1978","journal-title":"Foundations of secure computation"},{"year":"2009","key":"18"},{"key":"31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-13190-5_2"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-22792-9_29"},{"key":"30","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-13013-7_25"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-46800-5_24"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-32009-5_50"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-49896-5_26"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-53644-5_9"},{"key":"28","doi-asserted-by":"publisher","DOI":"10.1145\/1568318.1568324"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-40041-4_5"},{"volume":"163","journal-title":"IACR Cryptology ePrint Archive","year":"2017","key":"16"},{"journal-title":"Advances in Cryptology\u2013EUROCRYPT 2012","first-page":"719","year":"2012","key":"5"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539793244708"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-40041-4_4"},{"issue":"589","key":"1","volume":"2016","year":"2016","journal-title":"IACR Cryptology ePrint Archive"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-29011-4_41"}],"container-title":["Security and Communication Networks"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/scn\/2018\/5967635.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/scn\/2018\/5967635.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/scn\/2018\/5967635.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,19]],"date-time":"2020-05-19T09:58:03Z","timestamp":1589882283000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/scn\/2018\/5967635\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018]]},"references-count":17,"alternative-id":["5967635","5967635"],"URL":"https:\/\/doi.org\/10.1155\/2018\/5967635","relation":{},"ISSN":["1939-0114","1939-0122"],"issn-type":[{"type":"print","value":"1939-0114"},{"type":"electronic","value":"1939-0122"}],"subject":[],"published":{"date-parts":[[2018]]}}}