{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T00:06:37Z","timestamp":1764115597893,"version":"3.46.0"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T00:00:00Z","timestamp":1764115200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T00:00:00Z","timestamp":1764115200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Academicians in Chongqing lead the scientific and technological innovation guidance project","award":["2022YSZX-JCX0011CSTB,CSTB2023YSZX-JCX0008"],"award-info":[{"award-number":["2022YSZX-JCX0011CSTB,CSTB2023YSZX-JCX0008"]}]},{"name":"National Key Research and Development Project","award":["2020YFA0712300"],"award-info":[{"award-number":["2020YFA0712300"]}]},{"name":"Chongqing Talents Plan Youth Top-Notch Project","award":["2021000263"],"award-info":[{"award-number":["2021000263"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cybersecurity"],"abstract":"Abstract<\/jats:title>\n \n In the case of standard LWE samples\n \n \n $$({\\textbf {A}},{\\textbf {b = sA + e}})$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n A<\/mml:mi>\n ,<\/mml:mo>\n \n b<\/mml:mi>\n =<\/mml:mo>\n sA<\/mml:mi>\n +<\/mml:mo>\n e<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ,\n \n \n $${\\textbf {A}}$$<\/jats:tex-math>\n \n A<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is typically uniformly over\n \n \n $$\\mathbb {Z}_q^{n \\times m}$$<\/jats:tex-math>\n \n \n Z<\/mml:mi>\n q<\/mml:mi>\n \n n<\/mml:mi>\n \u00d7<\/mml:mo>\n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Under the\n \n \n $$\\textsf {DLWE}$$<\/jats:tex-math>\n \n DLWE<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n assumption, the conditional distribution of\n \n \n $${\\textbf {s}}|({\\textbf {A}}, {\\textbf {b}})$$<\/jats:tex-math>\n \n \n s<\/mml:mi>\n |<\/mml:mo>\n (<\/mml:mo>\n A<\/mml:mi>\n ,<\/mml:mo>\n b<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $${\\textbf {s}}$$<\/jats:tex-math>\n \n s<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is expected to be consistent. However, in the case where an adversary chooses\n \n \n $${\\textbf {A}}$$<\/jats:tex-math>\n \n A<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n adaptively, the disparity between the two entities may be larger. In this work, our primary focus is on the quantification of the Average Conditional Min-Entropy\n \n \n $$\\tilde{H}_\\infty ({\\textbf {s}}|{\\textbf {sA + e}})$$<\/jats:tex-math>\n \n \n \n \n H<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n \u221e<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n s<\/mml:mi>\n |<\/mml:mo>\n \n sA<\/mml:mi>\n +<\/mml:mo>\n e<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of\n \n \n $${\\textbf {s}}$$<\/jats:tex-math>\n \n s<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n $${\\textbf {A}}$$<\/jats:tex-math>\n \n A<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is chosen by the adversary. Brakerski and D\u00f6ttling answered the question in one case: they proved that when\n \n \n $${\\textbf {s}}$$<\/jats:tex-math>\n \n s<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is uniformly chosen from\n \n \n $$\\mathbb {Z}_q^n$$<\/jats:tex-math>\n \n \n Z<\/mml:mi>\n q<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , it holds that\n \n \n $$\\tilde{H}_\\infty ({\\textbf {s}}|{\\textbf {sA + e}}) \\varpropto \\rho _\\sigma (\\varLambda _q({\\textbf {A}}))$$<\/jats:tex-math>\n \n \n \n \n H<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n \u221e<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n s<\/mml:mi>\n |<\/mml:mo>\n \n sA<\/mml:mi>\n +<\/mml:mo>\n e<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \u221d<\/mml:mo>\n \n \u03c1<\/mml:mi>\n \u03c3<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n \u039b<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n A<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We prove that for any\n \n \n $$d \\le q$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n \u2264<\/mml:mo>\n q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , when\n \n \n $${\\textbf {s}}$$<\/jats:tex-math>\n \n s<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is uniformly chosen from\n \n \n $$\\mathbb {Z}_d^n$$<\/jats:tex-math>\n \n \n Z<\/mml:mi>\n d<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n or is sampled from a discrete Gaussian distribution, there are also similar results. As an application of the above results, we improved the multi-key fully homomorphic encryption and answered the question raised at the end of their work positively: we have GSW-type ciphertext rather than Dual-GSW, and the improved scheme has shorter keys and ciphertexts.\n <\/jats:p>","DOI":"10.1186\/s42400-025-00439-5","type":"journal-article","created":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T00:02:31Z","timestamp":1764115351000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lattice-based, LWE-leakage model for Gaussian and uniform secret and its application in decentralization"],"prefix":"10.1186","volume":"8","author":[{"given":"Xiaokang","family":"Dai","sequence":"first","affiliation":[]},{"given":"Jingwei","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Wenyuan","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Yong","family":"Feng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,11,26]]},"reference":[{"key":"439_CR1","doi-asserted-by":"crossref","unstructured":"Alperin-Sheriff J, Peikert C (2014) Faster bootstrapping with polynomial error. 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