{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:31:27Z","timestamp":1764981087257,"version":"3.46.0"},"reference-count":12,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2020,11,17]],"date-time":"2020-11-17T00:00:00Z","timestamp":1605571200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,11,17]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    A statistical framework applicable to Ring-LWE was outlined by Murphy and Player (IACR eprint 2019\/452). Its applicability was demonstrated with an analysis of the decryption failure probability for degree-1 and degree-2 ciphertexts in the homomorphic encryption scheme of Lyubashevsky, Peikert and Regev (IACR eprint 2013\/293). In this paper, we clarify and extend results presented by Murphy and Player. Firstly, we make precise the approximation of the discretisation of a Normal random variable as a Normal random variable, as used in the encryption process of Lyubashevsky, Peikert and Regev. Secondly, we show how to extend the analysis given by Murphy and Player to degree-\n                    <jats:italic>k<\/jats:italic>\n                    ciphertexts, by precisely characterising the distribution of the noise in these ciphertexts.\n                  <\/jats:p>","DOI":"10.1515\/jmc-2020-0073","type":"journal-article","created":{"date-parts":[[2020,11,30]],"date-time":"2020-11-30T15:54:49Z","timestamp":1606751689000},"page":"45-59","source":"Crossref","is-referenced-by-count":5,"title":["Discretisation and Product Distributions in Ring-LWE"],"prefix":"10.1515","volume":"15","author":[{"given":"Sean","family":"Murphy","sequence":"first","affiliation":[{"name":"Royal Holloway, University of London , Egham , United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rachel","family":"Player","sequence":"additional","affiliation":[{"name":"Royal Holloway, University of London , Egham , United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2020,11,17]]},"reference":[{"key":"2025120600293862477_j_jmc-2020-0073_ref_001","unstructured":"M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions Dover Publications, 1965."},{"key":"2025120600293862477_j_jmc-2020-0073_ref_002","unstructured":"R. Askey and A. Daalhuis and A. Olde, Meijer G-function NIST Handbook of Mathematical Functions (F. Olver et al. ed.), Cambridge University Press, 2010."},{"key":"2025120600293862477_j_jmc-2020-0073_ref_003","unstructured":"H. Bateman and A. Erd\u00e9lyi, Higher Transcendental Functions 1, McGraw-Hill, 1953."},{"key":"2025120600293862477_j_jmc-2020-0073_ref_004","doi-asserted-by":"crossref","unstructured":"R. Beals and J. Szmiglieski, Meijer G-Functions: A Gentle Introduction, Notices Amer. Math. Soc. 60 (2013), 886\u2013872.","DOI":"10.1090\/noti1016"},{"key":"2025120600293862477_j_jmc-2020-0073_ref_005","doi-asserted-by":"crossref","unstructured":"C. Gentry, Fully Homomorphic Encryption using Ideal Lattices, in: 41st Annual ACM Symposium on Theory of Computing, STOC 2009 Proceedings, ACM, (2009), 169\u2013178.","DOI":"10.1145\/1536414.1536440"},{"key":"2025120600293862477_j_jmc-2020-0073_ref_006","doi-asserted-by":"crossref","unstructured":"V. Lyubashevsky and C. Peikert and O. Regev, On Ideal Lattices and Learning with Errors over Rings, in: Advances in Cryptology - EUROCRYPT 2010 Lecture Notes in Comput. Sci. 6110, Springer, (2010), 1\u201323.","DOI":"10.1007\/978-3-642-13190-5_1"},{"key":"2025120600293862477_j_jmc-2020-0073_ref_007","doi-asserted-by":"crossref","unstructured":"V. Lyubashevsky and C. Peikert and O. Regev, A Toolkit for Ring-LWE Cryptography preprint (2013), https:\/\/eprint.iacr.org\/2013\/293","DOI":"10.1007\/978-3-642-38348-9_3"},{"key":"2025120600293862477_j_jmc-2020-0073_ref_008","doi-asserted-by":"crossref","unstructured":"V. Lyubashevsky and C. Peikert and O. Regev, A Toolkit for Ring-LWE Cryptography, in: Advances in Cryptology - EUROCRYPT 2013 Lecture Notes in Comput. Sci. 7881, Springer, (2013), 35\u201354.","DOI":"10.1007\/978-3-642-38348-9_3"},{"key":"2025120600293862477_j_jmc-2020-0073_ref_009","doi-asserted-by":"crossref","unstructured":"D. Micciancio and C. Peikert, Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller, in: Advances in Cryptology - EUROCRYPT 2012 Lecture Notes in Comput. Sci. 7237, Springer, (2012), 700\u2013718.","DOI":"10.1007\/978-3-642-29011-4_41"},{"key":"2025120600293862477_j_jmc-2020-0073_ref_010","doi-asserted-by":"crossref","unstructured":"S. Murphy and R. Player, -subgaussian Random Variables in Cryptography in Information Security and Privacy \u2013 24th Australasian Conference, ACISP 2019, Lecture Notes in Computing. Sci. 11547, Springer, (2019), 251\u2013268.","DOI":"10.1007\/978-3-030-21548-4_14"},{"key":"2025120600293862477_j_jmc-2020-0073_ref_011","unstructured":"S. Murphy and R. Player, A Central Limit Framework for Ring-LWE Decryption preprint (2019), https:\/\/eprint.iacr.org\/2019\/452"},{"key":"2025120600293862477_j_jmc-2020-0073_ref_012","doi-asserted-by":"crossref","unstructured":"D. Stehl\u00e9 and R. Steinfeld and K. Tanaka and K. Xagawa, Eflcient Public Key Encryption Based on Ideal Lattices, in: Advances in Cryptology - ASIACRYPT 2009 Lecture Notes in Comput. 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