{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T22:33:21Z","timestamp":1775774001611,"version":"3.50.1"},"publisher-location":"Berlin, Heidelberg","reference-count":34,"publisher":"Springer Berlin Heidelberg","isbn-type":[{"value":"9783642383472","type":"print"},{"value":"9783642383489","type":"electronic"}],"license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2013]]},"DOI":"10.1007\/978-3-642-38348-9_3","type":"book-chapter","created":{"date-parts":[[2013,5,8]],"date-time":"2013-05-08T20:46:38Z","timestamp":1368045998000},"page":"35-54","source":"Crossref","is-referenced-by-count":272,"title":["A Toolkit for Ring-LWE Cryptography"],"prefix":"10.1007","author":[{"given":"Vadim","family":"Lyubashevsky","sequence":"first","affiliation":[]},{"given":"Chris","family":"Peikert","sequence":"additional","affiliation":[]},{"given":"Oded","family":"Regev","sequence":"additional","affiliation":[]}],"member":"297","reference":[{"key":"3_CR1","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1007\/978-3-642-13190-5_28","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2010","author":"S. Agrawal","year":"2010","unstructured":"Agrawal, S., Boneh, D., Boyen, X.: Efficient lattice (H)IBE in the standard model. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol.\u00a06110, pp. 553\u2013572. Springer, Heidelberg (2010)"},{"issue":"3","key":"3_CR2","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1007\/s00224-010-9278-3","volume":"48","author":"J. Alwen","year":"2011","unstructured":"Alwen, J., Peikert, C.: Generating shorter bases for hard random lattices. Theory of Computing Systems\u00a048(3), 535\u2013553 (2011); Preliminary version in STACS 2009","journal-title":"Theory of Computing Systems"},{"key":"3_CR3","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1007\/978-3-642-03356-8_35","volume-title":"Advances in Cryptology - CRYPTO 2009","author":"B. Applebaum","year":"2009","unstructured":"Applebaum, B., Cash, D., Peikert, C., Sahai, A.: Fast cryptographic primitives and circular-secure encryption based on hard learning problems. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol.\u00a05677, pp. 595\u2013618. Springer, Heidelberg (2009)"},{"key":"3_CR4","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1007\/BFb0023990","volume-title":"STACS 85","author":"L. Babai","year":"1984","unstructured":"Babai, L.: On Lov\u00e1sz\u2019 lattice reduction and the nearest lattice point problem. Combinatorica 6(1), 1\u201313 (1986); Preliminary version in Mehlhorn, K. (ed.) STACS 1985. LNCS, vol.\u00a0182, pp. 13\u201320. Springer, Heidelberg (1984)"},{"key":"3_CR5","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"719","DOI":"10.1007\/978-3-642-29011-4_42","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2012","author":"A. Banerjee","year":"2012","unstructured":"Banerjee, A., Peikert, C., Rosen, A.: Pseudorandom functions and lattices. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol.\u00a07237, pp. 719\u2013737. Springer, Heidelberg (2012)"},{"key":"3_CR6","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/BF01810296","volume":"1","author":"W. Bosma","year":"1990","unstructured":"Bosma, W.: Canonical bases for cyclotomic fields. Appl. Algebra Eng. Commun. Comput.\u00a01, 125\u2013134 (1990)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"3_CR7","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1007\/978-3-642-13013-7_29","volume-title":"Public Key Cryptography \u2013 PKC 2010","author":"X. Boyen","year":"2010","unstructured":"Boyen, X.: Lattice mixing and vanishing trapdoors: A framework for fully secure short signatures and more. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol.\u00a06056, pp. 499\u2013517. Springer, Heidelberg (2010)"},{"key":"3_CR8","doi-asserted-by":"crossref","unstructured":"Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: ICTS, pp. 309\u2013325 (2012)","DOI":"10.1145\/2090236.2090262"},{"key":"3_CR9","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1007\/978-3-642-22792-9_29","volume-title":"Advances in Cryptology \u2013 CRYPTO 2011","author":"Z. Brakerski","year":"2011","unstructured":"Brakerski, Z., Vaikuntanathan, V.: Fully homomorphic encryption from ring-LWE and security for key dependent messages. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol.\u00a06841, pp. 505\u2013524. Springer, Heidelberg (2011)"},{"key":"3_CR10","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/978-3-642-13190-5_27","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2010","author":"D. Cash","year":"2010","unstructured":"Cash, D., Hofheinz, D., Kiltz, E., Peikert, C.: Bonsai trees, or how to delegate a lattice basis. J. Cryptology 25(4), 601\u2013639(2010); Preliminary version in Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol.\u00a06110, pp. 523\u2013552. Springer, Heidelberg (2010)"},{"key":"3_CR11","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1007\/978-3-642-30057-8_3","volume-title":"Public Key Cryptography \u2013 PKC 2012","author":"L. Ducas","year":"2012","unstructured":"Ducas, L., Durmus, A.: Ring-LWE in polynomial rings. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol.\u00a07293, pp. 34\u201351. Springer, Heidelberg (2012)"},{"issue":"2","key":"3_CR12","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1090\/S0002-9904-1946-08538-9","volume":"52","author":"P. Erd\u00f6s","year":"1946","unstructured":"Erd\u00f6s, P.: On the coefficients of the cyclotomic polynomial. Bulletin of the American Mathematical Society\u00a052(2), 179\u2013184 (1946)","journal-title":"Bulletin of the American Mathematical Society"},{"key":"3_CR13","doi-asserted-by":"crossref","unstructured":"Gentry, C.: A fully homomorphic encryption scheme. PhD thesis, Stanford University (2009), \n                  \n                    http:\/\/crypto.stanford.edu\/craig","DOI":"10.1145\/1536414.1536440"},{"key":"3_CR14","doi-asserted-by":"crossref","unstructured":"Gentry, C.: Fully homomorphic encryption using ideal lattices. In: STOC, pp. 169\u2013178 (2009)","DOI":"10.1145\/1536414.1536440"},{"key":"3_CR15","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1007\/978-3-642-14623-7_7","volume-title":"Advances in Cryptology \u2013 CRYPTO 2010","author":"C. Gentry","year":"2010","unstructured":"Gentry, C.: Toward basing fully homomorphic encryption on worst-case hardness. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol.\u00a06223, pp. 116\u2013137. Springer, Heidelberg (2010)"},{"key":"3_CR16","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1007\/978-3-642-32928-9_2","volume-title":"Security and Cryptography for Networks","author":"C. Gentry","year":"2012","unstructured":"Gentry, C., Halevi, S., Peikert, C., Smart, N.P.: Ring switching in BGV-style homomorphic encryption. In: Visconti, I., De Prisco, R. (eds.) SCN 2012. LNCS, vol.\u00a07485, pp. 19\u201337. Springer, Heidelberg (2012), Full version at \n                  \n                    http:\/\/eprint.iacr.org\/2012\/240"},{"key":"3_CR17","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1007\/978-3-642-29011-4_28","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2012","author":"C. Gentry","year":"2012","unstructured":"Gentry, C., Halevi, S., Smart, N.P.: Fully homomorphic encryption with polylog overhead. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol.\u00a07237, pp. 465\u2013482. Springer, Heidelberg (2012)"},{"key":"3_CR18","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1007\/978-3-642-32009-5_49","volume-title":"CRYPTO","author":"C. Gentry","year":"2012","unstructured":"Gentry, C., Halevi, S., Smart, N.P.: Homomorphic evaluation of the AES circuit. In: Safavi-Naini, R. (ed.) CRYPTO 2012. LNCS, vol.\u00a07417, pp. 850\u2013867. Springer, Heidelberg (2012)"},{"key":"3_CR19","doi-asserted-by":"crossref","unstructured":"Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: STOC, pp. 197\u2013206 (2008)","DOI":"10.1145\/1374376.1374407"},{"key":"3_CR20","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"530","DOI":"10.1007\/978-3-642-33027-8_31","volume-title":"Cryptographic Hardware and Embedded Systems \u2013 CHES 2012","author":"T. G\u00fcneysu","year":"2012","unstructured":"G\u00fcneysu, T., Lyubashevsky, V., P\u00f6ppelmann, T.: Practical lattice-based cryptography: A signature scheme for embedded systems. In: Prouff, E., Schaumont, P. (eds.) CHES 2012. LNCS, vol.\u00a07428, pp. 530\u2013547. Springer, Heidelberg (2012)"},{"key":"3_CR21","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"598","DOI":"10.1007\/978-3-642-10366-7_35","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2009","author":"V. Lyubashevsky","year":"2009","unstructured":"Lyubashevsky, V.: Fiat-Shamir with aborts: Applications to lattice and factoring-based signatures. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol.\u00a05912, pp. 598\u2013616. Springer, Heidelberg (2009)"},{"key":"3_CR22","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"738","DOI":"10.1007\/978-3-642-29011-4_43","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2012","author":"V. Lyubashevsky","year":"2012","unstructured":"Lyubashevsky, V.: Lattice signatures without trapdoors. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol.\u00a07237, pp. 738\u2013755. Springer, Heidelberg (2012)"},{"key":"3_CR23","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"144","DOI":"10.1007\/11787006_13","volume-title":"Automata, Languages and Programming","author":"V. Lyubashevsky","year":"2006","unstructured":"Lyubashevsky, V., Micciancio, D.: Generalized compact knapsacks are collision resistant. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006, Part II. LNCS, vol.\u00a04052, pp. 144\u2013155. Springer, Heidelberg (2006)"},{"key":"3_CR24","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/978-3-540-78524-8_3","volume-title":"Theory of Cryptography","author":"V. Lyubashevsky","year":"2008","unstructured":"Lyubashevsky, V., Micciancio, D.: Asymptotically efficient lattice-based digital signatures. In: Canetti, R. (ed.) TCC 2008. LNCS, vol.\u00a04948, pp. 37\u201354. Springer, Heidelberg (2008)"},{"key":"3_CR25","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1007\/978-3-540-71039-4_4","volume-title":"Fast Software Encryption","author":"V. Lyubashevsky","year":"2008","unstructured":"Lyubashevsky, V., Micciancio, D., Peikert, C., Rosen, A.: SWIFFT: A modest proposal for FFT hashing. In: Nyberg, K. (ed.) FSE 2008. LNCS, vol.\u00a05086, pp. 54\u201372. Springer, Heidelberg (2008)"},{"key":"3_CR26","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-642-13190-5_1","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2010","author":"V. Lyubashevsky","year":"2010","unstructured":"Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol.\u00a06110, pp. 1\u201323. Springer, Heidelberg (2010)"},{"issue":"4","key":"3_CR27","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/s00037-007-0234-9","volume":"16","author":"D. Micciancio","year":"2002","unstructured":"Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and efficient one-way functions. Computational Complexity\u00a016(4), 365\u2013411 (2002); Preliminary version in FOCS 2002","journal-title":"Computational Complexity"},{"key":"3_CR28","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"700","DOI":"10.1007\/978-3-642-29011-4_41","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2012","author":"D. Micciancio","year":"2012","unstructured":"Micciancio, D., Peikert, C.: Trapdoors for lattices: Simpler, tighter, faster, smaller. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol.\u00a07237, pp. 700\u2013718. Springer, Heidelberg (2012)"},{"key":"3_CR29","doi-asserted-by":"crossref","unstructured":"Peikert, C.: Public-key cryptosystems from the worst-case shortest vector problem. In: STOC, pp. 333\u2013342 (2009)","DOI":"10.1145\/1536414.1536461"},{"key":"3_CR30","doi-asserted-by":"crossref","unstructured":"Peikert, C., Rosen, A.: Lattices that admit logarithmic worst-case to average-case connection factors. In: STOC, pp. 478\u2013487 (2007)","DOI":"10.1145\/1250790.1250860"},{"issue":"6","key":"3_CR31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1568318.1568324","volume":"56","author":"O. Regev","year":"2005","unstructured":"Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM\u00a056(6), 1\u201340 (2005); Preliminary version in STOC","journal-title":"J. ACM"},{"key":"3_CR32","unstructured":"Smart, N.P., Vercauteren, F.: Fully homomorphic SIMD operations. Cryptology ePrint Archive, Report 2011\/133 (2011), \n                  \n                    http:\/\/eprint.iacr.org\/"},{"key":"3_CR33","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/978-3-642-20465-4_4","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2011","author":"D. Stehl\u00e9","year":"2011","unstructured":"Stehl\u00e9, D., Steinfeld, R.: Making NTRU as secure as worst-case problems over ideal lattices. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol.\u00a06632, pp. 27\u201347. Springer, Heidelberg (2011)"},{"key":"3_CR34","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1007\/978-3-642-10366-7_36","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2009","author":"D. Stehl\u00e9","year":"2009","unstructured":"Stehl\u00e9, D., Steinfeld, R., Tanaka, K., Xagawa, K.: Efficient public key encryption based on ideal lattices. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol.\u00a05912, pp. 617\u2013635. Springer, Heidelberg (2009)"}],"container-title":["Lecture Notes in Computer Science","Advances in Cryptology \u2013 EUROCRYPT 2013"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-642-38348-9_3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,19]],"date-time":"2019-05-19T21:42:05Z","timestamp":1558302125000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-642-38348-9_3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"ISBN":["9783642383472","9783642383489"],"references-count":34,"URL":"https:\/\/doi.org\/10.1007\/978-3-642-38348-9_3","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013]]}}}