{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,7]],"date-time":"2026-05-07T04:27:30Z","timestamp":1778128050335,"version":"3.51.4"},"publisher-location":"Cham","reference-count":29,"publisher":"Springer International Publishing","isbn-type":[{"value":"9783319636962","type":"print"},{"value":"9783319636979","type":"electronic"}],"license":[{"start":{"date-parts":[[2017,1,1]],"date-time":"2017-01-01T00:00:00Z","timestamp":1483228800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2017,1,1]],"date-time":"2017-01-01T00:00:00Z","timestamp":1483228800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017]]},"DOI":"10.1007\/978-3-319-63697-9_10","type":"book-chapter","created":{"date-parts":[[2017,8,1]],"date-time":"2017-08-01T04:07:12Z","timestamp":1501560432000},"page":"283-297","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["Middle-Product Learning with Errors"],"prefix":"10.1007","author":[{"given":"Miruna","family":"Ro\u015fca","sequence":"first","affiliation":[]},{"given":"Amin","family":"Sakzad","sequence":"additional","affiliation":[]},{"given":"Damien","family":"Stehl\u00e9","sequence":"additional","affiliation":[]},{"given":"Ron","family":"Steinfeld","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2017,8,2]]},"reference":[{"key":"10_CR1","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. 5677, pp. 595\u2013618. Springer, Heidelberg (2009). doi:10.1007\/978-3-642-03356-8_35"},{"key":"10_CR2","unstructured":"Alkim, E., Ducas, L., P\u00f6ppelmann, T., Schwabe, P.: Post-quantum key exchange - A new hope. In: Proceeding of USENIX, pp. 327\u2013343 (2016)"},{"key":"10_CR3","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1007\/978-3-642-22006-7_34","volume-title":"Automata, Languages and Programming","author":"S Arora","year":"2011","unstructured":"Arora, S., Ge, R.: New algorithms for learning in presence of errors. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 403\u2013415. Springer, Heidelberg (2011). doi:10.1007\/978-3-642-22006-7_34"},{"key":"10_CR4","doi-asserted-by":"crossref","unstructured":"Ajtai, M.: Generating hard instances of lattice problems (extended abstract). In: Proceeding of STOC, pp. 99\u2013108. ACM (1996)","DOI":"10.1145\/237814.237838"},{"key":"10_CR5","unstructured":"Bernstein, D.J., Chuengsatiansup, C., Lange, T., van Vredendaal, C.: NTRU Prime. Cryptology ePrint Archive (2016). http:\/\/eprint.iacr.org\/2016\/461"},{"key":"10_CR6","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1007\/978-3-662-49896-5_20","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2016","author":"R Cramer","year":"2016","unstructured":"Cramer, R., Ducas, L., Peikert, C., Regev, O.: Recovering short generators of principal ideals in cyclotomic rings. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 559\u2013585. Springer, Heidelberg (2016). doi:10.1007\/978-3-662-49896-5_20"},{"key":"10_CR7","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"324","DOI":"10.1007\/978-3-319-56620-7_12","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2017","author":"R Cramer","year":"2017","unstructured":"Cramer, R., Ducas, L., Wesolowski, B.: Short stickelberger class relations and application to ideal-SVP. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 324\u2013348. Springer, Cham (2017). doi:10.1007\/978-3-319-56620-7_12"},{"key":"10_CR8","unstructured":"Castryck, W., Iliashenko, I., Vercauteren, F.: On the tightness of the error bound in Ring-LWE. LMS J. Comput. Math. (2016)"},{"key":"10_CR9","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/978-3-662-49890-3_6","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2016","author":"W Castryck","year":"2016","unstructured":"Castryck, W., Iliashenko, I., Vercauteren, F.: Provably weak instances of ring-LWE revisited. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9665, pp. 147\u2013167. Springer, Heidelberg (2016). doi:10.1007\/978-3-662-49890-3_6"},{"key":"10_CR10","unstructured":"Chen, H., Lauter, K., Stange, K.E.: Attacks on search RLWE. Cryptology ePrint Archive (2015). http:\/\/eprint.iacr.org\/2015\/971"},{"key":"10_CR11","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/978-3-319-13051-4_11","volume-title":"Selected Areas in Cryptography \u2013 SAC 2014","author":"K Eisentr\u00e4ger","year":"2014","unstructured":"Eisentr\u00e4ger, K., Hallgren, S., Lauter, K.: Weak instances of PLWE. In: Joux, A., Youssef, A. (eds.) SAC 2014. LNCS, vol. 8781, pp. 183\u2013194. Springer, Cham (2014). doi:10.1007\/978-3-319-13051-4_11"},{"key":"10_CR12","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/978-3-662-47989-6_4","volume-title":"Advances in Cryptology \u2013 CRYPTO 2015","author":"Y Elias","year":"2015","unstructured":"Elias, Y., Lauter, K.E., Ozman, E., Stange, K.E.: Provably weak instances of ring-LWE. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 63\u201392. Springer, Heidelberg (2015). doi:10.1007\/978-3-662-47989-6_4"},{"key":"10_CR13","doi-asserted-by":"crossref","unstructured":"Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Proceeding of STOC, pp. 197\u2013206. ACM (2008)","DOI":"10.1145\/1374376.1374407"},{"key":"10_CR14","unstructured":"Lauter, K., Chen, H., Stange, K.E.: Vulnerable Galois RLWE families and improved attacks. Cryptology ePrint Archive (2016). http:\/\/eprint.iacr.org\/2016\/193"},{"key":"10_CR15","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1007\/BFb0054868","volume-title":"Algorithmic Number Theory","author":"J Hoffstein","year":"1998","unstructured":"Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: A ring-based public key cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267\u2013288. Springer, Heidelberg (1998). doi:10.1007\/BFb0054868"},{"issue":"6","key":"10_CR16","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1007\/s00200-003-0144-2","volume":"14","author":"G Hanrot","year":"2004","unstructured":"Hanrot, G., Quercia, M., Zimmermann, P.: The middle product algorithm I. Appl. Algebra Engrg. Comm. Comput. 14(6), 415\u2013438 (2004)","journal-title":"Appl. Algebra Engrg. Comm. Comput."},{"key":"10_CR17","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. LNCS, vol. 4052, pp. 144\u2013155. Springer, Heidelberg (2006). doi:10.1007\/11787006_13"},{"issue":"6","key":"10_CR18","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1145\/2535925","volume":"60","author":"V Lyubashevsky","year":"2013","unstructured":"Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. J. ACM 60(6), 43 (2013)","journal-title":"J. ACM"},{"key":"10_CR19","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. 5912, pp. 598\u2013616. Springer, Heidelberg (2009). doi:10.1007\/978-3-642-10366-7_35"},{"key":"10_CR20","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1007\/978-3-662-53890-6_7","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2016","author":"V Lyubashevsky","year":"2016","unstructured":"Lyubashevsky, V.: Digital signatures based on the hardness of ideal lattice problems in all rings. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10032, pp. 196\u2013214. Springer, Heidelberg (2016). doi:10.1007\/978-3-662-53890-6_7"},{"issue":"4","key":"10_CR21","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/s00037-007-0234-9","volume":"16","author":"D Micciancio","year":"2007","unstructured":"Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and efficient one-way functions. Comput. Complex. 16(4), 365\u2013411 (2007)","journal-title":"Comput. Complex."},{"key":"10_CR22","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. 7237, pp. 700\u2013718. Springer, Heidelberg (2012). doi:10.1007\/978-3-642-29011-4_41"},{"key":"10_CR23","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1007\/978-3-319-44618-9_22","volume-title":"Security and Cryptography for Networks","author":"C Peikert","year":"2016","unstructured":"Peikert, C.: How (Not) to instantiate ring-LWE. In: Zikas, V., Prisco, R. (eds.) SCN 2016. LNCS, vol. 9841, pp. 411\u2013430. Springer, Cham (2016). doi:10.1007\/978-3-319-44618-9_22"},{"key":"10_CR24","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1007\/11681878_8","volume-title":"Theory of Cryptography","author":"C Peikert","year":"2006","unstructured":"Peikert, C., Rosen, A.: Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 145\u2013166. Springer, Heidelberg (2006). doi:10.1007\/11681878_8"},{"key":"10_CR25","doi-asserted-by":"crossref","unstructured":"Peikert, C., Regev, O., Stephens-Davidowitz, N.: Pseudorandomness of Ring-LWE for any ring and modulus. In: Proceeding of STOC. ACM (2017)","DOI":"10.1145\/3055399.3055489"},{"issue":"6","key":"10_CR26","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1145\/1568318.1568324","volume":"56","author":"O Regev","year":"2009","unstructured":"Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM 56(6), 34 (2009)","journal-title":"J. ACM"},{"issue":"2\u20133","key":"10_CR27","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1016\/0304-3975(87)90064-8","volume":"53","author":"CP Schnorr","year":"1987","unstructured":"Schnorr, C.P.: A hierarchy of polynomial time lattice basis reduction algorithms. Theor. Comput. Sci. 53(2\u20133), 201\u2013224 (1987)","journal-title":"Theor. Comput. Sci."},{"key":"10_CR28","doi-asserted-by":"crossref","unstructured":"Shoup, V.: Efficient computation of minimal polynomials in algebraic extensions of finite fields. In: Proceeding of ISSAC, pp. 53\u201358. ACM (1999)","DOI":"10.1145\/309831.309859"},{"key":"10_CR29","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. 5912, pp. 617\u2013635. Springer, Heidelberg (2009). doi:10.1007\/978-3-642-10366-7_36"}],"container-title":["Lecture Notes in Computer Science","Advances in Cryptology \u2013 CRYPTO 2017"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-319-63697-9_10","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,13]],"date-time":"2024-03-13T20:07:30Z","timestamp":1710360450000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-319-63697-9_10"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017]]},"ISBN":["9783319636962","9783319636979"],"references-count":29,"URL":"https:\/\/doi.org\/10.1007\/978-3-319-63697-9_10","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017]]},"assertion":[{"value":"2 August 2017","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"CRYPTO","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Annual International Cryptology Conference","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Santa Barbara","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"USA","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2017","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"20 August 2017","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"24 August 2017","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"37","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"crypto2017","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/www.iacr.org\/conferences\/crypto2017\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}