{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T16:55:29Z","timestamp":1774976129129,"version":"3.50.1"},"publisher-location":"Cham","reference-count":34,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783031865985","type":"print"},{"value":"9783031865992","type":"electronic"}],"license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"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":[[2025]]},"DOI":"10.1007\/978-3-031-86599-2_1","type":"book-chapter","created":{"date-parts":[[2025,3,24]],"date-time":"2025-03-24T19:08:46Z","timestamp":1742843326000},"page":"3-34","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the\u00a0Structure of\u00a0the\u00a0Schur Squares of\u00a0Twisted Generalized Reed-Solomon Codes and\u00a0Application to\u00a0Cryptanalysis"],"prefix":"10.1007","author":[{"given":"Alain","family":"Couvreur","sequence":"first","affiliation":[]},{"given":"Rakhi","family":"Pratihar","sequence":"additional","affiliation":[]},{"given":"Nihan","family":"Tan\u0131sal\u0131","sequence":"additional","affiliation":[]},{"given":"Ilaria","family":"Zappatore","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,3,15]]},"reference":[{"key":"1_CR1","unstructured":"Albrecht, M., et al.: Classic McEliece (merger of Classic McEliece and NTS-KEM) (2022). https:\/\/classic.mceliece.org"},{"key":"1_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00145-014-9187-8","volume":"29","author":"M Baldi","year":"2016","unstructured":"Baldi, M., Bianchi, M., Chiaraluce, F., Rosenthal, J., Schipani, D.: Enhanced public key security for the McEliece cryptosystem. J. Cryptol. 29, 1\u201327 (2016)","journal-title":"J. Cryptol."},{"key":"1_CR3","doi-asserted-by":"crossref","unstructured":"Beelen, P., Bossert, M., Puchinger, S., Rosenkilde, J.: Structural properties of twisted Reed-Solomon codes with applications to cryptography. In: Proceedings of IEEE International Symposium on Information Theory - ISIT\u00a02018, pp. 946\u2013950. IEEE (2018)","DOI":"10.1109\/ISIT.2018.8437923"},{"key":"1_CR4","doi-asserted-by":"crossref","unstructured":"Beelen, P., Puchinger, S., n\u00e9\u00a0Nielsen, J.R.: Twisted Reed-Solomon codes. In: Proceedings of IEEE International Symposium on Information Theory - ISIT\u00a02017, pp. 336\u2013340. IEEE (2017)","DOI":"10.1109\/ISIT.2017.8006545"},{"issue":"5","key":"1_CR5","doi-asserted-by":"publisher","first-page":"3047","DOI":"10.1109\/TIT.2022.3146254","volume":"68","author":"P Beelen","year":"2022","unstructured":"Beelen, P., Puchinger, S., Rosenkilde, J.: Twisted Reed-Solomon codes. IEEE Trans. Inf. Theory 68(5), 3047\u20133061 (2022)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"1_CR6","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1080\/0025570X.2007.11953481","volume":"80","author":"AT Benjamin","year":"2007","unstructured":"Benjamin, A.T., Bennett, C.D.: The probability of relatively prime polynomials. Math. Mag. 80(3), 196\u2013202 (2007). https:\/\/doi.org\/10.1080\/0025570X.2007.11953481","journal-title":"Math. Mag."},{"key":"1_CR7","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/s10623-003-6151-2","volume":"35","author":"T Berger","year":"2005","unstructured":"Berger, T., Loidreau, P.: How to mask the structure of codes for a cryptographic use. Des. Codes Cryptogr. 35, 63\u201379 (2005)","journal-title":"Des. Codes Cryptogr."},{"key":"1_CR8","doi-asserted-by":"publisher","unstructured":"Cascudo, I., Cramer, R., Mirandola, D., Z\u00e9mor, G.: Squares of random linear codes. IEEE Trans. Inf. Theory 61(3), 1159\u20131173 (2015). https:\/\/doi.org\/10.1109\/TIT.2015.2393251","DOI":"10.1109\/TIT.2015.2393251"},{"key":"1_CR9","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1007\/s10623-014-9967-z","volume":"73","author":"A Couvreur","year":"2014","unstructured":"Couvreur, A., Gaborit, P., Gauthier-Uma\u00f1a, V., Otmani, A., Tillich, J.P.: Distinguisher-based attacks on public-key cryptosystems using Reed-Solomon codes. Des. Codes Cryptogr. 73, 641\u2013666 (2014)","journal-title":"Des. Codes Cryptogr."},{"issue":"1","key":"1_CR10","doi-asserted-by":"publisher","first-page":"632","DOI":"10.1109\/TIT.2021.3120440","volume":"68","author":"A Couvreur","year":"2022","unstructured":"Couvreur, A., Lequesne, M.: On the security of subspace subcodes of Reed-Solomon codes for public key encryption. IEEE Trans. Inf. Theory 68(1), 632\u2013648 (2022). https:\/\/doi.org\/10.1109\/TIT.2021.3120440","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1_CR11","series-title":"LNCS","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1007\/978-3-030-25510-7_8","volume-title":"Post-Quantum Cryptography 2019","author":"A Couvreur","year":"2019","unstructured":"Couvreur, A., Lequesne, M., Tillich, J.P.: Recovering short secret keys of RLCE in polynomial time. In: Ding, J., Steinwandt, R. (eds.) Post-Quantum Cryptography 2019. LNCS, vol. 11505, pp. 133\u2013152. Springer, Chongquing (2019). https:\/\/doi.org\/10.1007\/978-3-030-25510-7_8"},{"issue":"8","key":"1_CR12","doi-asserted-by":"publisher","first-page":"5404","DOI":"10.1109\/TIT.2017.2712636","volume":"63","author":"A Couvreur","year":"2017","unstructured":"Couvreur, A., M\u00e1rquez-Corbella, I., Pellikaan, R.: Cryptanalysis of McEliece cryptosystem based on algebraic geometry codes and their subcodes. IEEE Trans. Inf. Theory 63(8), 5404\u20135418 (2017)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1_CR13","series-title":"LNCS","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/978-3-642-55220-5_2","volume-title":"Advances in Cryptology - EUROCRYPT 2014","author":"A Couvreur","year":"2014","unstructured":"Couvreur, A., Otmani, A., Tillich, J.P.: Polynomial time attack on wild McEliece over quadratic extensions. In: Nguyen, P.Q., Oswald, E. (eds.) Advances in Cryptology - EUROCRYPT 2014. LNCS, vol. 8441, pp. 17\u201339. Springer, Heidelberg (2014). https:\/\/doi.org\/10.1007\/978-3-642-55220-5_2"},{"issue":"1","key":"1_CR14","doi-asserted-by":"publisher","first-page":"404","DOI":"10.1109\/TIT.2016.2574841","volume":"63","author":"A Couvreur","year":"2017","unstructured":"Couvreur, A., Otmani, A., Tillich, J.P.: Polynomial time attack on wild McEliece over quadratic extensions. IEEE Trans. Inf. Theory 63(1), 404\u2013427 (2017)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1_CR15","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/978-3-662-46447-2_8","volume-title":"Public-Key Cryptography \u2013 PKC 2015","author":"A Couvreur","year":"2015","unstructured":"Couvreur, A., Otmani, A., Tillich, J.-P., Gauthier\u2013Uma\u00f1a, V.: A polynomial-time attack on the BBCRS scheme. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 175\u2013193. Springer, Heidelberg (2015). https:\/\/doi.org\/10.1007\/978-3-662-46447-2_8"},{"issue":"10","key":"1_CR16","doi-asserted-by":"publisher","first-page":"6830","DOI":"10.1109\/TIT.2013.2272036","volume":"59","author":"JC Faug\u00e8re","year":"2013","unstructured":"Faug\u00e8re, J.C., Gauthier-Uma\u00f1a, V., Otmani, A., Perret, L., Tillich, J.P.: A distinguisher for high-rate McEliece cryptosystems. IEEE Trans. Inf. Theory 59(10), 6830\u20136844 (2013). https:\/\/doi.org\/10.1109\/TIT.2013.2272036","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1_CR17","doi-asserted-by":"publisher","unstructured":"Guruswami, V., Sudan, M.: Improved decoding of reed-solomon and algebraic-geometric codes. In: Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No. 98CB36280), pp. 28\u201337 (1998). https:\/\/doi.org\/10.1109\/SFCS.1998.743426","DOI":"10.1109\/SFCS.1998.743426"},{"issue":"3","key":"1_CR18","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1023\/A:1027351723034","volume":"8","author":"H Janwa","year":"1996","unstructured":"Janwa, H., Moreno, O.: McEliece public key cryptosystems using algebraic-geometric codes. Des. Codes Cryptogr. 8(3), 293\u2013307 (1996)","journal-title":"Des. Codes Cryptogr."},{"key":"1_CR19","doi-asserted-by":"publisher","unstructured":"Khathuria, K., Rosenthal, J., Weger, V.: Encryption scheme based on expanded Reed-Solomon codes. Adv. Math. Commun. 15(2), 207\u2013218 (2021). https:\/\/doi.org\/10.3934\/amc.2020053. http:\/\/aimsciences.org\/\/article\/id\/0f055199-6fe4-404f-b206-517ce7d02a58","DOI":"10.3934\/amc.2020053"},{"issue":"7","key":"1_CR20","doi-asserted-by":"publisher","first-page":"1285","DOI":"10.1007\/s10623-020-00747-6","volume":"88","author":"J Lavauzelle","year":"2020","unstructured":"Lavauzelle, J., Renner, J.: Cryptanalysis of a system based on twisted Reed-Solomon codes. Des. Codes Cryptogr. 88(7), 1285\u20131300 (2020)","journal-title":"Des. Codes Cryptogr."},{"key":"1_CR21","first-page":"85","volume":"3","author":"R McEliece","year":"1978","unstructured":"McEliece, R.: A public-key system based on algebraic coding theory. Jet Propul. Lab. California Inst. Technol. 3, 85\u201386 (1978)","journal-title":"Jet Propul. Lab. California Inst. Technol."},{"key":"1_CR22","doi-asserted-by":"publisher","unstructured":"Misoczki, R., Tillich, J.P., Sendrier, N., Barreto, P.S.L.M.: MDPC-McEliece: new McEliece variants from moderate density parity-check codes. In: Proceedings of IEEE International Symposium on Information Theory - ISIT, pp. 2069\u20132073 (2013). https:\/\/doi.org\/10.1109\/ISIT.2013.6620590","DOI":"10.1109\/ISIT.2013.6620590"},{"issue":"2","key":"1_CR23","first-page":"157","volume":"15","author":"H Niederreiter","year":"1986","unstructured":"Niederreiter, H.: Knapsack-type cryptosystems and algebraic coding theory. Prob. Contr. Inf. Theory 15(2), 157\u2013166 (1986)","journal-title":"Prob. Contr. Inf. Theory"},{"key":"1_CR24","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511808968","volume-title":"Introduction to Coding Theory","author":"R Roth","year":"2006","unstructured":"Roth, R.: Introduction to Coding Theory. Cambridge University Press, New York (2006)"},{"key":"1_CR25","doi-asserted-by":"publisher","unstructured":"Sheekey, J.: A new family of linear maximum rank distance codes. Adv. Math. Commun. 10(3), 475\u2013488 (2016). https:\/\/doi.org\/10.3934\/amc.2016019. https:\/\/www.aimsciences.org\/article\/id\/3ff311ae-18e0-47d8-8edc-ab169dbbd975","DOI":"10.3934\/amc.2016019"},{"issue":"3","key":"1_CR26","first-page":"191","volume":"4","author":"VM Sidelnikov","year":"1994","unstructured":"Sidelnikov, V.M.: A public-key cryptosytem based on Reed-Muller codes. Disc. Math. Appl. 4(3), 191\u2013207 (1994)","journal-title":"Disc. Math. Appl."},{"issue":"4","key":"1_CR27","first-page":"439","volume":"1","author":"VM Sidelnikov","year":"1992","unstructured":"Sidelnikov, V.M., Shestakov, S.: On the insecurity of cryptosystems based on generalized Reed-Solomon codes. Disc. Math. Appl. 1(4), 439\u2013444 (1992)","journal-title":"Disc. Math. Appl."},{"key":"1_CR28","unstructured":"Stein, W., et\u00a0al.: Sage Mathematics Software (Version 9.5). The Sage Development Team (2022). http:\/\/www.sagemath.org"},{"key":"1_CR29","doi-asserted-by":"publisher","unstructured":"Sudan, M.: Decoding of Reed-Solomon codes beyond the error-correction bound. J. Complex. 13(1), 180\u2013193 (1997). https:\/\/doi.org\/10.1006\/jcom.1997.0439. https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0885064X97904398","DOI":"10.1006\/jcom.1997.0439"},{"key":"1_CR30","doi-asserted-by":"publisher","unstructured":"Wang, Y.: Quantum resistant random linear code based public key encryption scheme RLCE. In: Proceedings of IEEE International Symposium on Information Theory - ISIT\u00a02016, pp. 2519\u20132523. IEEE, Barcelona (2016). https:\/\/doi.org\/10.1109\/ISIT.2016.7541753","DOI":"10.1109\/ISIT.2016.7541753"},{"key":"1_CR31","unstructured":"Wang, Y.: RLCE\u2013KEM (2017). http:\/\/quantumca.org"},{"key":"1_CR32","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1007\/11745853_2","volume-title":"Public Key Cryptography - PKC 2006","author":"C Wieschebrink","year":"2006","unstructured":"Wieschebrink, C.: An attack on a modified niederreiter encryption scheme. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) PKC 2006. LNCS, vol. 3958, pp. 14\u201326. Springer, Heidelberg (2006). https:\/\/doi.org\/10.1007\/11745853_2"},{"key":"1_CR33","doi-asserted-by":"crossref","unstructured":"Wieschebrink, C.: Two NP-complete problems in coding theory with an application in code based cryptography. In: Proceedings of IEEE International Symposium on Information Theory - ISIT\u00a02006, pp. 1733\u20131737. IEEE (2006)","DOI":"10.1109\/ISIT.2006.261651"},{"key":"1_CR34","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/978-3-642-12929-2_5","volume-title":"Post-Quantum Cryptography","author":"C Wieschebrink","year":"2010","unstructured":"Wieschebrink, C.: Cryptanalysis of the niederreiter public key scheme based on GRS subcodes. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 61\u201372. Springer, Heidelberg (2010). https:\/\/doi.org\/10.1007\/978-3-642-12929-2_5"}],"container-title":["Lecture Notes in Computer Science","Post-Quantum Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-86599-2_1","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,24]],"date-time":"2025-03-24T19:09:22Z","timestamp":1742843362000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-86599-2_1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"ISBN":["9783031865985","9783031865992"],"references-count":34,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-86599-2_1","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025]]},"assertion":[{"value":"15 March 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"PQCrypto","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Post-Quantum Cryptography","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Taipei","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Taiwan","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2025","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"9 April 2025","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"11 April 2025","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"16","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"pqcrypto2025","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/pqcrypto2025.iis.sinica.edu.tw\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}