{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T03:56:05Z","timestamp":1780545365280,"version":"3.54.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,1,6]],"date-time":"2020-01-06T00:00:00Z","timestamp":1578268800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,1,6]],"date-time":"2020-01-06T00:00:00Z","timestamp":1578268800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Cryptogr Eng"],"published-print":{"date-parts":[[2020,6]]},"DOI":"10.1007\/s13389-019-00221-7","type":"journal-article","created":{"date-parts":[[2020,2,3]],"date-time":"2020-02-03T14:29:19Z","timestamp":1580740159000},"page":"111-133","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Efficient modular operations using the adapted modular number system"],"prefix":"10.1007","volume":"10","author":[{"given":"Laurent-St\u00e9phane","family":"Didier","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fangan-Yssouf","family":"Dosso","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7610-6870","authenticated-orcid":false,"given":"Pascal","family":"V\u00e9ron","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,1,6]]},"reference":[{"key":"221_CR1","unstructured":"Abarz\u00faa, R., Valencia, C., L\u00f3pez, J.: Survey for performance and security problems of passive side-channel attacks countermeasures in ECC. Cryptology ePrint Archive, Report 2019\/010 (2019)"},{"issue":"5","key":"221_CR2","doi-asserted-by":"publisher","first-page":"629","DOI":"10.1093\/comjnl\/bxr119","volume":"55","author":"S Ant\u00e3o","year":"2012","unstructured":"Ant\u00e3o, S., Bajard, J.C., Sousa, L.: RNS based elliptic curve point multiplication for massive parallel architectures. Comput. J. 55(5), 629\u2013647 (2012)","journal-title":"Comput. J."},{"key":"221_CR3","doi-asserted-by":"crossref","unstructured":"Bajard, J.C., Duquesne, S., Ercegovac, M.: Combining leak-resistant arithmetic for elliptic curves defined over $$f_p$$. Publications Math\u00e9matiques de Besan\u00e7on. Algr\u00e8bre et Th\u00e9orie des Nombres, pp. 67\u201387 (2013). ISSN: 1958-7236","DOI":"10.5802\/pmb.a-152"},{"key":"221_CR4","doi-asserted-by":"crossref","unstructured":"Bajard, J.C., Eynard, J., Hasan, A., Zucca, V.: A full RNS variant of fv like somewhat homomorphic encryption schemes. In: SAC 2016, Selected Areas in Cryptography. St. John\u2019s, Newfoundland and Labrador, Canada (2016)","DOI":"10.1007\/978-3-319-69453-5_23"},{"issue":"6","key":"221_CR5","doi-asserted-by":"publisher","first-page":"769","DOI":"10.1109\/TC.2004.2","volume":"53","author":"JC Bajard","year":"2004","unstructured":"Bajard, J.C., Imbert, L.: A full RNS implementation of RSA. IEEE Trans. Comput. 53(6), 769\u2013774 (2004)","journal-title":"IEEE Trans. Comput."},{"key":"221_CR6","doi-asserted-by":"crossref","unstructured":"Bajard, J.C., Imbert, L., Liardet, P.Y., Teglia, Y.: Leak resistant arithmetic. In: Workshop on Cryptographic Hardware and Embedded Systems CHES 2004. Cambridge (Boston), USA. Lecture Notes in Computer Science, pp. 62\u201375. Springer (2004)","DOI":"10.1007\/978-3-540-28632-5_5"},{"key":"221_CR7","doi-asserted-by":"crossref","unstructured":"Bajard, J.C., Imbert, L., Plantard, T.: Modular number systems: beyond the Mersenne family. In: 11th International Workshop, Selected Areas in Cryptography, SAC 2004, Waterloo, Canada, pp. 159\u2013169 (2004)","DOI":"10.1007\/978-3-540-30564-4_11"},{"key":"221_CR8","unstructured":"Bajard, J.C., Imbert, L., Plantard, T.: Arithmetic operations in the polynomial modular number system. In: 17th IEEE Symposium on Computer Arithmetic (ARITH-17) 2005, Cape Cod, MA, USA, pp. 206\u2013213 (2005). Extended (complete) version: https:\/\/hal-lirmm.ccsd.cnrs.fr\/lirmm-00109201\/document. Accessed 2 Jan 2020"},{"key":"221_CR9","volume-title":"QC-LDPC Code-Based Cryptography. SpringerBriefs in Electrical and Computer Engineering","author":"M Baldi","year":"2014","unstructured":"Baldi, M.: QC-LDPC Code-Based Cryptography. SpringerBriefs in Electrical and Computer Engineering. Springer, Berlin (2014)"},{"key":"221_CR10","doi-asserted-by":"crossref","unstructured":"Barrett, P.: Implementing the Rivest Shamir and Adleman public key encryption algorithm on a standard digital signal processor. In: Odlyzko, A.M. (ed.) Advances in Cryptology\u2014CRYPTO\u201986, pp. 311\u2013323. Springer, Berlin (1987)","DOI":"10.1007\/3-540-47721-7_24"},{"key":"221_CR11","doi-asserted-by":"publisher","unstructured":"Didier, L.S., Dosso, F.Y., El\u00a0Mrabet, N., Marrez, J., V\u00e9ron, P.: Randomization of arithmetic over polynomial modular number system. In: 26th IEEE International Symposium on Computer Arithmetic, vol.\u00a01, pp. 199\u2013206. Kyoto, Japan (2019). https:\/\/doi.org\/10.1109\/ARITH.2019.00048","DOI":"10.1109\/ARITH.2019.00048"},{"key":"221_CR12","doi-asserted-by":"crossref","unstructured":"El Mrabet, N., Gama, N.: Efficient multiplication over extension fields. In: WAIFI. Lecture Notes in Computer Science, vol.\u00a07369, pp. 136\u2013151. Springer (2012)","DOI":"10.1007\/978-3-642-31662-3_10"},{"key":"221_CR13","doi-asserted-by":"crossref","unstructured":"El Mrabet, N., N\u00e8gre, C.: Finite field multiplication combining AMNS and DFT approach for pairing cryptography. In: ACISP. Lecture Notes in Computer Science, vol.\u00a05594, pp. 422\u2013436. Springer (2009)","DOI":"10.1007\/978-3-642-02620-1_29"},{"issue":"6","key":"221_CR14","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1109\/TEC.1959.5219515","volume":"EL 8","author":"HL Garner","year":"1959","unstructured":"Garner, H.L.: The residue number system. IRE Trans. Electr. Comput. EL 8(6), 140\u2013147 (1959)","journal-title":"IRE Trans. Electr. Comput."},{"issue":"5","key":"221_CR15","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1006\/jsco.1998.0228","volume":"26","author":"JVZ Gathen","year":"1998","unstructured":"Gathen, J.V.Z., Hartlieb, S.: Factoring modular polynomials. J. Symb. Comput. 26(5), 583\u2013606 (1998)","journal-title":"J. Symb. Comput."},{"key":"221_CR16","doi-asserted-by":"crossref","unstructured":"Goubin, L.: A refined power-analysis attack on elliptic curve cryptosystems. In: International Workshop on Public Key Cryptography, pp. 199\u2013211. Springer (2003)","DOI":"10.1007\/3-540-36288-6_15"},{"key":"221_CR17","unstructured":"Granlund, T., et al.: GNU multiple precision arithmetic library 6.1.2 (2016). https:\/\/gmplib.org\/. Accessed 2 Jan 2020"},{"key":"221_CR18","first-page":"308","volume":"109","author":"WG Horner","year":"1819","unstructured":"Horner, W.G.: A new method of solving numerical equations of all orders, by continuous approximation. Philos. Trans. R. Soc. Lond. 109, 308\u2013335 (1819)","journal-title":"Philos. Trans. R. Soc. Lond."},{"key":"221_CR19","unstructured":"Johnston, A.M.: A generalized qth root algorithm. In: Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms. SODA\u201999, Society for Industrial and Applied Mathematics, pp. 929\u2013930 (1999)"},{"key":"221_CR20","doi-asserted-by":"crossref","unstructured":"Joye, M., Tymen, C.: Protections against differential analysis for elliptic curve cryptography\u2014an algebraic approach. In: Ko\u00e7, \u00c7.K., Naccache, D., Paar, C. (eds.) Cryptographic Hardware and Embedded Systems\u2014CHES 2001, pp. 377\u2013390. Springer, Berlin (2001)","DOI":"10.1007\/3-540-44709-1_31"},{"key":"221_CR21","doi-asserted-by":"crossref","unstructured":"Kocher, P.C.: Timing attacks on implementations of Diffie\u2013Hellman, RSA, DSS, and other systems. In: Koblitz, N. (ed.) Advances in Cryptology\u2014CRYPTO\u201996, pp. 104\u2013113. Springer, Berlin (1996)","DOI":"10.1007\/3-540-68697-5_9"},{"key":"221_CR22","doi-asserted-by":"crossref","unstructured":"Kocher, P.C., Jaffe, J., Jun, B.: Differential power analysis. In: CRYPTO. Lecture Notes in Computer Science, vol.\u00a01666, pp. 388\u2013397. Springer (1999)","DOI":"10.1007\/3-540-48405-1_25"},{"issue":"4","key":"221_CR23","doi-asserted-by":"publisher","first-page":"515","DOI":"10.1007\/BF01457454","volume":"261","author":"AK Lenstra","year":"1982","unstructured":"Lenstra, A.K., Lenstra, H.W., Lov\u00e1sz, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261(4), 515\u2013534 (1982)","journal-title":"Mathematische Annalen"},{"key":"221_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4344-1","volume-title":"Algorithmic Algebra","author":"B Mishra","year":"1993","unstructured":"Mishra, B.: Algorithmic Algebra. Springer, Berlin (1993)"},{"issue":"170","key":"221_CR25","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1090\/S0025-5718-1985-0777282-X","volume":"44","author":"PL Montgomery","year":"1985","unstructured":"Montgomery, P.L.: Modular multiplication without trial division. Math. Comput. 44(170), 519\u2013521 (1985)","journal-title":"Math. Comput."},{"key":"221_CR26","doi-asserted-by":"crossref","unstructured":"Negre, C., Plantard, T.: Efficient modular arithmetic in adapted modular number system using Lagrange representation. In: Information Security and Privacy, 13th Australasian Conference, ACISP 2008, Wollongong, Australia, pp. 463\u2013477 (2008)","DOI":"10.1007\/978-3-540-70500-0_34"},{"key":"221_CR27","unstructured":"Plantard, T.: Arithm\u00e9tique modulaire pour la cryptographie. Ph.D. thesis, Montpellier 2 University, France (2005)"},{"key":"221_CR28","unstructured":"Project, T.O.: Openssl (1999). https:\/\/www.openssl.org\/. Accessed 2 Jan 2020"},{"key":"221_CR29","unstructured":"Shoup, V., et al.: NTL: a library for doing number theory (1990). https:\/\/www.shoup.net\/ntl\/. Accessed 2 Jan 2020"},{"key":"221_CR30","doi-asserted-by":"publisher","unstructured":"Solinas, J., Fu, D.E.: Elliptic curve groups modulo a prime (ECP Groups) for IKE and IKEv2. RFC 5903 (2010). https:\/\/doi.org\/10.17487\/RFC5903. https:\/\/rfc-editor.org\/rfc\/rfc5903.txt. Accessed 2 Jan 2020","DOI":"10.17487\/RFC5903"},{"key":"221_CR31","unstructured":"Stein, W., et al.: Sagemath (2005). http:\/\/www.sagemath.org\/index.html. Accessed 2 Jan 2020"}],"container-title":["Journal of Cryptographic Engineering"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s13389-019-00221-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s13389-019-00221-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s13389-019-00221-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,5]],"date-time":"2021-01-05T00:50:03Z","timestamp":1609807803000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s13389-019-00221-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,6]]},"references-count":31,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,6]]}},"alternative-id":["221"],"URL":"https:\/\/doi.org\/10.1007\/s13389-019-00221-7","relation":{},"ISSN":["2190-8508","2190-8516"],"issn-type":[{"value":"2190-8508","type":"print"},{"value":"2190-8516","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,6]]},"assertion":[{"value":"30 October 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 December 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 January 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}