{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T08:06:00Z","timestamp":1774944360926,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,5,20]],"date-time":"2025-05-20T00:00:00Z","timestamp":1747699200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,5,20]],"date-time":"2025-05-20T00:00:00Z","timestamp":1747699200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Quantum Science Strategic Project of Guangdong Province","award":["GDZX2306004"],"award-info":[{"award-number":["GDZX2306004"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Cryptogr Eng"],"published-print":{"date-parts":[[2025,6]]},"DOI":"10.1007\/s13389-025-00376-6","type":"journal-article","created":{"date-parts":[[2025,5,20]],"date-time":"2025-05-20T15:27:57Z","timestamp":1747754877000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A flexible efficient and configurable number theoretic transform architecture implemented on FPGA"],"prefix":"10.1007","volume":"15","author":[{"given":"Zhenhong","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Min","family":"Gu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,20]]},"reference":[{"key":"376_CR1","doi-asserted-by":"publisher","unstructured":"Ajtai, M.: Generating hard instances of lattice problems. In: Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, pp. 99\u2013108 (1996). https:\/\/doi.org\/10.1145\/237814.237838","DOI":"10.1145\/237814.237838"},{"key":"376_CR2","doi-asserted-by":"publisher","unstructured":"Ajtai, M., Dwork, C.: A public-key cryptosystem with worst-case\/average-case equivalence. In: Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, pp. 284\u2013293 (1997). https:\/\/doi.org\/10.1145\/258533.258604","DOI":"10.1145\/258533.258604"},{"key":"376_CR3","doi-asserted-by":"publisher","unstructured":"Halevi, S., Shoup, V.: Algorithms in helib. In: Advances in Cryptology\u2013CRYPTO 2014: 34th Annual Cryptology Conference, Santa Barbara, CA, USA, August 17-21, 2014, Proceedings, Part I 34, pp. 554\u2013571 (2014). https:\/\/doi.org\/10.1007\/978-3-662-44371-2_31. Springer","DOI":"10.1007\/978-3-662-44371-2_31"},{"key":"376_CR4","doi-asserted-by":"publisher","unstructured":"Aguilar-Melchor, C., Barrier, J., Guelton, S., Guinet, A., Killijian, M.-O., Lepoint, T.: Nfllib: Ntt-based fast lattice library. In: Cryptographers\u2019 Track at the RSA Conference, pp. 341\u2013356 (2016). https:\/\/doi.org\/10.1007\/978-3-319-29485-8_20. Springer","DOI":"10.1007\/978-3-319-29485-8_20"},{"key":"376_CR5","doi-asserted-by":"publisher","unstructured":"Dai, W., Sunar, B.: cuhe: A homomorphic encryption accelerator library. In: Cryptography and Information Security in the Balkans: Second International Conference, BalkanCryptSec 2015, Koper, Slovenia, September 3-4, 2015, Revised Selected Papers 2, pp. 169\u2013186 (2016). https:\/\/doi.org\/10.1007\/978-3-319-29172-7_11. Springer","DOI":"10.1007\/978-3-319-29172-7_11"},{"key":"376_CR6","doi-asserted-by":"publisher","unstructured":"Nguyen, D.T., Dang, V.B., Gaj, K.: A high-level synthesis approach to the software\/hardware codesign of ntt-based post-quantum cryptography algorithms. In: 2019 International Conference on Field-Programmable Technology (ICFPT), pp. 371\u2013374 (2019). https:\/\/doi.org\/10.1109\/ICFPT47387.2019.00070. IEEE","DOI":"10.1109\/ICFPT47387.2019.00070"},{"key":"376_CR7","doi-asserted-by":"publisher","unstructured":"Nguyen, D.T., Dang, V.B., Gaj, K.: High-level synthesis in implementing and benchmarking number theoretic transform in lattice-based post-quantum cryptography using software\/hardware codesign. In: Applied Reconfigurable Computing. Architectures, Tools, and Applications: 16th International Symposium, ARC 2020, Toledo, Spain, April 1\u20133, 2020, Proceedings 16, pp. 247\u2013257 (2020). https:\/\/doi.org\/10.1007\/978-3-030-44534-8_19. Springer","DOI":"10.1007\/978-3-030-44534-8_19"},{"key":"376_CR8","doi-asserted-by":"publisher","unstructured":"Yang, J., Li, Z., Ren, J., Wang, X., Ni, W., Du, G.-M.: A polynomial multiplication accelerator for homomorphic encryption using dgt. In: 2021 IEEE 15th International Conference on Anti-counterfeiting, Security, and Identification (ASID), pp. 111\u2013115 (2021). https:\/\/doi.org\/10.1109\/ASID52932.2021.9651679. IEEE","DOI":"10.1109\/ASID52932.2021.9651679"},{"key":"376_CR9","doi-asserted-by":"publisher","unstructured":"El-Kady, A., Fournaris, A.P., Tsakoulis, T., Haleplidis, E., Paliouras, V.: High-level synthesis design approach for number-theoretic transform implementations. In: 2021 IFIP\/IEEE 29th International Conference on Very Large Scale Integration (VLSI-SoC), pp. 1\u20136 (2021). https:\/\/doi.org\/10.1109\/VLSI-SoC53125.2021.9607003. IEEE","DOI":"10.1109\/VLSI-SoC53125.2021.9607003"},{"key":"376_CR10","doi-asserted-by":"publisher","unstructured":"Soni, D., Karri, R.: Efficient hardware implementation of pqc primitives and pqc algorithms using high-level synthesis. In: 2021 IEEE Computer Society Annual Symposium on VLSI (ISVLSI), pp. 296\u2013301 (2021). https:\/\/doi.org\/10.1109\/ISVLSI51109.2021.00061. IEEE","DOI":"10.1109\/ISVLSI51109.2021.00061"},{"key":"376_CR11","doi-asserted-by":"publisher","unstructured":"Kim, S., Lee, K., Cho, W., Nam, Y., Cheon, J.H., Rutenbar, R.A.: Hardware architecture of a number theoretic transform for a bootstrappable rns-based homomorphic encryption scheme. In: 2020 IEEE 28th Annual International Symposium on Field-Programmable Custom Computing Machines (FCCM), pp. 56\u201364 (2020). https:\/\/doi.org\/10.1109\/FCCM48280.2020.00017. IEEE","DOI":"10.1109\/FCCM48280.2020.00017"},{"key":"376_CR12","doi-asserted-by":"publisher","unstructured":"Renter\u00eda-Mej\u00eda, C., Velasco-Medina, J.: Hardware design of an ntt-based polynomial multiplier. In: 2014 IX Southern Conference on Programmable Logic (SPL), pp. 1\u20135 (2014). https:\/\/doi.org\/10.1109\/SPL.2014.7002209. IEEE","DOI":"10.1109\/SPL.2014.7002209"},{"issue":"11","key":"376_CR13","doi-asserted-by":"publisher","first-page":"1637","DOI":"10.1109\/TC.2018.2816640","volume":"67","author":"SS Roy","year":"2018","unstructured":"Roy, S.S., J\u00e4rvinen, K., Vliegen, J., Vercauteren, F., Verbauwhede, I.: Hepcloud: An fpga-based multicore processor for fv somewhat homomorphic function evaluation. IEEE Transactions on Computers 67(11), 1637\u20131650 (2018). https:\/\/doi.org\/10.1109\/TC.2018.2816640","journal-title":"IEEE Transactions on Computers"},{"key":"376_CR14","doi-asserted-by":"publisher","unstructured":"Song, S., Tang, W., Chen, T., Zhang, Z.: Leia: A 2.05 mm 2 140mw lattice encryption instruction accelerator in 40nm cmos. In: 2018 IEEE Custom Integrated Circuits Conference (CICC), pp. 1\u20134 (2018). https:\/\/doi.org\/10.1109\/CICC.2018.8357070. IEEE","DOI":"10.1109\/CICC.2018.8357070"},{"key":"376_CR15","doi-asserted-by":"publisher","unstructured":"Fritzmann, T., Sep\u00falveda, J.: Efficient and flexible low-power ntt for lattice-based cryptography. In: 2019 IEEE International Symposium on Hardware Oriented Security and Trust (HOST), pp. 141\u2013150 (2019). https:\/\/doi.org\/10.1109\/HST.2019.8741027. IEEE","DOI":"10.1109\/HST.2019.8741027"},{"key":"376_CR16","doi-asserted-by":"publisher","unstructured":"Roy, S.S., Vercauteren, F., Mentens, N., Chen, D.D., Verbauwhede, I.: Compact ring-lwe cryptoprocessor. In: Cryptographic Hardware and Embedded Systems\u2013CHES 2014: 16th International Workshop, Busan, South Korea, September 23-26, 2014. Proceedings 16, pp. 371\u2013391 (2014). https:\/\/doi.org\/10.1007\/978-3-662-44709-3_21. Springer","DOI":"10.1007\/978-3-662-44709-3_21"},{"key":"376_CR17","doi-asserted-by":"publisher","unstructured":"Banerjee, U., Ukyab, T.S., Chandrakasan, A.P.: Sapphire: A configurable crypto-processor for post-quantum lattice-based protocols. arXiv preprint arXiv:1910.07557 (2019) https:\/\/doi.org\/10.46586\/tches.v2019.i4.17-61","DOI":"10.46586\/tches.v2019.i4.17-61"},{"key":"376_CR18","doi-asserted-by":"publisher","unstructured":"Ye, T., Yang, Y., Kuppannagari, S.R., Kannan, R., Prasanna, V.K.: Fpga acceleration of number theoretic transform. In: High Performance Computing: 36th International Conference, ISC High Performance 2021, Virtual Event, June 24\u2013July 2, 2021, Proceedings 36, pp. 98\u2013117 (2021). https:\/\/doi.org\/10.1007\/978-3-030-78713-4_6 . Springer","DOI":"10.1007\/978-3-030-78713-4_6"},{"key":"376_CR19","doi-asserted-by":"publisher","unstructured":"Bisheh-Niasar, M., Azarderakhsh, R., Mozaffari-Kermani, M.: High-speed ntt-based polynomial multiplication accelerator for post-quantum cryptography. In: 2021 IEEE 28th Symposium on Computer Arithmetic (ARITH), pp. 94\u2013101 (2021). https:\/\/doi.org\/10.1109\/ARITH51176.2021.00028 . IEEE","DOI":"10.1109\/ARITH51176.2021.00028"},{"key":"376_CR20","unstructured":"Franz, W.: Polynomial algorithms in computer algebra. Texts and Monographs in Symb. Comput. (1996)"},{"key":"376_CR21","doi-asserted-by":"publisher","unstructured":"P\u00f6ppelmann, T., G\u00fcneysu, T.: Towards efficient arithmetic for lattice-based cryptography on reconfigurable hardware. In: Progress in Cryptology\u2013LATINCRYPT 2012: 2nd International Conference on Cryptology and Information Security in Latin America, Santiago, Chile, October 7-10, 2012. Proceedings 2, pp. 139\u2013158 (2012). https:\/\/doi.org\/10.1007\/978-3-642-33481-8_8. Springer","DOI":"10.1007\/978-3-642-33481-8_8"},{"key":"376_CR22","doi-asserted-by":"publisher","unstructured":"Gentleman, W.M., Sande, G.: Fast fourier transforms: for fun and profit. In: Proceedings of the November 7-10, 1966, Fall Joint Computer Conference, pp. 563\u2013578 (1966). https:\/\/doi.org\/10.1145\/1464291.1464352","DOI":"10.1145\/1464291.1464352"},{"key":"376_CR23","doi-asserted-by":"publisher","unstructured":"Mert, A.C., \u00d6zt\u00fcrk, E., Sava\u015f, E.: Design and implementation of a fast and scalable ntt-based polynomial multiplier architecture. In: 2019 22nd Euromicro Conference on Digital System Design (DSD), pp. 253\u2013260 (2019). https:\/\/doi.org\/10.1109\/DSD.2019.00045. IEEE","DOI":"10.1109\/DSD.2019.00045"},{"issue":"170","key":"376_CR24","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. Mathematics of computation 44(170), 519\u2013521 (1985)","journal-title":"Mathematics of computation"},{"issue":"9","key":"376_CR25","doi-asserted-by":"publisher","first-page":"1879","DOI":"10.1109\/TVLSI.2013.2281786","volume":"22","author":"W Wang","year":"2013","unstructured":"Wang, W., Huang, X., Emmart, N., Weems, C.: Vlsi design of a large-number multiplier for fully homomorphic encryption. IEEE Transactions on Very Large Scale Integration (VLSI) Systems 22(9), 1879\u20131887 (2013). https:\/\/doi.org\/10.1109\/TVLSI.2013.2281786","journal-title":"IEEE Transactions on Very Large Scale Integration (VLSI) Systems"},{"key":"376_CR26","doi-asserted-by":"publisher","unstructured":"Hu, X., Li, M., Tian, J., Wang, Z.: Darm: A low-complexity and fast modular multiplier for lattice-based cryptography. In: 2021 IEEE 32nd International Conference on Application-specific Systems, Architectures and Processors (ASAP), pp. 175\u2013178 (2021). https:\/\/doi.org\/10.1109\/ASAP52443.2021.00033. IEEE","DOI":"10.1109\/ASAP52443.2021.00033"},{"issue":"8","key":"376_CR27","doi-asserted-by":"publisher","first-page":"2927","DOI":"10.1109\/TCSII.2021.3064232","volume":"68","author":"W Tan","year":"2021","unstructured":"Tan, W., Case, B.M., Wang, A., Gao, S., Lao, Y.: High-speed modular multiplier for lattice-based cryptosystems. IEEE Transactions on Circuits and Systems II: Express Briefs 68(8), 2927\u20132931 (2021). https:\/\/doi.org\/10.1109\/TCSII.2021.3064232","journal-title":"IEEE Transactions on Circuits and Systems II: Express Briefs"},{"key":"376_CR28","doi-asserted-by":"publisher","unstructured":"Yaman, F., Mert, A.C., \u00d6zt\u00fcrk, E., Sava\u015f, E.: A hardware accelerator for polynomial multiplication operation of crystals-kyber pqc scheme. In: 2021 Design, Automation & Test in Europe Conference & Exhibition (DATE), pp. 1020\u20131025 (2021). https:\/\/doi.org\/10.23919\/DATE51398.2021.9474139. IEEE","DOI":"10.23919\/DATE51398.2021.9474139"},{"key":"376_CR29","doi-asserted-by":"publisher","unstructured":"Zhang, N., Yang, B., Chen, C., Yin, S., Wei, S., Liu, L.: Highly efficient architecture of newhope-nist on fpga using low-complexity ntt\/intt. IACR Transactions on Cryptographic Hardware and Embedded Systems, 49\u201372 (2020) https:\/\/doi.org\/10.13154\/tches.v2020.i2.49-72","DOI":"10.13154\/tches.v2020.i2.49-72"},{"key":"376_CR30","doi-asserted-by":"publisher","unstructured":"Kim, S., Jung, W., Park, J., Ahn, J.H.: Accelerating number theoretic transformations for bootstrappable homomorphic encryption on gpus. In: 2020 IEEE International Symposium on Workload Characterization (IISWC), pp. 264\u2013275 (2020). https:\/\/doi.org\/10.1109\/IISWC50251.2020.00033. IEEE","DOI":"10.1109\/IISWC50251.2020.00033"},{"issue":"1","key":"376_CR31","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1109\/TC.2016.2574340","volume":"66","author":"E \u00d6zt\u00fcrk","year":"2016","unstructured":"\u00d6zt\u00fcrk, E., Dor\u00f6z, Y., Sava\u015f, E., Sunar, B.: A custom accelerator for homomorphic encryption applications. IEEE Transactions on Computers 66(1), 3\u201316 (2016). https:\/\/doi.org\/10.1109\/TC.2016.2574340","journal-title":"IEEE Transactions on Computers"},{"issue":"7","key":"376_CR32","doi-asserted-by":"publisher","first-page":"1297","DOI":"10.1109\/TVLSI.2021.3077885","volume":"29","author":"M Bisheh-Niasar","year":"2021","unstructured":"Bisheh-Niasar, M., Azarderakhsh, R., Mozaffari-Kermani, M.: Cryptographic accelerators for digital signature based on ed25519. IEEE Transactions on Very Large Scale Integration (VLSI) Systems 29(7), 1297\u20131305 (2021). https:\/\/doi.org\/10.1109\/TVLSI.2021.3077885","journal-title":"IEEE Transactions on Very Large Scale Integration (VLSI) Systems"},{"key":"376_CR33","doi-asserted-by":"publisher","unstructured":"Kawamura, K., Yanagisawa, M., Togawa, N.: A loop structure optimization targeting high-level synthesis of fast number theoretic transform. In: 2018 19th International Symposium on Quality Electronic Design (ISQED), pp. 106\u2013111 (2018). https:\/\/doi.org\/10.1109\/ISQED.2018.8357273. IEEE","DOI":"10.1109\/ISQED.2018.8357273"},{"key":"376_CR34","unstructured":"Brutzkus, A., Gilad-Bachrach, R., Elisha, O.: Low latency privacy preserving inference. In: International Conference on Machine Learning, pp. 812\u2013821 (2019). PMLR"},{"key":"376_CR35","doi-asserted-by":"publisher","unstructured":"Roy, S.S., Turan, F., Jarvinen, K., Vercauteren, F., Verbauwhede, I.: Fpga-based high-performance parallel architecture for homomorphic computing on encrypted data. In: 2019 IEEE International Symposium on High Performance Computer Architecture (HPCA), pp. 387\u2013398 (2019). https:\/\/doi.org\/10.1109\/HPCA.2019.00052. IEEE","DOI":"10.1109\/HPCA.2019.00052"},{"key":"376_CR36","doi-asserted-by":"publisher","unstructured":"Ye, Z., Cheung, R.C., Huang, K.: Pipentt: A pipelined number theoretic transform architecture. IEEE Transactions on Circuits and Systems II: Express Briefs 69(10), 4068\u20134072 (2022). https:\/\/doi.org\/10.1109\/TCSII.2022.3184703","DOI":"10.1109\/TCSII.2022.3184703"}],"container-title":["Journal of Cryptographic Engineering"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13389-025-00376-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s13389-025-00376-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s13389-025-00376-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,20]],"date-time":"2025-06-20T18:03:13Z","timestamp":1750442593000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s13389-025-00376-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,20]]},"references-count":36,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["376"],"URL":"https:\/\/doi.org\/10.1007\/s13389-025-00376-6","relation":{},"ISSN":["2190-8508","2190-8516"],"issn-type":[{"value":"2190-8508","type":"print"},{"value":"2190-8516","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,20]]},"assertion":[{"value":"22 July 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 April 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 May 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}},{"value":"Not applicable.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"Not applicable.","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Materials availability"}},{"value":"Not applicable.","order":6,"name":"Ethics","group":{"name":"EthicsHeading","label":"Code availability"}}],"article-number":"13"}}