{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T05:27:50Z","timestamp":1775107670951,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,7,3]],"date-time":"2025-07-03T00:00:00Z","timestamp":1751500800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Cryptography"],"abstract":"<jats:p>Over years of development in secure multi-party computation (MPC), many sophisticated functionalities have been made practical, and multi-dimensional operations occur more and more frequently in MPC protocols, especially in protocols involving datasets of vector elements, such as privacy-preserving biometric identification and privacy-preserving machine learning. In this paper, we introduce a new kind of correlation, called tensor triples, which is designed to make multi-dimensional MPC protocols more efficient. We will discuss the generation process, the usage, and the applications of tensor triples and show that they can accelerate privacy-preserving biometric identification protocols, such as FingerCode, Eigenfaces, and FaceNet, by more than 1000 times, with reasonable offline costs, and grant pre-computability for the secure matrix multiplication process in privacy-preserving machine learning protocols, such as SecureML and SecureNN, while achieving similar efficiency.<\/jats:p>","DOI":"10.3390\/cryptography9030050","type":"journal-article","created":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T03:47:26Z","timestamp":1751600846000},"page":"50","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Efficient Secure Multi-Party Computation for Multi-Dimensional Arithmetics and Its Applications"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9300-4075","authenticated-orcid":false,"given":"Dongyu","family":"Wu","sequence":"first","affiliation":[{"name":"Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8622-8596","authenticated-orcid":false,"given":"Bei","family":"Liang","sequence":"additional","affiliation":[{"name":"Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-2509-917X","authenticated-orcid":false,"given":"Zijie","family":"Lu","sequence":"additional","affiliation":[{"name":"Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1257-7598","authenticated-orcid":false,"given":"Jintai","family":"Ding","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Xi\u2019an Jiaotong-Liverpool University, Suzhou 215123, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Erkin, Z., Franz, M., Guajardo, J., Katzenbeisser, S., Lagendijk, I., and Toft, T. (2009, January 5\u20137). Privacy-Preserving Face Recognition. Proceedings of the 9th Privacy Enhancing Technologies Symposium (PETS 2009), Seattle, WA, USA.","DOI":"10.1007\/978-3-642-03168-7_14"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Blanton, M., and Gasti, P. (2011, January 12\u201314). Secure and Efficient Protocols for Iris and Fingerprint Identification. Proceedings of the 16th European Symposium on Research in Computer Security, Leuven, Belgium.","DOI":"10.1007\/978-3-642-23822-2_11"},{"key":"ref_3","unstructured":"Huang, Y., Malka, L., Evans, D., and Katz, J. (2011, January 6\u20139). Efficient privacy-preserving biometric identification. Proceedings of the 17th Conference Network and Distributed System Security Symposium, San Diego, CA, USA."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Shahandashti, S.F., Safavi-Naini, R., and Ogunbona, P. (2012, January 9\u201311). Private Fingerprint Matching. Proceedings of the 17th Australasian Conference on Information Security and Privacy, Wollongong, Australia.","DOI":"10.1007\/978-3-642-31448-3_32"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1109\/MSP.2012.2230218","article-title":"Privacy-Preserving Biometric Identification Using Secure Multiparty Computation: An Overview and Recent Trends","volume":"30","author":"Bringer","year":"2013","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_6","unstructured":"Bringer, J., Chabanne, H., Favre, M., Patey, A., Schneider, T., and Zohner, M. (2014, January 11\u201313). GSHADE: Faster Privacy-Preserving Distance Computation and Biometric Identification. Proceedings of the 2nd ACM Workshop on Information Hiding and Multimedia Security, Salzburg, Austria."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.icte.2016.08.006","article-title":"Efficient and privacy-preserving biometric identification in cloud","volume":"2","author":"Hahn","year":"2016","journal-title":"ICT Express"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"8606","DOI":"10.1109\/ACCESS.2017.2691578","article-title":"Privacy-Preserving Comparison of Variable-Length Data with Application to Biometric Template Protection","volume":"5","author":"Galbally","year":"2017","journal-title":"IEEE Access"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5778","DOI":"10.1109\/JIOT.2019.2905555","article-title":"Lightweight Privacy-Preserving Ensemble Classification for Face Recognition","volume":"6","author":"Ma","year":"2019","journal-title":"IEEE Internet Things J."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Nikolaenko, V., Weinsberg, U., Ioannidis, S., Joye, M., Boneh, D., and Taft, N. (2013, January 19\u201322). Privacy-Preserving Ridge Regression on Hundreds of Millions of Records. Proceedings of the IEEE Symposium on Security and Privacy, San Francisco, CA, USA.","DOI":"10.1109\/SP.2013.30"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Bonawitz, K., Ivanov, V., Kreuter, B., Marcedone, A., McMahan, H.B., Patel, S., Ramage, D., Segal, A., and Seth, K. (November, January 30). Practical Secure Aggregation for Privacy-Preserving Machine Learning. Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, Dallas, TX, USA.","DOI":"10.1145\/3133956.3133982"},{"key":"ref_12","unstructured":"Mohassel, P., and Rindal, P. (22018, January 15\u201319). ABY3: A Mixed Protocol Framework for Machine Learning. Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, Toronto, ON, Canada."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"414","DOI":"10.2478\/popets-2020-0080","article-title":"Practical Privacy-Preserving K-means Clustering","volume":"2020","author":"Mohassel","year":"2020","journal-title":"Proc. Priv. Enhancing Technol."},{"key":"ref_14","unstructured":"Koti, N., Pancholi, M., Patra, A., and Suresh, A. (2021, January 11\u201313). SWIFT: Super-fast and Robust Privacy-Preserving Machine Learning. Proceedings of the 30th USENIX Security Symposium (USENIX Security 21), Online."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Chen, H., Kim, M., Razenshteyn, I.P., Rotaru, D., Song, Y., and Wagh, S. (2020, January 7\u201311). Maliciously Secure Matrix Multiplication with Applications to Private Deep Learning. Proceedings of the Advances in Cryptology\u2014ASIACRYPT 2020, Daejeon, Republic of Korea.","DOI":"10.1007\/978-3-030-64840-4_2"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Abspoel, M., Cramer, R., Damgard, I., Escudero, D., Rambaud, M., Xing, C., and Yuan, C. (2020, January 7\u201311). Asymptotically Good Multiplicative LSSS over Galois Rings and Applications to MPC over Z\/pkZ. Proceedings of the Advances in Cryptology\u2014ASIACRYPT 2020, Daejeon, Republic of Korea.","DOI":"10.1007\/978-3-030-64840-4_6"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1137\/S0097539704383633","article-title":"Oblivious Polynomial Evaluation","volume":"35","author":"Naor","year":"2006","journal-title":"SIAM J. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Applebaum, B., Damgard, I., Ishai, Y., Nielsen, M., and Zichron, L. (2017, January 20\u201324). Secure Arithmetic Computation with Constant Computational Overhead. Proceedings of the Advances in Cryptology\u2014CRYPTO 2017, Santa Barbara, CA, USA.","DOI":"10.1007\/978-3-319-63688-7_8"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Boyle, E., Couteau, G., Gilboa, N., and Ishai, Y. (2018, January 15\u201319). Compressing Vector OLE. Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, Toronto, ON, Canada.","DOI":"10.1145\/3243734.3243868"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Boyle, E., Couteau, G., Gilboa, N., Ishai, Y., Kohl, L., Rindal, P., and Scholl, P. (2019, January 11\u201315). Efficient Two-Round OT Extension and Silent Non-Interactive Secure Computation. Proceedings of the 2019 ACM SIGSAC Conference on Computer and Communications Security, London, UK.","DOI":"10.1145\/3319535.3354255"},{"key":"ref_21","unstructured":"Jain, A.K., Prabhakar, S., Hong, L., and Pankanti, S. (1999, January 23\u201325). FingerCode: A filterbank for fingerprint representation and matching. Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149), Fort Collins, CO, USA."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1162\/jocn.1991.3.1.71","article-title":"Eigenfaces for Recognition","volume":"3","author":"Turk","year":"1991","journal-title":"J. Cogn. Neurosci."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Schroff, F., Kalenichenko, D., and Philbin, J. (2015, January 7\u201312). FaceNet: A unified embedding for face recognition and clustering. Proceedings of the 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Boston, MA, USA.","DOI":"10.1109\/CVPR.2015.7298682"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Mohassel, P., and Zhang, Y. (2017, January 22\u201326). SecureML: A System for Scalable Privacy-Preserving Machine Learning. Proceedings of the 2017 IEEE Symposium on Security and Privacy (SP), San Jose, CA, USA.","DOI":"10.1109\/SP.2017.12"},{"key":"ref_25","unstructured":"Wagh, S., Gupta, D., and Chandran, N. (2025, July 01). SecureNN: Efficient and Private Neural Network Training. IACR Cryptol. ePrint Arch., Available online: https:\/\/eprint.iacr.org\/2018\/442."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Wu, D., Liang, B., Lu, Z., and Ding, J. (2024, January 24\u201327). Efficient Secure Multi-party Computation for Multi-dimensional Arithmetics and Its Application in Privacy-Preserving Biometric Identification. Proceedings of the 23rd International Conference on Cryptology and Network Security, Cambridge, UK.","DOI":"10.1007\/978-981-97-8013-6_1"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mono, J., and G\u00fcneysu, T. (2023, January 10\u201314). Implementing and Optimizing Matrix Triples with Homomorphic Encryption. Proceedings of the 2023 ACM Asia Conference on Computer and Communications Security, Melbourne, Australia.","DOI":"10.1145\/3579856.3590344"},{"key":"ref_28","unstructured":"Rabin, M.O. (2025, July 01). How To Exchange Secrets with Oblivious Transfer. IACR Cryptol. ePrint Arch., Available online: https:\/\/eprint.iacr.org\/2005\/187.pdf."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Ishai, Y., Kilian, J., Nissim, K., and Petrank, E. (2003, January 17\u201321). Extending Oblivious Transfers Efficiently. Proceedings of the Advances in Cryptology\u2014CRYPTO 2003, Santa Barbara, CA, USA.","DOI":"10.1007\/978-3-540-45146-4_9"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Kolesnikov, V., and Kumaresan, R. (2013, January 18\u201322). Improved OT Extension for Transferring Short Secrets. Proceedings of the Advances in Cryptology\u2014CRYPTO 2013, Santa Barbara, CA, USA.","DOI":"10.1007\/978-3-642-40084-1_4"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Kolesnikov, V., Kumaresan, R., Rosulek, M., and Trieu, N. (2016, January 24\u201328). Efficient Batched Oblivious PRF with Applications to Private Set Intersection. Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, Vienna, Austria.","DOI":"10.1145\/2976749.2978381"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Asharov, G., Lindell, Y., Schneider, T., and Zohner, M. (2013, January 4\u20138). More Efficient Oblivious Transfer and Extensions for Faster Secure Computation. Proceedings of the 2013 ACM SIGSAC Conference on Computer & Communications Security, Berlin, Germany.","DOI":"10.1145\/2508859.2516738"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Yang, K., Weng, C., Lan, X., Zhang, J., and Wang, X. (2020, January 9\u201313). Ferret: Fast Extension for Correlated OT with Small Communication. Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security, Virtual.","DOI":"10.1145\/3372297.3417276"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Schoppmann, P., Gasc\u00f3n, A., Reichert, L., and Raykova, M. (2019, January 11\u201315). Distributed Vector-OLE: Improved Constructions and Implementation. Proceedings of the 2019 ACM SIGSAC Conference on Computer and Communications Security, London, UK.","DOI":"10.1145\/3319535.3363228"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Couteau, G., Rindal, P., and Raghuraman, S. (2021, January 16\u201320). Silver: Silent VOLE and Oblivious Transfer from Hardness of Decoding Structured LDPC Codes. Proceedings of the Advances in Cryptology\u2014CRYPTO 2021, Virtual.","DOI":"10.1007\/978-3-030-84252-9_17"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Raghuraman, S., Rindal, P., and Tanguy, T. (2023, January 20\u201324). Expand-Convolute Codes for Pseudorandom Correlation Generators from LPN. Proceedings of the Advances in Cryptology\u2014CRYPTO 2023, Santa Barbara, CA, USA.","DOI":"10.1007\/978-3-031-38551-3_19"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Gilboa, N. (1999, January 15\u201319). Two Party RSA Key Generation. Proceedings of the Advances in Cryptology\u2014CRYPTO\u2019 99, Santa Barbara, CA, USA.","DOI":"10.1007\/3-540-48405-1_8"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Demmler, D., Schneider, T., and Zohner, M. (2015, January 8\u201311). ABY\u2014A Framework for Efficient Mixed-Protocol Secure Two-Party Computation. Proceedings of the Network and Distributed System Security Symposium, San Diego, CA, USA.","DOI":"10.14722\/ndss.2015.23113"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Smart, N.P., and Tanguy, T. (2019, January 11\u201315). TaaS: Commodity MPC via Triples-as-a-Service. Proceedings of the 2019 ACM SIGSAC Conference on Cloud Computing Security Workshop, London, UK.","DOI":"10.1145\/3338466.3358918"},{"key":"ref_40","unstructured":"Muth, P., and Katzenbeisser, S. (2025, July 01). Assisted MPC. Cryptol. ePrint Arch., Available online: https:\/\/eprint.iacr.org\/2022\/1453."},{"key":"ref_41","unstructured":"Wu, D. (2025, July 01). Highly Efficient Server-Aided Multiparty Subfield VOLE Distribution Protocol. Cryptol. ePrint Arch., Available online: https:\/\/eprint.iacr.org\/2025\/029."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Rathee, D., Schneider, T., and Shukla, K.K. (2019, January 25\u201327). Improved Multiplication Triple Generation over Rings via RLWE-Based AHE. Proceedings of the Cryptology and Network Security, Fuzhou, China.","DOI":"10.1007\/978-3-030-31578-8_19"},{"key":"ref_43","unstructured":"Damg<b>a<\/b>rd, I., Pastro, V., Smart, N., and Zakarias, S. (2012, January 19\u201323). Multiparty Computation from Somewhat Homomorphic Encryption. Proceedings of the Advances in Cryptology\u2014CRYPTO 2012, Santa Barbara, CA, USA."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Naresh Boddeti, V. (2018, January 22\u201325). Secure Face Matching Using Fully Homomorphic Encryption. Proceedings of the 2018 IEEE 9th International Conference on Biometrics Theory, Applications and Systems (BTAS), Redondo Beach, CA, USA.","DOI":"10.1109\/BTAS.2018.8698601"}],"container-title":["Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2410-387X\/9\/3\/50\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:04:13Z","timestamp":1760033053000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2410-387X\/9\/3\/50"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,3]]},"references-count":44,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["cryptography9030050"],"URL":"https:\/\/doi.org\/10.3390\/cryptography9030050","relation":{},"ISSN":["2410-387X"],"issn-type":[{"value":"2410-387X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,3]]}}}