{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T12:18:27Z","timestamp":1774613907811,"version":"3.50.1"},"reference-count":18,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T00:00:00Z","timestamp":1774310400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Shenzhen Science and Technology Program","award":["KJZD20240903104301003"],"award-info":[{"award-number":["KJZD20240903104301003"]}]},{"name":"Shenzhen Science and Technology Program","award":["SYSPG20241211173609009"],"award-info":[{"award-number":["SYSPG20241211173609009"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t              https:\/\/ror.org\/01h0zpd94","doi-asserted-by":"publisher","award":["62272131"],"award-info":[{"award-number":["62272131"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Shenzhen Social Science Fund","award":["SZ2025B011"],"award-info":[{"award-number":["SZ2025B011"]}]},{"name":"PCL-CMCC Foundation for Science and Innovation","award":["2024ZY2B0050"],"award-info":[{"award-number":["2024ZY2B0050"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Cryptography"],"abstract":"<jats:p>Fully homomorphic encryption (FHE) offers a promising solution for privacy-preserving machine learning by enabling arbitrary computations on encrypted data. However, the efficient evaluation of non-linear functions\u2014such as the ReLU activation function over large integers\u2014remains a major obstacle in practical deployments, primarily due to high bootstrapping overhead and limited precision support in existing schemes. In this paper, we propose LargeIntReLU, a novel framework that enables efficient homomorphic ReLU evaluation over large integers (7\u201311 bits) via full-domain bootstrapping. Central to our approach is a signed digit decomposition algorithm, SignedDecomp, that partitions a large integer ciphertext into signed 6-bit segments using three new low-level primitives: LeftShift, HomMod, and CipherClean. This decomposition preserves arithmetic consistency, avoids cross-segment carry propagation, and allows parallelized bootstrapping. By segmenting the large integer and processing each chunk independently with optimized small-integer bootstrapping, we achieve homomorphic ReLU with full-domain bootstrapping, which significantly reduces the total number of sequential bootstrapping operations required. The security of our scheme is guaranteed by TFHE. Experimental results demonstrate that the proposed method reduces the bootstrapping cost by an average of 28.58% compared to state-of-the-art approaches while maintaining 95.2% accuracy. With execution times ranging from 1.16 s to 1.62 s across 7\u201311 bit integers, our work bridges a critical gap toward a scalable and efficient homomorphic ReLU function, which is useful in privacy-preserving machine learning. Furthermore, an end-to-end encrypted inference test on a CNN model with the MNIST dataset confirms its practicality, achieving 88.85% accuracy and demonstrating a complete pipeline for privacy-preserving neural network evaluation.<\/jats:p>","DOI":"10.3390\/cryptography10020021","type":"journal-article","created":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T09:11:14Z","timestamp":1774429874000},"page":"21","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Homomorphic ReLU with Full-Domain Bootstrapping"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-5751-3169","authenticated-orcid":false,"given":"Yuqun","family":"Lin","sequence":"first","affiliation":[{"name":"The School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China"}]},{"given":"Yi","family":"Huang","sequence":"additional","affiliation":[{"name":"The School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen 518055, China"}]},{"given":"Xiaomeng","family":"Tang","sequence":"additional","affiliation":[{"name":"The School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen 518055, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3203-311X","authenticated-orcid":false,"given":"Jingjing","family":"Fan","sequence":"additional","affiliation":[{"name":"The School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen 518055, China"},{"name":"The School of Computing and Data Science, University of Hong Kong, Hong Kong SAR, China"}]},{"given":"Qifei","family":"Xu","sequence":"additional","affiliation":[{"name":"The School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8944-7444","authenticated-orcid":false,"given":"Zoe-Lin","family":"Jiang","sequence":"additional","affiliation":[{"name":"The School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen 518055, China"}]},{"given":"Xiaosong","family":"Zhang","sequence":"additional","affiliation":[{"name":"The School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0730-8732","authenticated-orcid":false,"given":"Junbin","family":"Fang","sequence":"additional","affiliation":[{"name":"Department of Optoelectronic Engineering, Jinan University, Guangzhou 510632, China"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,24]]},"reference":[{"key":"ref_1","unstructured":"Gentry, C. (June, January 31). Fully Homomorphic Encryption Using Ideal Lattices. Proceedings of the 41st Annual ACM Symposium on Theory of Computing, Bethesda, MD, USA."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1572","DOI":"10.1109\/JPROC.2022.3205665","article-title":"Survey on Fully Homomorphic Encryption, Theory, and Applications","volume":"110","author":"Marcolla","year":"2022","journal-title":"Proc. IEEE"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2454","DOI":"10.1109\/TIFS.2024.3350909","article-title":"Secure and Efficient Similarity Retrieval in Cloud Computing Based on Homomorphic Encryption","volume":"19","author":"Wang","year":"2024","journal-title":"IEEE Trans. Inf. Forensics Secur."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1109\/TCC.2025.3561586","article-title":"Secure kNN for Distributed Cloud Environment Using Fully Homomorphic Encryption","volume":"13","author":"Fukuchi","year":"2025","journal-title":"IEEE Trans. Cloud Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3740","DOI":"10.1109\/TIFS.2021.3090959","article-title":"DOReN: Toward Efficient Deep Convolutional Neural Networks with Fully Homomorphic Encryption","volume":"16","author":"Meftah","year":"2021","journal-title":"IEEE Trans. Inf. Forensics Secur."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Chillotti, I., Gama, N., Georgieva, M., and Izabachene, M. (2016, January 4\u20138). Faster Fully Homomorphic Encryption: Bootstrapping in Less Than 0.1 Seconds. Proceedings of the 22nd International Conference on the Theory and Application of Cryptology and Information Security, Hanoi, Vietnam.","DOI":"10.1007\/978-3-662-53887-6_1"},{"key":"ref_7","unstructured":"Lou, Q., and Jiang, L. (2019, January 8\u201314). SHE: A Fast and Accurate Deep Neural Network for Encrypted Data. Proceedings of the 33rd Neural Information Processing Systems, Vancouver, BC, Canada."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bourse, F., Minelli, M., Minihold, M., and Paillier, P. (2018, January 19\u201323). Fast Homomorphic Evaluation of Deep Discretized Neural Networks. Proceedings of the 38th Annual International Cryptology Conference, Santa Barbara, CA, USA.","DOI":"10.1007\/978-3-319-96878-0_17"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Okada, H., Kiyomoto, S., and Cid, C. (2020, January 16\u201318). Integerwise Functional Bootstrapping on TFHE. Proceedings of the 23rd International Conference on Information Security, Bali, Indonesia.","DOI":"10.1007\/978-3-030-62974-8_7"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"501","DOI":"10.46586\/tches.v2023.i1.501-537","article-title":"FDFB: Full Domain Functional Bootstrapping Towards Practical Fully Homomorphic Encryption","volume":"2023","author":"Kluczniak","year":"2022","journal-title":"IACR Trans. Cryptogr. Hardw. Embed. Syst."},{"key":"ref_11","unstructured":"Clet, P.E., Zuber, M., Boudguiga, A., Sirdey, R., and Gouy-Pailler, C. (2022). Putting Up the Swiss Army Knife of Homomorphic Calculations by Means of TFHE Functional Bootstrapping. Cryptol. Eprint Arch., Available online: https:\/\/eprint.iacr.org\/2022\/149.pdf."},{"key":"ref_12","unstructured":"Yang, Z., Xie, X., Shen, H., Chen, S., and Zhou, J. (2021). TOTA: Fully Homomorphic Encryption with Smaller Parameters and Stronger Security. Cryptol. Eprint Arch., Available online: https:\/\/eprint.iacr.org\/2021\/1347."},{"key":"ref_13","first-page":"161","article-title":"An Efficient Integer-Wise ReLU on TFHE","volume":"14895","author":"Zhu","year":"2024","journal-title":"Proceedings of the Information Security and Privacy\u201429th Australasian Conference, ACISP 2024, Sydney, NSW, Australia, 15\u201317 July 2024"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1007\/s00145-023-09463-5","article-title":"Parameter Optimization and Larger Precision for (T) FHE","volume":"36","author":"Bergerat","year":"2023","journal-title":"J. Cryptol."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Chillotti, I., Ligier, D., Orfila, J.B., and Tap, S. (2021, January 6\u201310). Improved Programmable Bootstrapping with Larger Precision and Efficient Arithmetic Circuits for TFHE. Proceedings of the 27th International Conference on the Theory and Application of Cryptology and Information Security, Singapore.","DOI":"10.1007\/978-3-030-92078-4_23"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1568318.1568324","article-title":"On Lattices, Learning with Errors, Random Linear Codes, and Cryptography","volume":"56","author":"Regev","year":"2009","journal-title":"J. ACM"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/s00145-019-09319-x","article-title":"TFHE: Fast Fully Homomorphic Encryption over the Torus","volume":"33","author":"Chillotti","year":"2020","journal-title":"J. Cryptol."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Liu, Z., Micciancio, D., and Polyakov, Y. (2022, January 5\u20139). Large-Precision Homomorphic Sign Evaluation Using FHEW\/TFHE Bootstrapping. Proceedings of the 28th International Conference on the Theory and Application of Cryptology and Information Security, Taiwan, China.","DOI":"10.1007\/978-3-031-22966-4_5"}],"container-title":["Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2410-387X\/10\/2\/21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T11:33:55Z","timestamp":1774611235000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2410-387X\/10\/2\/21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,24]]},"references-count":18,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,4]]}},"alternative-id":["cryptography10020021"],"URL":"https:\/\/doi.org\/10.3390\/cryptography10020021","relation":{},"ISSN":["2410-387X"],"issn-type":[{"value":"2410-387X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,24]]}}}