{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T18:41:30Z","timestamp":1785436890501,"version":"3.56.0"},"reference-count":22,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T00:00:00Z","timestamp":1778457600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004731","name":"Zhejiang Provincial Natural Science Foundation of China","doi-asserted-by":"publisher","award":["LMS26F020025"],"award-info":[{"award-number":["LMS26F020025"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]},{"award":["LMS26F020025"],"award-info":[{"award-number":["LMS26F020025"]}],"id":[{"id":"https:\/\/ror.org\/04aqat463","id-type":"ROR","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007834","name":"Ningbo Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2025J185"],"award-info":[{"award-number":["2025J185"]}],"id":[{"id":"10.13039\/100007834","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Ningbo University of Finance & Economics","award":["1320263702"],"award-info":[{"award-number":["1320263702"]}]},{"award":["1320263702"],"award-info":[{"award-number":["1320263702"]}],"id":[{"id":"https:\/\/ror.org\/0085x3f43","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Deploying neural networks on encrypted data requires efficient evaluation of nonlinear activations, especially the ReLU function, without decryption. While the CKKS homomorphic encryption scheme supports packed arithmetic over approximate numbers efficiently, its approximate semantics make direct nonlinear evaluation difficult, and polynomial surrogates often introduce approximation error and non-discrete outputs. In this work, we present a task-specific, non-interactive construction for discrete ReLU evaluation in CKKS by combining modulus-switch-based discretization with interpolation-driven lookup-table (LUT) evaluation. We instantiate this design in two complementary schemes. The first uses trigonometric Hermite interpolation and functional bootstrapping to compute a discrete sign indicator, which is then combined with the encrypted input through conditional multiplication to obtain the ReLU output; this variant is compact and suitable for lightweight settings. The second uses iterative most-significant-bit (MSB) bootstrapping to support larger plaintext moduli and higher-precision regimes through repeated digit extraction. A common enabler of both schemes is a discretization step that maps approximate CKKS plaintexts to a finite integer representation; exactness in our setting therefore refers to exact evaluation over this discretized representation, while the deviation from the original CKKS plaintext is governed by the discretization error analyzed in Lemma 1. Experiments on encrypted MNIST inference and the accompanying LUT\/storage analysis indicate that the proposed schemes preserve competitive accuracy relative to polynomial-approximation baselines while maintaining manageable auxiliary storage under the reported parameter settings. These results suggest that interpolation-based discrete activation is a promising alternative to polynomial approximation for selected CKKS-based encrypted inference tasks.<\/jats:p>","DOI":"10.3390\/e28050542","type":"journal-article","created":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T16:07:26Z","timestamp":1779206846000},"page":"542","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Efficient Non-Interactive Discrete ReLU over CKKS Using Interpolation Look-Up Table"],"prefix":"10.3390","volume":"28","author":[{"given":"Zhigang","family":"Chen","sequence":"first","affiliation":[{"name":"College of Artificial Intelligence, Ningbo University of Finance and Economics, Ningbo 315175, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinxia","family":"Song","sequence":"additional","affiliation":[{"name":"School of Big Data and Software Engineering, Zhejiang Wanli University, Ningbo 315100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Liqun","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Surrey, Guildford GU2 7XH, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,5,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Brakerski, Z., Gentry, C., and Vaikuntanathan, V. (2012). (Leveled) fully homomorphic encryption without bootstrapping. Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, Cambridge, MA, USA, 8\u201310 January 2012, ACM.","DOI":"10.1145\/2090236.2090262"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Safavi-Naini, R., and Canetti, R. (2012). Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP. Advances in Cryptology\u2014CRYPTO 2012, Springer.","DOI":"10.1007\/978-3-642-32009-5"},{"key":"ref_3","unstructured":"Fan, J., and Vercauteren, F. (2026, April 01). Somewhat Practical Fully Homomorphic Encryption. Cryptology ePrint Archive. Available online: https:\/\/eprint.iacr.org\/2012\/144.pdf."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Cheon, J.H., Kim, A., Kim, M., and Song, Y. (2017). Homomorphic Encryption for Arithmetic of Approximate Numbers. 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