[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84306-en":3,"doc-seo-84306-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84306,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Degree Constrained Interval Optimization for Minimax Polynomial Approximation in Homomorphic Encryption","Homomorphic encryption enables privacy-preserving inference under arithmetic constraints that allow only additions and multiplications, forcing non-polynomial neural components to be replaced by polynomial approximations. Minimax approximation, commonly computed via the Remez algorithm, depends critically on the approximation interval: wider intervals improve robustness to large inputs but increase minimax error for a fixed degree. The work formulates a distribution-aware interval optimization using MSE over a target pre-activation distribution, and mitigates outside-interval behavior via HE-realizable domain extension polynomials. Experiments on sigmoid, tanh, and GELU show substantial MSE reductions versus conventional baselines.","Degree-Constrained Interval Optimization for Minimax Polynomial Approximation in Homomorphic Encryption  \nJiheon Woo, Donggyun Ryu, and Yongjune Kim  \narXiv :2607 .08042v 1 [ cs .CR] 9 Jul 2026  \nAbstract—Homomorphic encryption (HE) enables privacypreserving inference under arithmetic constraints that restrict encrypted evaluation to additions and multiplications. As a result, non-polynomial activation functions must be replaced by polynomial approximations. Among polynomial approximation methods, minimax approximation, typically computed by the Remez algorithm, is a standard approach because it minimizes the maximum approximation error over a given design interval. For minimax polynomial design, the approximation interval is a critical hyperparameter: a wider interval improves robustness to large-magnitude inputs while increasing the minimax approximation error under a fixed degree budget. In this paper, we formulate this trade-off as a distribution-aware interval optimization problem, where the approximation interval is chosen to minimize the mean-squared error (MSE) with respect to the pre-activation distribution of interest. To effectively control outside-interval inputs, we combine minimax polynomials with domain extension functions (DEFs) and their HE-realizable polynomial counterparts, domain extension polynomials (DEPs), which approximate a clipping operation outside the design interval and thereby suppress uncontrolled polynomial extrapolation. We first derive an analytically tractable DEF-based proxy objective that captures the trade-off between within-interval minimax approximation error and outside-interval clipping error. We then connect this idealized objective to HE-realizable DEP constructions through an implementation-error decomposition with an accompanying upper bound. Numerical experiments on representative nonpolynomial activation functions show that the proposed interval optimization achieves significantly lower MSE than conventional minimax baselines, with particularly large gains for sigmoid, tanh, and GELU, while the minimizer of the analytical proxy closely matches that of the numerical ideal objective.  \nIndex Terms—Homomorphic encryption, Privacy-preserving machine learning, Polynomial approximation, Interval optimization  \nI. INTRODUCTION  \nThe rapid advancement of artificial intelligence (AI) has accelerated the adoption of machine learning as a service (MLaaS), in which clients delegate inference tasks to remote servers. However, this paradigm raises significant privacy concerns with respect to the handling of sensitive user data. To address these concerns, privacy-preserving machine learning (PPML) has emerged as a key research area, aiming to perform inference while keeping user data confidential [1]–[4] .  \nAmong various approaches to PPML, homomorphic encryption (HE) is regarded as one of the most promising  \nJ. Woo, D. Ryu, and Y. Kim are with the Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, South Korea (e-mail: {jhwoo1997, dgryu, [yongjune](yongjune}@postech.ac.kr)[}](yongjune}@postech.ac.kr)[@postech.ac.kr](yongjune}@postech.ac.kr)).  \nsolutions [5], [6] . HE allows a server to perform computations directly on encrypted data without decryption. This property naturally supports a non-interactive inference setting: once the user provides the encrypted input, the server can execute the entire inference pipeline and return the encrypted result [7],[8] . Compared with interactive protocols such as multi-party computation (MPC) [9], this setting avoids repeated clientserver communication and active client participation during inference.  \nA major practical challenge in HE-based PPML is that most HE schemes support only a limited set of operations: addition, multiplication, and rotation. Consequently, non-polynomial operations in neural networks, such as activation functions, cannot be directly evaluated under HE. A common approach is theref","cbCaihUQrzNtxeKo","https://ap.wps.com/l/cbCaihUQrzNtxeKo","pdf",541663,5,1,11,"English","en",105,"# Abstract\n# Introduction\n## Privacy-preserving machine learning and HE basics\n## Polynomial approximation under HE constraints\n## Minimax approximation limitations outside the interval\n## Proposed distribution-aware interval optimization","[{\"question\":\"Why must activation functions be replaced with polynomial approximations in homomorphic encryption inference?\",\"answer\":\"Most HE schemes support only addition, multiplication, and rotation, so non-polynomial operations like activation functions cannot be evaluated directly. Replacing them with polynomials turns the network into an arithmetic circuit compatible with HE.\"},{\"question\":\"What is the key trade-off when choosing the minimax approximation interval?\",\"answer\":\"A wider interval increases robustness to larger-magnitude inputs, but it also raises the minimax approximation error under a fixed polynomial degree budget.\"},{\"question\":\"How does the proposed method control errors for inputs outside the design interval?\",\"answer\":\"It combines minimax polynomials with domain extension functions and HE-realizable domain extension polynomials that approximate clipping outside the interval, suppressing uncontrolled polynomial extrapolation.\"}]",1784194706,28,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"degree-constrained-interval-optimization-for-minimax-polynomial-approximation-in-homomorphic-encryption","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/degree-constrained-interval-optimization-for-minimax-polynomial-approximation-in-homomorphic-encryption/84306/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why must activation functions be replaced with polynomial approximations in homomorphic encryption inference?","Question",{"text":76,"@type":77},"Most HE schemes support only addition, multiplication, and rotation, so non-polynomial operations like activation functions cannot be evaluated directly. Replacing them with polynomials turns the network into an arithmetic circuit compatible with HE.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the key trade-off when choosing the minimax approximation interval?",{"text":81,"@type":77},"A wider interval increases robustness to larger-magnitude inputs, but it also raises the minimax approximation error under a fixed polynomial degree budget.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the proposed method control errors for inputs outside the design interval?",{"text":85,"@type":77},"It combines minimax polynomials with domain extension functions and HE-realizable domain extension polynomials that approximate clipping outside the interval, suppressing uncontrolled polynomial extrapolation.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]