{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:49:03Z","timestamp":1760237343976,"version":"build-2065373602"},"reference-count":62,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,5,8]],"date-time":"2020-05-08T00:00:00Z","timestamp":1588896000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100010418","name":"Institute for Information and Communications Technology Promotion","doi-asserted-by":"publisher","award":["2018-0-00251"],"award-info":[{"award-number":["2018-0-00251"]}],"id":[{"id":"10.13039\/501100010418","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This work is mainly interested in ensuring users\u2019 privacy in asymmetric computing, such as cloud computing. In particular, because lots of user data are expressed in non-integer data types, privacy-enhanced applications built on fully homomorphic encryption (FHE) must support real-valued comparisons due to the ubiquity of real numbers in real-world applications. However, as FHE schemes operate in specific domains, such as that of congruent integers, most FHE-based solutions focus only on homomorphic comparisons of integers. Attempts to overcome this barrier can be grouped into two classes. Given point numbers in the form of approximate real numbers, one class of solution uses a special-purpose encoding to represent the point numbers, whereas the other class constructs a dedicated FHE scheme to encrypt point numbers directly. The solutions in the former class may provide depth-efficient arithmetic (i.e., logarithmic depth in the size of the data), but not depth-efficient comparisons between FHE-encrypted point numbers. The second class may avoid this problem, but it requires the precision of point numbers to be determined before the FHE setup is run. Thus, the precision of the data cannot be controlled once the setup is complete. Furthermore, because the precision accuracy is closely related to the sizes of the encryption parameters, increasing the precision of point numbers results in increasing the sizes of the FHE parameters, which increases the sizes of the public keys and ciphertexts, incurring more expensive computation and storage. Unfortunately, this problem also occurs in many of the proposals that fall into the first class. In this work, we are interested in depth-efficient comparison over FHE-encrypted point numbers. In particular, we focus on enabling the precision of point numbers to be manipulated after the system parameters of the underlying FHE scheme are determined, and even after the point numbers are encrypted. To this end, we encode point numbers in continued fraction (CF) form. Therefore, our work lies in the first class of solutions, except that our CF-based approach allows depth-efficient homomorphic comparisons (more precisely, the complexity of the comparison is     O ( log \u03ba + log n )     for a number of partial quotients n and their bit length    \u03ba   , which is normally small) while allowing users to determine the precision of the encrypted point numbers when running their applications. We develop several useful applications (e.g., sorting) that leverage our CF-based homomorphic comparisons.<\/jats:p>","DOI":"10.3390\/sym12050788","type":"journal-article","created":{"date-parts":[[2020,5,8]],"date-time":"2020-05-08T11:26:00Z","timestamp":1588937160000},"page":"788","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Homomorphic Comparison for Point Numbers with User-Controllable Precision and Its Applications"],"prefix":"10.3390","volume":"12","author":[{"given":"Heewon","family":"Chung","sequence":"first","affiliation":[{"name":"Department of Mathematics, Hanyang University, Seoul 04763, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0461-3053","authenticated-orcid":false,"given":"Myungsun","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Information Security, College of ICT Convergence, The University of Suwon, Hwaseong 18323, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7759-7368","authenticated-orcid":false,"given":"Ahmad Al","family":"Badawi","sequence":"additional","affiliation":[{"name":"Institute for Infocomm Research (I2R), A*STAR, Singapore 138634, Singapore"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5652-3455","authenticated-orcid":false,"given":"Khin Mi Mi","family":"Aung","sequence":"additional","affiliation":[{"name":"Institute for Infocomm Research (I2R), A*STAR, Singapore 138634, Singapore"}]},{"given":"Bharadwaj","family":"Veeravalli","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, The National University of Singapore, Singapore 117576, Singapore"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Yao, A.C. 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