{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:29:03Z","timestamp":1760149743594,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,9,12]],"date-time":"2023-09-12T00:00:00Z","timestamp":1694476800000},"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>In this paper, we present new variants of Newton\u2013Raphson-based protocols for the secure computation of the reciprocal and the (reciprocal) square root. The protocols rely on secure fixed-point arithmetic with arbitrary precision parameterized by the total bit length of the fixed-point numbers and the bit length of the fractional part. We perform a rigorous error analysis aiming for tight accuracy claims while minimizing the overall cost of the protocols. Due to the nature of secure fixed-point arithmetic, we perform the analysis in terms of absolute errors. Whenever possible, we allow for stochastic (or probabilistic) rounding as an efficient alternative to deterministic rounding. We also present a new protocol for secure integer division based on our protocol for secure fixed-point reciprocals. The resulting protocol is parameterized by the bit length of the inputs and yields exact results for the integral quotient and remainder. The protocol is very efficient, minimizing the number of secure comparisons. Similarly, we present a new protocol for integer square roots based on our protocol for secure fixed-point square roots. The quadratic convergence of the Newton\u2013Raphson method implies a logarithmic number of iterations as a function of the required precision (independent of the input value). The standard error analysis of the Newton\u2013Raphson method focuses on the termination condition for attaining the required precision, assuming sufficiently precise floating-point arithmetic. We perform an intricate error analysis assuming fixed-point arithmetic of minimal precision throughout and minimizing the number of iterations in the worst case.<\/jats:p>","DOI":"10.3390\/cryptography7030043","type":"journal-article","created":{"date-parts":[[2023,9,12]],"date-time":"2023-09-12T21:41:12Z","timestamp":1694554872000},"page":"43","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Divisions and Square Roots with Tight Error Analysis from Newton\u2013Raphson Iteration in Secure Fixed-Point Arithmetic"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3630-1679","authenticated-orcid":false,"given":"Stan","family":"Korzilius","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6273-8930","authenticated-orcid":false,"given":"Berry","family":"Schoenmakers","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/3-540-45708-9_27","article-title":"Efficient computation modulo a shared secret with application to the generation of shared safe-prime products","volume":"Volume 2442","author":"Algesheimer","year":"2002","journal-title":"Advances in Cryptology\u2014CRYPTO 2002"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1007\/978-3-642-15497-3_9","article-title":"Secure multiparty linear programming using fixed-point arithmetic","volume":"Volume 6345","author":"Catrina","year":"2010","journal-title":"Computer Security\u2014ESORICS 2010"},{"key":"ref_3","first-page":"35","article-title":"Secure computation with fixed-point numbers","volume":"Volume 6052","author":"Catrina","year":"2010","journal-title":"Financial Cryptography and Data Security\u2014FC 2010"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/978-3-642-29101-2_19","article-title":"Secure distributed computation of the square root and applications","volume":"Volume 7232","author":"Liedel","year":"2012","journal-title":"Information Security Practice and Experience\u2014ISPEC 2012"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1007\/978-3-030-21568-2_25","article-title":"Benchmarking privacy preserved scientific operations","volume":"Volume 11464","author":"Aly","year":"2019","journal-title":"Applied Cryptography and Network Security\u2014ACNS 2019"},{"key":"ref_6","unstructured":"Knuth, D.E. (1997). The Art of Computer Programming (Vol. 2: Seminumerical Algorithms), Addison Wesley. [3rd ed.]."},{"key":"ref_7","unstructured":"Wilkinson, J.H. (1963). Rounding Errors in Algebraic Processes, Prentice Hall."},{"key":"ref_8","unstructured":"Wilkinson, J.H. (1965). Monographs on Numerical Analysis, Clarendon Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1007\/978-3-031-20974-1_3","article-title":"Through the looking-glass: Benchmarking secure multi-party computation comparisons for ReLU\u2019s","volume":"Volume 13641","author":"Aly","year":"2022","journal-title":"Cryptology and Network Security\u2014CANS 2022"},{"key":"ref_10","first-page":"285","article-title":"Unconditionally secure constant-rounds multi-party computation for equality, comparison, bits and exponentiation","volume":"Volume 3876","author":"Fitzi","year":"2006","journal-title":"Theory of Cryptography Conference\u2014TCC 2006"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/978-3-540-45146-4_15","article-title":"Universally composable efficient multiparty computation from threshold homomorphic encryption","volume":"Volume 2729","author":"Nielsen","year":"2003","journal-title":"Advances in Cryptology\u2014CRYPTO 2003"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1358","DOI":"10.1093\/imanum\/drac012","article-title":"Effects of round-to-nearest and stochastic rounding in the numerical solution of the heat equation in low precision","volume":"43","author":"Croci","year":"2022","journal-title":"IMA J. Numer. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Na, T., Ko, J.H., Kung, J., and Mukhopadhyay, S. (2017, January 14\u201319). On-chip training of recurrent neural networks with limited numerical precision. Proceedings of the 2017 International Joint Conference on Neural Networks (IJCNN), Anchorage, AK, USA.","DOI":"10.1109\/IJCNN.2017.7966324"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1215","DOI":"10.1175\/JCLI-D-21-0343.1","article-title":"Climate modeling in low precision: Effects of both deterministic and stochastic rounding","volume":"35","author":"Paxton","year":"2022","journal-title":"J. Clim."},{"key":"ref_15","unstructured":"Wang, N., Choi, J., Brand, D., Chen, C., and Gopalakrishnan, K. (2002, January 18\u201322). Training deep neural networks with 8-bit floating point numbers. Proceedings of the 32nd International Conference on Neural Information Processing Systems\u2014NIPS 2018, Santa Barbara, CA, USA."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"211631","DOI":"10.1098\/rsos.211631","article-title":"Stochastic rounding: Implementation, error analysis and applications","volume":"9","author":"Croci","year":"2022","journal-title":"R. Soc. Open Sci."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ryaben\u2019kii, V.S., and Tsynkov, S.V. (2006). A Theoretical Introduction to Numerical Analysis, Chapman and Hall\/CRC.","DOI":"10.1201\/9781420011166"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0377-0427(00)00417-9","article-title":"Historical developments in convergence analysis for Newton\u2019s and Newton-like methods","volume":"124","author":"Yamamoto","year":"2000","journal-title":"J. Comput. Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ercegovac, M., and Lang, T. (2004). Digital Arithmetic, Morgan Kaufmann.","DOI":"10.1016\/B978-155860798-9\/50011-7"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/978-3-031-34671-2_22","article-title":"New approach for sine and cosine in secure fixed-point arithmetic","volume":"Volume 13914","author":"Korzilius","year":"2023","journal-title":"Cyber Security, Cryptology, and Machine Learning\u2014CSCML 2023"},{"key":"ref_21","unstructured":"Schoenmakers, B. (2023, September 07). MPyC Package for Secure Multiparty Computation in Python. Available online: github.com\/lschoe\/mpyc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/978-3-031-34671-2_3","article-title":"Efficient Extended GCD and Class Groups from Secure Integer Arithmetic","volume":"Volume 13914","author":"Schoenmakers","year":"2023","journal-title":"Cyber Security, Cryptology, and Machine Learning\u2014CSCML 2023"}],"container-title":["Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2410-387X\/7\/3\/43\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:49:43Z","timestamp":1760129383000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2410-387X\/7\/3\/43"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,12]]},"references-count":22,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["cryptography7030043"],"URL":"https:\/\/doi.org\/10.3390\/cryptography7030043","relation":{},"ISSN":["2410-387X"],"issn-type":[{"type":"electronic","value":"2410-387X"}],"subject":[],"published":{"date-parts":[[2023,9,12]]}}}