Towards the Sign Function Best Approximation for Secure Outsourced Computations and Control

Mikhail Babenko, Andrei Tchernykh, Bernardo Pulido-Gaytan, Arutyun Avetisyan, Sergio Nesmachnow, Xinheng Wang and Fabrizio Granelli
Additional contact information
Mikhail Babenko: North-Caucasus Center for Mathematical Research, North-Caucasus Federal University, 1 Pushkin Street, 355017 Stavropol, Russia
Andrei Tchernykh: Control/Management and Applied Mathematics, Ivannikov Institute for System Programming, 109004 Moscow, Russia
Bernardo Pulido-Gaytan: Computer Science Department, CICESE Research Center, Ensenada 22800, Mexico
Arutyun Avetisyan: Control/Management and Applied Mathematics, Ivannikov Institute for System Programming, 109004 Moscow, Russia
Sergio Nesmachnow: Faculty of Engineering, Universidad de la República, Montevideo 11300, Uruguay
Xinheng Wang: Department of Mechatronics and Robotics, Xi’an Jiaotong-Liverpool University, Suzhou 215123, China
Fabrizio Granelli: Department of Information Engineering and Computer Science, University of Trento, 38150 Trento, Italy

Mathematics, 2022, vol. 10, issue 12, 1-22

Abstract: Homomorphic encryption with the ability to compute over encrypted data without access to the secret key provides benefits for the secure and powerful computation, storage, and communication of resources in the cloud. One of its important applications is fast-growing robot control systems for building lightweight, low-cost, smarter robots with intelligent brains consisting of data centers, knowledge bases, task planners, deep learning, information processing, environment models, communication support, synchronous map construction and positioning, etc. It enables robots to be endowed with secure, powerful capabilities while reducing sizes and costs. Processing encrypted information using homomorphic ciphers uses the sign function polynomial approximation, which is a widely studied research field with many practical results. State-of-the-art works are mainly focused on finding the polynomial of best approximation of the sign function (PBAS) with the improved errors on the union of the intervals [ − 1 , − ϵ ] ∪ [ ϵ , 1 ] . However, even though the existence of the single PBAS with the minimum deviation is well known, its construction method on the complete interval [ − 1 , 1 ] is still an open problem. In this paper, we provide the PBAS construction method on the interval [ − 1 , 1 ] , using as a norm the area between the sign function and the polynomial and showing that for a polynomial degree n ≥ 1 , there is (1) unique PBAS of the odd sign function, (2) no PBAS of the general form sign function if n is odd, and (3) an uncountable set of PBAS, if n is even.

Keywords: minimax approximate polynomial; Chebyshev polynomials of the second kind; Bernstein polynomial; sign function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/12/2006/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/12/2006/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:10:y:2022:i:12:p:2006-:d:835967

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().