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, Fabrizio Granelli

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

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-theart 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, −∊] ∪ [∊, 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.

Original languageEnglish
Article number2006
JournalMathematics
Volume10
Issue number12
DOIs
Publication statusPublished - 1 Jun 2022

Keywords

  • Bernstein polynomial
  • Chebyshev polynomials of the second kind
  • Minimax approximate polynomial
  • Sign function

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