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

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,-epsilon]?[epsilon,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.