Piecewise Function Approximation With Private Data

Abstract

Abstract-We present two Secure Two Party Computation (STPC) protocols for piecewise function approximation on private data. The protocols rely on a piecewise approximation of the to-be-computed function easing the implementation in a STPC setting. The first protocol relies entirely on Garbled Circuit (GC) theory, while the second one exploits a hybrid construction where GC and Homomorphic Encryption (HE) are used together. In addition to piecewise constant and linear approximation, polynomial interpolation is also considered. From a communication complexity perspective, the full-GC implementation is preferable when the input and output variables can be represented with a small number of bits, while the hybrid solution is preferable otherwise. With regard to computational complexity, the full-GC solution is generally more convenient.

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“…Therefore, many researchers have been using interpolation methods to analyze data of different applications. Some of the interpolation methods are as following: B-spline interpolation using linear approximations for setting up viewing and projection matrices and describing complex objects were studied in reference [1], implementing a cubic spline interpolation algorithm on DSP was studied in reference [2], an iterative linear interpolation based on fuzzy gradient model for low-cost VLSI implementation was introduced by reference [3], a polynomial interpolation for space-efficient verifiable secret sharing was given in reference [4], an interpolation method to investigate the improvement in image quality for ground penetrating radar (GPR) acquisition was highlighted in reference [5], gauss interpolation algorithms for nonlinear rational parameters were presented in reference [6], four image interpolation methods for 2-D AR modeling were given in reference [7], a polynomial approach to evaluate the fragmented function approach for two secure two-party computation (STPC) was introduced in reference [8], a bivariate splines in piecewise constant tension as the solution for a functional minimization problem was given in reference [9], a cubic spline interpolation algorithm for smoothness interpolation model of non-circular curve mechanical mold was studied in reference [10], a spline interpolation functions for solution of a non-linear equation was given in reference [11], a view interpolation method from defocused stereo images using linear filtering was introduced in reference [12], a linear interpolation effects on signal transferring was highlighted in reference [13], a novel and fast cubic B-splines algorithm for cancellation of random valued impulse noise was given in reference [14], a real time implementation of cubic B-spline algorithm for electro optical tracking system was studied in reference [15], a cubic B-spline curves based research of the approximate algorithm was given in reference [16], and a fast algorithm for quadratic and cubic spline wavelets was provided in reference [17]. Among these, the spline curve makes it easy to build an interface that will allow designing and controlling the shape of complex curves and surfaces by using low-degree polynomials in each of the intervals and by choosing the polynomial pieces such that they fit smoothly together.…”