Transformation-Based Outsourcing of Linear Equation Systems over Real Numbers

This paper studies the possibility of achieving indistinguishability-based security in privately outsourcing linear equation systems over real numbers. The particular task is to solve a full-rank (n x n) system Ax = b. Since the most complex part of this task is inverting A, the problem can be reduced to outsourcing of a square matrix inverse computation. Although outsourcing matrix inverse is trivial for matrices over finite fields, it is not so easy for matrices over real numbers. We study the class of affine transformations for matrices over real numbers, find out which forms are possible at all, and state some properties that the transformation and the initial matrices must satisfy in order to make the initial matrices perfectly (or statistically) indistinguishable after applying the transformation. This paper provides both possibility and impossibility results