Oblivious evaluation of multivariate polynomials

One of the fundamental problems of multi-party computation is Oblivious Polynomial Evaluation. In that problem, that was introduced by Naor and Pinkas, Alice has a polynomial P(x) and Bob has a point alpha. The goal is to allow Bob to compute P(alpha) so that Alice remains oblivious of alpha and Bob of P(x), apart from what is implied by P(alpha) and alpha. We introduce the multivariate version of this problem, where x and alpha are vectors, and offer an efficient secure protocol. In addition, we discuss several applications that may be solved efficiently using oblivious multivariate polynomial evaluation, such as private linear algebraic computations and private support vector machines (SVM).