Gram – Schmidt Orthogonalization on Encrypted Vectors

In this paper we address a privacy preserving version of the well known Gram – Schmidt Orthogonalization procedure. Specifically, we propose a building block for secure multiparty computation, that is able to orthogonalize a set of componentwise encrypted vectors. Our setting is the following: Bob needs to compute this orthogonalization on some vectors encrypted with the public key of Alice. Hence our intent is not to propose a stand-alone protocol to solve a specific scenario or a specific application, but rather to develop a sub-protocol to be embedded in more complex algorithms or protocols where the vectors to be orthogonalized can be the result of previous computations. We show that our protocol is secure in the honest but curious model and evaluate its computation complexity.