Secure computation of scalar product is of considerable importance due to its central role in many practical computation scenarios with privacy and security requirements. This paper includes new results about the secure two-party scalar product. Specifically, a perfectly secure and universally composable two-party split scalar product (SSP) protocol is proposed in the preprocessing model. In addition to full security, the proposed SSP protocol enjoys the advantage of optimal efficiency. To show the optimality of this SSP protocol, information theoretic lower bounds on the amount of communicated data in a secure two-party computation in the preprocessing model are derived. These bounds are not limited to SSP functionality but apply to a large class of two-party functionalities. A part of this paper is devoted to applications of the proposed SSP protocol in secure cloud computing. Specifically, based on this protocol, a cloud-assisted privacy-preserving profile-matching scheme and a secure remote health monitoring scheme are proposed. Both of the solutions are highly efficient and significantly improve the previous work.