The MapReduce programming paradigm allows to process big data sets in parallel on a large cluster of commodity machines. The MapReduce users often outsource their data and computations to a public cloud provider. We focus on the fundamental problem of matrix multiplication, and address the inherent security and privacy concerns that occur when outsourcing to a public cloud. Our goal is to enhance the two state-of-the-art algorithms for MapReduce matrix multiplication with privacy guarantees such as: none of the nodes storing an input matrix can learn the other input matrix or the output matrix, and moreover, none of the nodes computing an intermediate result can learn the input or the output matrices. To achieve our goal, we rely on the well-known Paillier's cryptosystem and we use its partially homomorphic property to develop efficient algorithms that satisfy our problem statement. We develop two different approaches called Secure-Private (SP) and Collision-Resistant-Secure-Private (CRSP), and compare their trade-offs with respect to three fundamental criteria: computation cost, communication cost, and privacy guarantees. Finally, we give security proofs of our protocols.