We present here new mechanisms for hashing data via binary embeddings. Contrary to most of the techniques presented before, the embedding matrix of our mechanism is highly structured. That enables us to perform hashing more efficiently and use less memory. What is crucial and nonintuitive is the fact that imposing structured mechanism does not affect the quality of the produced hash. To the best of our knowledge, we are the first to give strong theoretical guarantees of the proposed binary hashing method by proving the efficiency of the mechanism for several classes of structured projection matrices. As a corollary, we obtain binary hashing mechanisms with strong concentration results for circulant and Topelitz matrices. Our approach is however much more general.