Novel methods and systems for quantization based data embedding and reading in host signals, such as image, audio and video signals. To embed auxiliary data in a host signal, an embedder maps the host signal from a first domain into a mapped signal in a second domain that is robust to a class of amplitude modifications that are order preserving. The embedder performs quantization based embedding of auxiliary data into the host signal using quantization bins derived from the second domain that are robust to the amplitude modifications. The mapping improves the robustness of the data embedding method by increasing the chances that the embedded data can be recovered by an auxiliary data reader after amplitude modifications. A related embedding method projects the mapped signal unto a vector, and specifically, a pseudorandom vector. It performs quantization based embedding on the projected signal. The use of this projection provides added robustness of the embedded data to noise and other forms of distortion.