<p>Many scientific and security protocols rely on sources of unpredictable and private random numbers. Device-independent quantum random number generation is a framework that makes use of the intrinsic randomness of quantum processes to generate numbers that are fundamentally unpredictable according to our current understanding of physics. However, the difficulty of controlling quantum systems makes it challenging to carry out device-independent protocols in practice. It is, therefore, desirable to harness the full power of the quantum degrees of freedom that one can control. It is known that no more than 2 log(<i>d</i>) bits of private device-independent randomness can be extracted from a quantum system of local dimension <i>d</i>. Here we demonstrate that this bound can be achieved for all <i>d</i> by providing a family of explicit protocols. To obtain our result, we develop certification techniques that may be of wider interest in device-independent applications for scenarios in which complete certification by self-testing is impossible or impractical.</p>