  We describe a protocol for generating random numbers based on the existence
of quantum violations of a free Clauser-Horne-Shimony-Holt inequality, namely
CHSH-3. Our method uses semidefinite programming relaxations to compute such
violations. In a standard setting the CHSH-3 inequality involves two separated
qutrits and compatible measurement, that is, commuting with each other,
yielding the known quantum bound of $1+\sqrt{11/3} \approx 2.9149$. In our
framework, $d$-dimensional quantum systems (qudits) where $d$ is not fixed a
priori, and measurement operators possibly not compatible, are allowed. This
loss of constraints yields a higher value for the maximum expectation of the
CHSH-3 inequality. Based on such upper bound on the violation of CHSH-3, we
develop a random number generator of type prepare-and-measure, but with one
part. Our protocol generates a maximal entropy and its security is based,
through self testing arguments, on the attainability of the maximal violation
of the free CHSH-3 for quantum systems.
