  Bell nonlocality as a resource for device independent certification schemes
has been studied extensively in recent years. The strongest form of device
independent certification is referred to as self-testing, which given a device
certifies the promised quantum state as well as quantum measurements performed
on it without any knowledge of the internal workings of the device. In spite of
various results on self-testing protocols, it remains a highly nontrivial
problem to propose a certification scheme of qudit-qudit entangled states based
on violation of a single $d$-outcome Bell inequality. Here we address this
problem and propose a self-testing protocol for the maximally entangled state
of any local dimension using the minimum number of measurements possible, i.e.,
two per subsystem. Our self-testing result can be used to establish unbounded
randomness expansion, $\log_2d$ perfect random bits, while it requires only one
random bit to encode the measurement choice.
