In this paper, entirely new discrete-time variants of the super-twisting algorithm are presented. The development of these discrete-time equivalents is based on certain criteria, e.g., the approximation of the continuous-time super-twisting algorithm as the discretization time tends to zero. In contrast to the classical explicit Euler discretized super-twisting dynamics, the proposed schemes are exact in the sense that in the unperturbed case, the controllers ensure local convergence to the origin. Oscillations of the system states caused by the discrete-time implementation of the super-twisting algorithm are avoided. The superiority of the developed control laws is demonstrated in simulation examples as well as in a real-world application. These examples reveal that the standard accuracy of the homogeneous second-order sliding mode is preserved and, in contrast to the explicit Euler discretized algorithm, in the presence of exact discrete measurements, the precision of the controlled variable is insensitive to oversized control gains.