We consider a problem of implementing dynamic controllers over encrypted data for asymptotic stabilization of closed-loop systems. Though a time-varying quantizer is used and it can be infinitesimally fine with time, a major issue is that the underlying space for encrypted messages is unavoidably finite and the controller receives a limited amount of quantized data. To resolve this issue, the proposed method takes advantage of the state matrix consisting of integers, which enables the controller to generate only lower bits of the same output without computing the upper bits. Whenever a portion of the upper bits of output has converged, the computation scope can be moved further lower, receiving only lower bits of the measurement. The quantization is scheduled and the size of the message space is predetermined from the convergence rate, so that the feedback input is restored from the outcome of the lower bits, no matter how fine quantization is performed in the end. As a consequence, asymptotic stabilization can be achieved by encrypted operation, despite the limited controller capacity.