This paper proposes a new debiasing method for a stable and efficient extraction of uniform random binary responses from physically unclonable functions (PUFs). The proposed method handles multiple-valued (i.e., ternary) responses from PUF responses, including unstable response bits, and stably extracts uniform random-bit responses from them. In this paper, we evaluate the stability and effectiveness of the proposed method with two experiments with simulated and actual responses of latch PUFs implemented on an FPGA. We demonstrate that the proposed method can obtain longer debiased random-bit responses than the conventional method. In addition, we apply the proposed method to the construction of a fuzzy extractor (FE), and show the advantages of the proposed FE in terms of response length and authentication success rate in an experimental evaluation.