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We have developed a new two-way time transfer modem to improve the time transfer precision of remote clock comparison. As a timing signal, we apply a binary offset carrier, which is similar to those signals used for the next-generation Global Navigation Satellite Systems. We took advantage of versatile A/D and D/A converters, and most of the digital signal processing stages were realized by software, running on an off-the-shelf PC. This enabled us to realize the complete system with cheaper equipment, leading to an affordable low-cost modem. For the real-time digital signal processing stages implemented in software, we relied on a graphics processing unit (GPU) developed for computer game enthusiast. The developed modem can receive four channels at the same time with a single GPU card. We performed two-way satellite time transfer experiments using these modems between Japan and Taiwan. The obtained results are consistent within 200 ps with respect to the results of GPS carrier phase time transfer. As a consequence, we improved the time transfer precision by nearly one order of magnitude as compared to a conventional two-way modem without increasing the connection fees caused by commercial communication satellites.