A new silicon-strip readout chip named "SliT" has been developed for the measurement of the muon anomalous magnetic moment and electric dipole moment (EDM) of the muon at Japan Proton Accelerator Research Complex (J-PARC). The SliT chip is designed in the Silterra 180-nm CMOS technology with mixed-signal integrated circuits. An analog circuit incorporates a conventional charge-sensitive amplifier, shaping amplifiers, and two distinct discriminators for each of the 128 identical channels. A digital part includes storage memories, an event building block, a serializer, and low voltage differential signaling (LVDS) drivers. A distinct feature of the SliT is utilization of the zero-crossing architecture, which consists of a CR-RC filter followed by a CR circuit as a voltage differentiator. This architecture allows generating hit signals with subnanosecond amplitude-independent time walk, which is the primary requirement for the experiment. The test results show a time walk of 0.38 +/- 0.16 ns between 0.5 and 3 MIP signals. The equivalent noise charge is 1547 +/- 75 e(-) (rms) at C-det = 33 pF as a strip-sensor capacitance. SliT128C satisfies all requirements of the J-PARC muon g - 2/EDM experiment.