The TORCH detector is proposed for the low-momentum particle identification upgrade of the LHCb experiment. It combines Time-Of-Flight and Cherenkov techniques to achieve positive pi/K/p separation up to 10 GeV/c. This requires a timing resolution of 70 ps for single photons. This paper reports on the electronics developed for such measurements, using commercial Micro Channel Plate (MCP) devices and custom ASICs (NINO and HPTDC). The intrinsic timing resolution of the electronics measured with electrical test pulses is 40 ps. With the MCP photon detector and a pulsed laser, a resolution of 90 ps has been recorded in laboratory tests and 130 ps in test beams.