Quantum key distribution (QKD) generates symmetric keys between two remote parties over an untrusted, authenticated public communication channel with information security. With the development of high-speed QKD technology to widen the applicable areas of the QKD technology, higher requirements for the data post-processing bandwidth of the authentication protocol are put forward. The design requirements of QKD system authentication protocol are analyzed in detail, then a high-speed authentication scheme based on Toeplitz matrix using linear feedback shift register (LFSR) is proposed. This scheme is implemented on FPGA hardware platform using the parallel processing capability of FPGA and pipeline data processing scheme. Simulation results show that the proposed scheme can achieve data post-processing bandwidth up to 10 Gbps, which is a promising candidate for ultra-high-speed QKD system implementation in the future.