The Lagrange Coded Computing (LCC), proposed recently by Yu et al., is regarded as a promising solution for most distributed learning algorithms thanks to its good tradeoff between resiliency, security, and privacy over cloud servers. As a kind of coded computing, LCC also costs extra computations in a local computer for encoding and decoding, which contains many complex operations, such as the continued product operations and divisions. In this paper, we present an efficient high-speed LCC codec architecture based on the linear regression algorithm for the first time. By analyzing the formulas and evaluating the hardware resource, we select a set of optimal parameters and remove most of the complex operations by storing the precomputed coefficients. Besides, the proposed architecture is inherently scalable and can be fully utilized and reused for encoding and decoding. The experimental results on an FPGA show that a significant speedup is achieved compared with the prior art.