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Brain-inspired HyperDimensional (HD) computing emulates cognitive tasks by computing with long binary vectors–aka hypervectors–as opposed to computing with numbers. However, we observed that in order to provide acceptable classification accuracy on practical applications, HD algorithms need to be trained and tested on non-binary hypervectors. In this paper, we propose SearcHD, a fully binarized HD computing algorithm with a fully binary training. SearcHD maps every data points to a high-dimensional space with binary elements. Instead of training an HD model with non-binary elements, SearcHD implements a full binary training method which generates multiple binary hypervectors for each class. We also use the analog characteristic of non-volatile memories (NVMs) to perform all encoding, training, and inference computations in memory. We evaluate the efficiency and accuracy of SearcHD on a wide range of classification applications. Our evaluation shows that SearcHD can provide on average 31.1× higher energy efficiency and 12.8× faster training as compared to the state-of-the-art HD computing algorithms.