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We classify those cubic m-variable Boolean functions which are (m-4)-resilient. We prove that there are four types of such functions, depending on the structure of the support of their Walsh spectra. We are able to determine, for each type, the Walsh spectrum and, then, the nonlinearity of the corresponding functions. We also give the dimension of their linear space. This dimension equals m-k where k=3 for the first type, k=4 for the second type, k=5 for the third type, and 5≤k≤9 for the fourth type.