A recursive algorithm for computing a lower bound on the all-terminal reliability of an n-dimensional hypercube is presented. The recursive step decomposes an n-dimensional hypercube into lower dimension hypercubes that are linked together. As an illustration of the effectiveness and power of this method, a lower bound is computed on the all-terminal reliability of the 16-dimensional hypercube (Connection Machine architecture) whose links number 2/sup 19/. The notation and assumptions are defined, and background information on bounding the reliability polynomial is provided. Methods for tightening these bounds for the analysis of the hypercube architecture are discussed.< >