De novo assembly is the process of reconstructing genomes from DNA fragments (reads), which may contain redundancy and errors. Longer reads simplify assembly and improve contiguity of the output, but current long-read technologies come with high error rates. A crucial step of de novo genome assembly for long reads consists of finding overlapping reads. We present Berkeley Long-Read to Long-Read Aligner and Overlapper (BELLA), which implement a novel approach to compute overlaps using Sparse Generalized Matrix Multiplication (SpGEMM). We present a probabilistic model which demonstrates the soundness of using short, fixed length k-mers to detect overlaps, avoiding expensive pairwise alignment of all reads against all others. We then introduce a notion of reliable k-mers based on our probabilistic model. The use of reliable k-mers eliminates both the k-mer set explosion that would otherwise happen with highly erroneous reads and the spurious overlaps due to k-mers originating from repetitive regions. Finally, we present a new method to separate true alignments from false positives depending on the alignment score. Using this methodology, which is employed in BELLAtextquoterights precise mode, the probability of false positives drops exponentially as the length of overlap between sequences increases. On simulated data, BELLA achieves an average of 2.26% higher recall than state-of-the-art tools in its sensitive mode and 18.90% higher precision than state-of-the-art tools in its precise mode, while being performance competitive.