The symmetric rank-$k$ update (SYRK), $C = AA^\top$, computes the dot product of each pair of rows of $A$, producing the Gram matrix $C$. Its sparse variant underpins similarity search in machine learning, graph analytics, and genomics, including Jaccard similarity on datasets too large for a single node. Despite the symmetry in its inputs and outputs, existing distributed sparse matrix multiplication algorithms such as Sparse SUMMA treat sparse SYRK as generic multiplication, computing the full output and materializing the explicit transpose even when the calling application uses only one triangle. Prior distributed $AA^\top$ computations in similarity search and genome assembly inherit this overhead from the underlying SpGEMM. This paper presents SpSYRK and CommSpSYRK, two distributed sparse SYRK algorithms that exploit symmetry. The first, SpSYRK, partitions the off-diagonal blocks of the output between the upper and lower triangular regions of the process grid and computes only the lower-triangular part of each diagonal block, halving per-process computation compared with state-of-the-art distributed SpGEMM. The second, CommSpSYRK, further reorders communication to avoid forming $A^\top$, which reduces per-process communication volume. On 32 nodes of the Perlmutter supercomputer, SpSYRK achieves a 2$\times$ speedup over an optimized Sparse SUMMA on matrices where local multiplication dominates the runtime; the advantage narrows on communication-bound inputs, a dependence that the cost model predicts from the arithmetic intensity. CommSpSYRK fixes this and consistently achieves superior scaling at high process counts. The approach is a drop-in replacement for any application computing $C = AA^\top$ via a distributed SpGEMM routine, and its triangular output can be consumed directly by subsequent operations, reducing both computation and memory footprint.

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