  We study the relation between streaming algorithms and linear sketching
algorithms, in the context of binary updates. We show that for inputs in $n$
dimensions, the existence of efficient streaming algorithms which can process
$\Omega(n^2)$ updates implies efficient linear sketching algorithms with
comparable cost. This improves upon the previous work of Li, Nguyen and
Woodruff [LNW14] and Ai, Hu, Li and Woodruff [AHLW16] which required a
triple-exponential number of updates to achieve a similar result for updates
over integers. We extend our results to updates modulo $p$ for integers $p \ge
2$, and to approximation instead of exact computation.
