  We study security-latency bounds for Nakamoto consensus, i.e., how secure a
block is after it becomes $k$-deep in the chain. We improve the
state-of-the-art bounds by analyzing the race between adversarial and honest
chains in three different phases. We find the probability distribution of the
growth of the adversarial chains under models similar to those in [Guo, Ren;
AFT 2022] when a target block becomes $k$-deep in the chain. We analyze certain
properties of this race to model each phase with random walks that provide
tighter bounds than the existing results. Combining all three phases provides
novel upper and lower bounds for blockchains with small $\lambda\Delta$.
