But if you have a signal overlaid with random noise [1], and you know what your signal's looking like and when it's happening, you can correlate. For example, a small delay that occurs at certain known points (or not), will introduce a bias into a timer measuring it, no matter how noisy or coarsely quantized that timer is.

Similar techniques have been used in other fields for decades to pull useful signals from far below the noise floor (e.g. a lock in amplifier can go dozens of dB below the noise floor, because it, essentially, correlates frequency and phase and thereby eliminates all but a tiny sliver of noise. E.g. GPS signals are typically 20 dB below the noise floor.

[1] It doesn't have to be random.

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So these mitigations just make the attacks harder, hopefully hard enough that they become not feasible to be exploited widely.