How can differential power analysis (DPA) still work without "sufficient" measurement resolution?

Question

Referring to Introduction to differential power analysis (Paul Kocher, Joshua Jaffe, Benjamin Jun, Pankaj Rohatgi)

[...] Because the amount of power used by a device is influenced by the data being processed, power consumption measurements contain information about a circuit’s calculations. Even the effects of a single transistor, while not directly observable in power measurements from a large devices, do appear as weak correlations. When a device is processing cryptographic secrets, its data-dependent power usage can expose these secrets to attack. [...] DPA can accomplish in minutes or days what decades of cryptanalytic work cannot: the extraction of secret keys from devices using completely correct implementations of strong primitives. Even if the amount of information in each trace is orders of magnitude below the resolution of the measurement apparatus, this additional information can convert the computationally infeasible problem of breaking a cipher using brute force into a computation that can be performed quickly on a PC. […]

Could someone explain to me how it's possible to observe correlation to power fluctuations which aren't captured by the measurement, optimally with an example? This sounds unintuitive / non-logical to me. Maybe, if the fluctuation just happens to change the measured value(s), while itself not being captured at the higher resolution, sure, but what if the measured value(s) remain exactly the same regardless of the small fluctuation(s)?

Answer 1

The article states the following example in a foot note:

A biased coin provides a more familiar example of how a signal can be identified with precision exceeding the measurement system. Given enough measurements, the coin’s bias can be determined with arbitrarily fine accuracy—even though the individual measurements each have only one bit of resolution (heads or tails).

However, this means that the (small as it may be) bias of the coin still has an influence on the outcome of the measurement (the coin flip result). Therefore, the assumption must be, that over many measurements, the small fluctuations in power which are much smaller than the measurement resolution, still will move/influence the measured digital sample on some occasions, which at the end can be detected by DPA. But this is the key point, the fluctuations must somehow influence the measured value, at least in some traces among the many thousands, even if the fluctuation itself cannot be captured by individual measurements.