V_V comments on Can noise have power? - Less Wrong
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Sure, but at this point they can also gain information on your keys or the data you wish to encrypt.
Not necessarily. Think wider, not only PCs use encrypted communications. Consider a router, for example, or a remote sensor.
Still, if they can compromise the RNG state in the router/sensor/whatever, they could probably compromise its CPU and/or RAM.
That's not self-evident to me. Passively observing power consumption is much easier than, say, getting inside a SOC in tamper-resistant packaging.